Really? • An infinite number of individually infinite universes in an infinite expanding background? Surely I can’t be serious! • Eternal inflation is a direct consequence of: non-unique vacuum Quantum field accelerated state theory expansion (possible in standard model) (works fantastically) (observed: dark energy) (common in BSM physics) (inferred: inflation) (inevitable in string theory) Thursday, 4 July, 13
Observational Tests of Eternal Inflation • Strong theoretical motivation, but is eternal inflation experimentally verifiable? Thursday, 4 July, 13
Observational Tests of Eternal Inflation • Strong theoretical motivation, but is eternal inflation experimentally verifiable? Our bubble does not evolve in isolation.... The collision of our bubble with others provides an observational test of eternal inflation. Aguirre, MCJ, Shomer Thursday, 4 July, 13
Making predictions and testing models Thursday, 4 July, 13
Bubble collisions today t Slow-roll ” g n a B g i x B “ Thursday, 4 July, 13
Bubble collisions ? today t Slow-roll ” ” g g n n a a B B g g i i x B B “ “ Thursday, 4 July, 13
Bubble collisions ? today t Slow-roll ” ” g g n n a a B B g g i i x B B “ “ • Collisions are always in our past. • The outcome is fixed by the potential and kinematics. Thursday, 4 July, 13
Bubble collisions ? today t Slow-roll ” ” g g n n a a B B g g i i x B B “ “ • Collisions are always in our past. • The outcome is fixed by the potential and kinematics. • To study what happens, need full GR. Thursday, 4 July, 13
Bubble collisions ? today t Slow-roll ” ” g g n n a a B B g g i i x B B “ “ • Collisions are always in our past. • The outcome is fixed by the potential and kinematics. • To study what happens, need full GR. • We want to find the post-collision cosmology: GR. • Huge center of mass energy in the collision. • Non-linear potential, non-linear field equations. Thursday, 4 July, 13
Numerical solutions • Numerical simulations with full GR: full dynamics. ds 2 = − α ( x, z ) dz 2 + a ( x, z ) dx 2 + z 2 dH 2 φ ( x, z ) 2 ' 3. ⇥ 10 � 10 V ( φ ) 2.8 ⇥ 10 � 10 2.6 ⇥ 10 � 10 2.4 ⇥ 10 � 10 2.2 ⇥ 10 � 10 2. ⇥ 10 � 10 1.8 ⇥ 10 � 10 � 0.005 0.000 0.005 0.010 Thursday, 4 July, 13
Numerical solutions • Numerical simulations with full GR: full dynamics. ds 2 = − α ( x, z ) dz 2 + a ( x, z ) dx 2 + z 2 dH 2 φ ( x, z ) 2 ' 3. ⇥ 10 � 10 V ( φ ) 2.8 ⇥ 10 � 10 2.6 ⇥ 10 � 10 2.4 ⇥ 10 � 10 2.2 ⇥ 10 � 10 2. ⇥ 10 � 10 1.8 ⇥ 10 � 10 � 0.005 0.000 0.005 0.010 φ A φ B φ C • 2 types of bubbles from false vacuum. Thursday, 4 July, 13
Numerical solutions • Numerical simulations with full GR: full dynamics. ds 2 = − α ( x, z ) dz 2 + a ( x, z ) dx 2 + z 2 dH 2 φ ( x, z ) 2 ' 3. ⇥ 10 � 10 V ( φ ) 3. ⇥ 10 � 10 2.8 ⇥ 10 � 10 2.5 ⇥ 10 � 10 2.6 ⇥ 10 � 10 2.4 ⇥ 10 � 10 2. ⇥ 10 � 10 2.2 ⇥ 10 � 10 1.5 ⇥ 10 � 10 2. ⇥ 10 � 10 1.8 ⇥ 10 � 10 1. ⇥ 10 � 10 � 0.005 0.000 0.005 0.010 5. ⇥ 10 � 11 φ A φ B φ C 0 0.0 0.5 1.0 1.5 2.0 2.5 • 2 types of bubbles from false vacuum. • Slow roll inflation inside one, starting near . φ C Thursday, 4 July, 13
Numerical solutions • Colliding identical bubbles. 2( φ C − φ B ) φ ' x 3. ⇥ 10 � 10 V ( φ ) 2.8 ⇥ 10 � 10 2.6 ⇥ 10 � 10 φ C 2.4 ⇥ 10 � 10 2.2 ⇥ 10 � 10 2. ⇥ 10 � 10 1.8 ⇥ 10 � 10 � 0.005 0.000 0.005 0.010 φ B φ C φ B • After the collision, fields linearly superpose: potential key. • Dynamics necessary! Thursday, 4 July, 13
Numerical solutions • Colliding identical bubbles. 2( φ C − φ B ) ' 3. ⇥ 10 � 10 V ( φ ) 2.8 ⇥ 10 � 10 2.6 ⇥ 10 � 10 φ C 2.4 ⇥ 10 � 10 2.2 ⇥ 10 � 10 2. ⇥ 10 � 10 1.8 ⇥ 10 � 10 � 0.005 0.000 0.005 0.010 φ B φ C φ B • After the collision, fields linearly superpose: potential key. • Dynamics necessary! Thursday, 4 July, 13
Numerical solutions • Colliding identical bubbles. 2( φ C − φ B ) ' 3. ⇥ 10 � 10 V ( φ ) 2.8 ⇥ 10 � 10 2.6 ⇥ 10 � 10 φ C 2.4 ⇥ 10 � 10 2.2 ⇥ 10 � 10 2. ⇥ 10 � 10 1.8 ⇥ 10 � 10 � 0.005 0.000 0.005 0.010 φ B φ C φ B • After the collision, fields linearly superpose: potential key. • Dynamics necessary! Thursday, 4 July, 13
Numerical solutions • Colliding different bubbles. φ C φ B φ φ A x Thursday, 4 July, 13
Numerical solutions • Colliding different bubbles. φ C φ B φ φ A x Thursday, 4 July, 13
Numerical solutions • Colliding different bubbles. φ C φ B φ φ A x Thursday, 4 July, 13
Numerical solutions • Colliding different bubbles. φ C φ B φ φ A x • Inflation does not end, there are new perturbations! Thursday, 4 July, 13
Observational Signatures • Bubble collisions perturb the epoch of inflation inside our bubble. ' V ( φ ) φ Thursday, 4 July, 13
Observational Signatures • Bubble collisions perturb the epoch of inflation inside our bubble. ' V ( φ ) time φ space Thursday, 4 July, 13
Observational Signatures • Bubble collisions perturb the epoch of inflation inside our bubble. ' V ( φ ) time φ space Thursday, 4 July, 13
Observational Signatures • Bubble collisions perturb the epoch of inflation inside our bubble. ' V ( φ ) time φ space stretching by φ inflation perturbed x unperturbed y Thursday, 4 July, 13
Observational Signatures surface of last scattering Thursday, 4 July, 13
Observational Signatures 2 θ c surface of last scattering Thursday, 4 July, 13
Observational Signatures 2 θ c surface of last scattering Symmetry+causality: effects confined to a disc. Thursday, 4 July, 13
Observational Signatures 2 θ c surface of last scattering Symmetry+causality: effects confined to a disc. • Generic signature (thanks inflation!): f ∆ T ( ˆ n ) ' f ( ˆ n ) + δ Λ CDM ( ˆ n ) T z � crit 0 � f : analytic arguments and numerics z crit Feeney, MCJ, Mortlock, Peiris Chang, Kleban, Levi Gobetti & Kleban Thursday, 4 July, 13 �
Observational Signatures 2 θ c surface of last scattering Symmetry+causality: effects confined to a disc. • Generic signature (thanks inflation!): Thursday, 4 July, 13
Counting collisions • How many collisions are there? constant FRW time Bubble wall Begin Inflation Nucleation surface False Vacuum Thursday, 4 July, 13
Counting collisions • How many collisions are there? Present T= " /2 Slow-roll Reheating End Inflation Begin Inflation Reheating T= #" /2 ! = " ! = 0 Thursday, 4 July, 13
Counting collisions • How many collisions are there? Present T= " /2 Past Light Cone Reheating End Inflation Begin Inflation Initial Conditions T= #" /2 ! = " ! = 0 Thursday, 4 July, 13
Counting collisions • How many collisions are there? Present T= " /2 Past Light Cone Reheating End Inflation Begin Inflation Initial Conditions Bubbles that nucleate in here are in principle observable. N = λ V past 4 T= #" /2 ! = " ! = 0 Thursday, 4 July, 13
Counting collisions • Counting only collisions whose disc of influence is smaller than the whole sky: τ o � H 2 ξ ls N ⌅ 16 πλ ⇥ ⇤ F τ ls Ω c 3 H 4 H 2 + R F I 0 − R 0 also Kleban et. al. Thursday, 4 July, 13
Counting collisions • Counting only collisions whose disc of influence is smaller than the whole sky: τ o � H 2 ξ ls N ⌅ 16 πλ ⇥ ⇤ F τ ls Ω c 3 H 4 H 2 + R F I 0 − R 0 also Kleban et. al. dN d Ψ d Φ n d � � cos Θ n � 3.5 • The collisions are very nearly 3.0 2.5 isotropic, and the distribution of disc 2.0 1.5 sizes on the CMB sky relatively flat: 1.0 0.5 Ψ Π 3 Π 0 2 Π Π 2 2 Thursday, 4 July, 13
Bubble collisions model • The model: ' V ( φ ) φ Thursday, 4 July, 13
Bubble collisions model • The model: ' V ( φ ) φ Thursday, 4 July, 13
Bubble collisions model • The model: ' V ( φ ) . . . φ Thursday, 4 July, 13
Bubble collisions model • The model: generic signature ' V ( φ ) . . . φ Thursday, 4 July, 13
Bubble collisions model • The model: generic signature ' V ( φ ) . . . φ ¯ N s expected number of collisions m parameters characterizing each collision Pr( N s , m ) How many of each type do I expect to find? Thursday, 4 July, 13
Collisions (exaggerated) + CMB + instrumental noise Thursday, 4 July, 13
Collisions (realistic) + CMB + instrumental noise Thursday, 4 July, 13
Collisions (realistic) + CMB + instrumental noise Does the data prefer a theory with collisions? Thursday, 4 July, 13
Searching for collisions Feeney, MCJ, Mortlock, Peiris • Lambda-CDM: very successful at describing the CMB power spectrum. WMAP 7-year data Thursday, 4 July, 13
Searching for collisions Feeney, MCJ, Mortlock, Peiris • Lambda-CDM: very successful at describing the CMB power spectrum. WMAP 7-year data • Are there anomalies? Thursday, 4 July, 13
Searching for collisions Feeney, MCJ, Mortlock, Peiris • Lambda-CDM: very successful at describing the CMB power spectrum. WMAP 7-year data • Are there anomalies? • Frequentist statistics: how discrepant is the data assuming the null hypothesis? Thursday, 4 July, 13
Searching for collisions Feeney, MCJ, Mortlock, Peiris • Lambda-CDM: very successful at describing the CMB power spectrum. WMAP 7-year data • Are there anomalies? • Frequentist statistics: how discrepant is the data assuming the null hypothesis? • Bayesian model selection: does one model fit the data better than another? Thursday, 4 July, 13
Bayesian statistics • The goal: P (Model , Θ | data) How should I bet? Thursday, 4 July, 13
Bayesian statistics • The goal: P (Model , Θ | data) How should I bet? • Bayes’ Theorem: P (Model , Θ | data) = P ( Θ ) P (data | Model , Θ ) P (data | Model) Thursday, 4 July, 13
Bayesian statistics • The goal: P (Model , Θ | data) How should I bet? • Bayes’ Theorem: P (Model , Θ | data) = P ( Θ ) P (data | Model , Θ ) P (data | Model) • Theory prior: Z P ( Θ ) P ( Θ ) d Θ = 1 • Likelihood: P (data | Model , Θ ) • Evidence (model averaged likelihood): P (data | Model) Z P (data | Model) = d Θ P ( Θ ) P (data | Model , Θ ) Thursday, 4 July, 13
Bayesian statistics • The likelihood is used to quantify how consistent data is with a set of model parameters. P (data | Model , Θ ) Thursday, 4 July, 13
Bayesian statistics • The likelihood is used to quantify how consistent data is with a set of model parameters. P (data | Model , Θ ) exclusion plots Thursday, 4 July, 13
Bayesian statistics • The likelihood is used to quantify how consistent data is with a set of model parameters. P (data | Model , Θ ) exclusion plots • This does NOT tell us how we should rank competing theories trying to describe the same data. Thursday, 4 July, 13
Bayesian statistics • The likelihood is used to quantify how consistent data is with a set of model parameters. P (data | Model , Θ ) exclusion plots • This does NOT tell us how we should rank competing theories trying to describe the same data. • To do so, we can apply Bayes’ theorem at the level of Models: P (Model | data) = P (Model) P (data | Model) P (data) Thursday, 4 July, 13
Bayesian model selection • Let’s say I have a model that fits the data fairly well, should I introduce a more complicated model that might fit it even better? Thursday, 4 July, 13
Bayesian model selection • Let’s say I have a model that fits the data fairly well, should I introduce a more complicated model that might fit it even better? • We can decide by looking at the evidence ratio: P (Model 1 | data) P (Model 0 | data) = P (Model 1) P (data | Model 1) P (Model 0) P (data | Model 0) = P (data | Model 1) P (data | Model 0) Thursday, 4 July, 13
Bayesian model selection • Let’s say I have a model that fits the data fairly well, should I introduce a more complicated model that might fit it even better? • We can decide by looking at the evidence ratio: P (Model 1 | data) P (Model 0 | data) = P (Model 1) P (data | Model 1) P (Model 0) P (data | Model 0) = P (data | Model 1) P (data | Model 0) • The evidence naturally implements Occam’s razor: the simpler model should be favored. Tension between volume of parameter space and goodness of fit. Z P (data | Model) = d Θ P ( Θ ) P (data | Model , Θ ) Thursday, 4 July, 13
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