Big Picture Suppose the microphysics behind inflation is random E.g. potential is some realization from a distribution ! !
e.g. Frazer & Liddle; Aazami & Easther; Tye et al.; Easther & McAllister; Agarwal et al.; …
! ! ! !
Are there universal properties that emerge? !
How are these scenarios constrained by data?
Motivation 1: Strings and Glasses Some string models take the form of brane motion ! !
D7
D7
! !
D3
! ! !
Fluxes
D7
O7
!
E.g. Motion of the brane D3 (inflation) controlled by potential generated by a large number of sources
Motivation 1: Strings and Glasses Qualitatively similar to an electron in a material ! !
SiO2
SiO2
! !
e-
! ! !
E-Field
SiO2
CaO
!
E.g. Motion controlled by atoms, E-field, etc. !
Disorder induced by random inhomogeneities
Motivation 1: Strings and Glasses Disorder in solids has interesting universality !
In , disorder localizes electrons (even for arbitrarily weak disorder) !
For
, localization occurs with enough disorder
!
Explains many observed properties of materials e.g. why is glass transparent and so easy to make? e.g. Mott’s Nobel Lecture
Motivation 1: Strings and Glasses Is there an analogy in the string landscape / inflation? !
Some similarities between glasses and de Sitter !
Denef; Anninos & Denef
Some authors have claimed the analogy is precise !
Tye; Podolsky et al.
In this analogy, inflation requires 3 or more fields (would make single-field inflation incredibly delicate) !
Fortunately, we will see the analogy is not precise
Motivation 2: Phenomenology CMB data analyses can be broken into 2 categories: • Targeted searches (e.g. searches for shapes of NG, Neff,…) • Broad consistency tests (e.g.
, total NG, isotropy)
I have tended to dismiss the latter My logic: targeted searches are more sensitive so we will discover something there first !
Are there situations where the broad tests fail without a compelling feature in a targeted search?
Strategy Break model into 2 parts: ! !
No disorder: we have some given model of inflation !
Disorder acts as random localized “features”
(No light states associated with the “features”)
!
E.g. for a random potential
Formalism
Disorder in the EFT of Inflation EFT allows us to start with perturbations: ! ! ! ! !
Creminelli et al. ; Cheung et al.
Split coefficient functions into fixed + stochastic
Disorder in the EFT of Inflation Disorder at quadratic order (in decoupling limit): ! ! !
From statistics
are like dim
“fields”
!
Leading order disorder is irrelevant !
I.e. Disorder is weak at low energies !
We are justified in treating disorder perturbatively
How do we calculate? First compute quantum expectation !
usual in-in formula
! ! ! ! !
Then average over realizations
How do we calculate? First compute quantum expectation ! ! ! ! !
The denominator depends on the realization
!
Then average over realizations ! ! !
Integral is non-trivial when denominator matters
How do we calculate? In practice, perturbative calculations are easy !
We expand and then use stochastic correlators ! ! ! !
Connection to Resonance Also useful to take the Fourier transform ! ! !
This is just a bunch of copies of “resonance” !
Weak coupling requires that !
Behbahani et al. Flauger et al.
Cutting off high frequency smooths out -function !
This is necessary for getting sensible results ! !
Comparison with electrons How does this compare with Anderson localization? !
Electron couples to disorder via ! ! !
Electron is dim.
and disorder is dim.
!
Disorder is relevant for !
Disorder always strong at low enough energy
Comparison with electrons Where does the analogy with D3-brane fail? !
1. A space filling brane has effectively ! ! !
Motion of brane is classical !
Stochastic parameters only depend on time !
Nothing qualitative changes with
(or # of fields)
Comparison with electrons Where does the analogy with D3-brane fail? !
2. We probe the system at finite energy !
Fluctuations are produced at energy !
We can choose the strength of disorder at this scale !
Can be perturbative for relevant or irrelevant disorder !
(Different order of limits from application to materials)
Observational Consequences
Noisy Power Spectra For simplicity consider ! ! !
A given realization will have 2 effects: 1. Modulates the amplitude of the fluctuations 2. Excites the quantum state Both effects average to zero at linear order, !
It will look like extra noise (trispectrum) ! !
Noisy Power Spectra As a trispectrum, we can think of it in diagrams ! ! ! ! ! !
No momentum “transfer” ! !
Not captured by
-like constraints
Noisy Power Spectra If we assume ! ! ! ! ! ! ! ! !
excited state
Divergence when resolved by frequency cutoff (in the resonance language)
Noisy Power Spectra If we assume ! ! ! ! ! ! ! ! !
varying amplitude
Divergence when resolved by frequency cutoff (in the resonance language)
Noisy Power Spectra Work in terms of CMB temperature multipoles ! !
where
and
!
No off-diagonal terms: homogeneity and isotropy are preserve by every realization !
Otherwise, very similar to lensing, modulations, etc. !
QML / optimal estimator follows from same formalism
Seljak & Hirata; Hanson & Lewis
Noisy Power Spectra Intuitively, the estimator (for CL limited modes) is given by ! ! !
This is the noise in each bin minus expected noise !
We can get a reasonably good constraint by using ! ! !
This constrains
Noisy Bispectra Additional noise may also arise in the bispectrum !
There are two possible sources 1. Gaussian disorder coupled to cubic terms ! !
2. Non-gaussian disorder ! !
Both produce a non-gaussian 6-point function
Noisy Bispectra Additional noise may also arise in the bispectrum !
Form of 6-point functions is different 1. Gaussian disorder coupled to cubic terms ! !
2. Non-gaussian disorder ! !
Arises because disorder carries no-momentum
Noisy Bispectra Constrained by overall NG measurements e.g. Planck XXIV ! ! !
n ! !
Length of an n-dimensional random walk !
Constrains NG from disorder at
Future Directions
Summary Disorder is a very general framework !
We have looked at one corner 1. Localized statistics ( - functions) 2. Scale invariant (statistics uniform in time) 3. Perturbative (small correction to a fixed model) !
These choices are motivated by lack of imagination
Future Directions Given these choices: ! •
What is the optimal constraint (and is it worth doing)?
! •
Does this approach avoid look elsewhere effects often associated with features, etc ?
!
What if be break scale invariance? ! •
Are there well motivated examples?
! •
Tempting because of low multipoles?
Future Directions Are there more natural settings for disorder? DG, Horn, Senatore & Silverstein • Trapped Inflation ? Already have localized light degrees of freedom Make the locations random !
• Solid Inflation
Gruzinov ; Endlich et al.
This is really the natural home of disorder (exact analogy with real-world solids) !
Calculations are more challenging !
Isotropy and homogeneity are broken in each realization
