slides

Cosmological Perturbation Theory of Shapes
Sean Gryb
(with Tom Zlo´snik)
Radboud Universiteit Nijmegen
Institute for Mathematics, Astrophysics
and Particle Physics
UNB Fredericton
Shape Dynamics Workshop
May 8, 2014
Intro
Standard Cosmo
Cosmo Shape Perts
Conclusions
Brief Introduction
What does cosmological perturbation theory look like in Shape
Dynamics?
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Intro
Standard Cosmo
Cosmo Shape Perts
Conclusions
Motivation
Shape Dynamics (SD) ⇒ clean separation of symmetry principle and
evolution equation. Could this be a useful tool for Cosmology?
Our fundamental equations — the Lichnerowicz–York equation and the
Lapse Fixing equation — are conveniently treated using a perturbative
expansion.
In cosmology, SD might be relevant as a tool for studying current
experiments (B-modes, non-Gaussianity).
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Intro
Standard Cosmo
Cosmo Shape Perts
Conclusions
Extended Motivation: Some Basics of Cosmological Perturbation Theory
Pre-History
Fixing gauges in GR is hard!!
⇒ Just fixing coordinates leads to a number of problems:
Caustics can develop.
Globally not well-defined.
Residual gauge freedom can exist.
Physical observables are hard to separate from coordinate artefacts.
⇒ in the olden days, most approaches to cosmological perturbation theory
(e.g., Newtonian gauge) suffered from these problems and many papers were
written confusing coordinate artefacts with physical effects.
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Intro
Standard Cosmo
Cosmo Shape Perts
Conclusions
Mukanov’s Solution
Gauge Invariant Approach
Bardeen, others, and — decisively — Mukanov developed a gauge-invariant
approach to observables in perturbation theory.
⇒ wrote down a set of gauge invariant observables for perturbative cosmology
(Mukanov variables) and worked out their equations of motion.
Observation: Mukanov’s original paper is a tour de force ⇒ basic underlying
structure was not particularly clear. (Now a much simpler derivation is known.)
Alternative History
If the York decomposition (or SD) had been known earlier to cosmologists, this
would have been a graduate level exercise.
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Intro
Standard Cosmo
Cosmo Shape Perts
Conclusions
Canonical Version (Langlois ’93):
Sketch of Canonical derivation of Mukanov variable
Expand (gab , φ; π ab , πφ ) into a background + small fluctuations.
0th order H(0) + flow ⇒ Friedmann equations for background.
(2)
2nd order H(2) and Ha treat as HJ-eq’ns:
H(gab , φ;
∂S ∂S
,
)≈0
∂gab ∂φ
Ha (gab , φ;
∂S ∂S
,
) ≈ 0.
∂gab ∂φ
(1)
The scalar part of these equations gives a generating functional, S, for a
degenerate canonical transformation that leads to a set of reduced
variables.
For a certain choice of partial observables you get a set of complete
observables (not explicit about this terminology), which are the canonical
analogues of the Mukanov variables.
⇒ this just gives a particular parametrization of the reduced phase space.
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Intro
Standard Cosmo
Cosmo Shape Perts
Conclusions
Other (Modern) Approaches
1
2
Dittrich / Tambornino (2007): worked out explicitly partial/complete
observables for cosmology in different sectors and at higher orders.
Construction gets bulky at higher orders.
Maldacena (2003): ⇒ state of the art for higher orders. This is the
current tool for dealing with non-Gaussianity.
Limited: only valid for scalar modes.
Messy: relies on ‘funny’ expansion and approximations ⇒ not so
transparent.
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Intro
Standard Cosmo
Cosmo Shape Perts
Conclusions
Summary (and End of Extended Motivation)
Gauge-invariant observables were tricky at 1st order, but not in SD.
Higher order terms are still a bit tricky, although some proposals exist.
Main Problem: Finding gauge invariant observables is intimately tied to
solving the dynamics!
Can Shape Dynamics clean any of this up? (i.e., can SD provide a better
tool than known techniques?)
Important because higher orders are relevant for current experiments.
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Intro
Standard Cosmo
Cosmo Shape Perts
Conclusions
Cosmological Shape Perturbations: Basic Strategy
Two Separate Tasks:
1
Isolate gauge-invariant degrees of freedom ⇒ solve local constraints:
√
Conformal Constraint (D = π − hπi g ≈ 0): algebraic (simple trick) to all
orders.
Diff constraint (Ha = ∇b πa b ): same as standard picture.
2
Compute eq’ns of motion ⇒ compute HSD perturbatively and evolve
observables.
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Intro
Standard Cosmo
Cosmo Shape Perts
Conclusions
Zeroth Order: FRW
Assumptions
Take k = +1 and Λ > 0.
Make the homogeneous/isotropic Ansatz for background:
(0)
gab = V 2/3 Ωab
φ(0) (x, t) = φ(0) (t)
V 1/3 τ ab
Ω
2
(0)
(0)
πφ (x, t) = πφ (t) ,
ab
π(0)
=
(2)
(3)
where τ ≡ “York time”, and {V , τ } = 1.
LYE ⇒ unique algebraic solution: 1st Friedmann eq’n (after some
messaging)!
Hamilton’s equations then lead to 2nd Friedmann eq’n (after more
messaging).
⇒ Reproduces FRW
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Intro
Standard Cosmo
Cosmo Shape Perts
Conclusions
Conformal Constraint (I)
Idea: Solve the conformal constraint using a clever gauge fixing that makes use
of background solution.
Perturbation-Inspired Transformation
Perform an expansion inspired by the decomposition gab = V 2/3 (Ωab + hab ):
p
1
τ
hab = 2/3 gab − Ωab
π
¯ ab = V 2/3 π ab − g ab (h, V ) g (h, V )
(4)
2
V
So that (gab ; π ab ) → (hab , V ; π ab , τ ) is canonical.
Note: g ab (h, V ) means that you must express g ab in terms of a hab and V
(can be done in an h-expansion).
New conformal constraint
D=π
¯ ab (Ωab + hab ) ≈ 0.
(5)
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Intro
Standard Cosmo
Cosmo Shape Perts
Conclusions
Conformal Constraint (II)
Trick
Use the gauge fixing:
1
hab Ωab ≈ 0,
D
which is such that {D(x), G (y )} = δ(x, y ).
G =
(6)
Solutions (to all orders!!):
GI
hab
= hab − 13 hcd Ωcd Ωab
ab
πGI
=π
¯
ab
−
(7)
p
1 cd
π gcd (h, V ) g ab (h, V )
3
g (h, V ) .
(8)
⇒ This gauge fixing makes use of the Weyl invariance:
Off the ADM intersection!! (only at order > 1)
First Order
Solutions are traceless wrt background:
GI
hab
= hab −
h
Ωab
3
ab
πGI
=π
¯ ab −
π
¯ ab √
Ω
Ω,
3
(9)
where h = hab Ωab and π
¯=π
¯ ab Ωab .
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Intro
Standard Cosmo
Cosmo Shape Perts
Conclusions
Diff Constraint (I)
Kindergarten:
1st order
Fourier Transform on S 3 ⇒ spin-decomposition on k-space:
ab
(1)
(1)
hab (x) → hi (k) = A−1
hab (k)
φ(x) → φ(k)
(1)
πab (x)
i
→ p (k) =
i
i
ab
Aab π
¯(1)
πφ (x) → πφ (k).
(10)
(11)
where Ωab Aiab = 0 for (i = 3, . . . , 6), k a k b Aiab = 0 for (i = 3, 4) k a Aiab = 0 for
(i = 5, 6).
Then i = 1, 2 ⇒ scalar modes, i = 3, 4 ⇒ vector modes, and i = 5, 6 ⇒ tensor
modes.
Diff constraint kills vector modes and one linear combination of scalar
modes (i = 2 and φ(k)).
Conformal constraint kills (i = 1) scalar mode.
⇒ Mukanov variable (in CMC gauge) is the left-over scalar mode!!
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Intro
Standard Cosmo
Cosmo Shape Perts
Conclusions
Diff Constraint (II)
Higher Orders
Perturbative: Can use standard perturbative techniques (e.g., diagramatic
expansion) for Diff constraint.
Non-Perturbative: Can solve the elliptic PDE (see, e.g., York ’72) using
standard techniques from spectral analysis or numerically.
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Intro
Standard Cosmo
Cosmo Shape Perts
Conclusions
Evolution
Compute Hsd order by order in perturbation theory in k-space.
Use result to propagate observables.
Solving the conformal constraint simplifies many terms.
An analytic recursion relation can be obtained at all orders of magnitude.
⇒ just follow a straightforward algorithm!!
Note
Don’t yet have an expression that I’m confident with (free of minus sign errors,
factors, etc...)
⇒ coming soon!!
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Intro
Standard Cosmo
Cosmo Shape Perts
Conclusions
Advantages
Perhaps a cleaner (simple expansion) and more powerful (applies to all
modes) tool for studying higher order perturbations than Maldacena or
complete/partial observables?
Clean identification of observables.
Straightforward algorithm for determining evolution.
No “funny” expansions.
Take advantage of our new symmetry: Weyl invariance.
⇒ we must explore what this extra Weyl invariance buys us!!
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Intro
Standard Cosmo
Cosmo Shape Perts
Conclusions
Conclusions
Can SD serve as a useful tool for computing observable effects for
non-Gaussianity?
⇒ If so: what can we do to bring the theory into a form that is
suitable for and attractive to cosmologists?
THANK YOU!!
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