Toward a theory of Toward a theory of de Sitter space?de Sitter space?

Donald MarolfDonald Marolf

May 25, 2007May 25, 2007Based on work Based on work

w/Steve Giddings.w/Steve Giddings.

ResultsResults• dS:dS: A laboratory to study locality (& more?) in perturbative gravity A laboratory to study locality (& more?) in perturbative gravity

• Constraints Constraints eacheach state dS invariant state dS invariant

• Finite # of pert states for Finite # of pert states for eternaleternal dS (pert. theory valid everywhere) dS (pert. theory valid everywhere) Limit ``energy’’ of seed states to avoid strong gravity. Limit ``energy’’ of seed states to avoid strong gravity. (Any Frame)(Any Frame)

Compact & finite F Compact & finite F finite N. S = ln N ~ ( finite N. S = ln N ~ (l/ll/lpp)) (d-2)(d-1)/d (d-2)(d-1)/d < S< SdSdS

neckneckConsider F = q TConsider F = q Tabab n naannbb

Observables?Observables?

Try Try O O = -g A(x) = -g A(x)x dSx dS

Finite (Finite (HH00) matrix elements <) matrix elements <11||OO||22> >

for appropriate A(x), |for appropriate A(x), |ii>.>.

Also dS-invariant to preserve Also dS-invariant to preserve HHphysphys..

A composite, VeV of A =0A composite, VeV of A =0

Relational observablesRelational observablesrecover local physicsrecover local physics

let let OO = -g A(x), = -g A(x),x dSx dS

A(x) =A(x) = (x) (x) (x) (x) (x)(x)

Given scalars Given scalars , , ,,

I.e., I.e., OO scans spacetime for intersection (“observer”), scans spacetime for intersection (“observer”),reports value of reports value of ..

If |If |> has 1 > has 1 -particle and 1 -particle and 1 -particle,-particle,,, then <then <||OO||> ~ <> ~ <||(x)|(x)|>>

Proto-local?Proto-local?

But fluctuations diverge!But fluctuations diverge!Work with seed states; Work with seed states;

Recall |0> is an attractor….Recall |0> is an attractor….

<<11||OO11OO22||22> = dx> = dx11 dx dx22 < <11|A|A11(x(x11)A)A22(x(x22)|)|22> >

~ dx~ dx11 dx dx22 < <|A|A11(x(x11)A)A22(x(x22)|)|>> ~ const(V~ const(VdSdS))

Note: <Note: <11||OO11OO22||22> = > = ii < <11||OO11|i><i||i><i|OO22||22> .> .control intermediate states?control intermediate states? OO = = P O P P O P for for PP a finite-dim projection; e.g. F < f. a finite-dim projection; e.g. F < f.dS UV/IR: Use “Energy” cut-off to control spacetime volumedS UV/IR: Use “Energy” cut-off to control spacetime volumeOO is insensitive to details of long time dynamics, as desired. is insensitive to details of long time dynamics, as desired.

Choose f to control “noise;” safe for f ~ MChoose f to control “noise;” safe for f ~ MmaxBHmaxBH..

Heavy reference object (“observer”) Heavy reference object (“observer”) safe for f ~ exp(S safe for f ~ exp(SdSdS),),

V < V < ll(d-1)(d-1) S SdSdS

~~

(vacuum noise, BBs)(vacuum noise, BBs)

~~

OO Proto-local Proto-local~~

Fundamental Lessons for Fundamental Lessons for cosmology?cosmology?

• No fundamental ``classical observers.”No fundamental ``classical observers.” Study Study quantumquantum observers & observables. observers & observables. Study fluctuations. Study fluctuations.

• Locality is approximate; no absolute HamiltonianLocality is approximate; no absolute Hamiltonian (no surprise, but no ``hot box’’)(no surprise, but no ``hot box’’)

• Approx. local physics over V Approx. local physics over V << exp(S exp(SdSdS) )

(smaller for light observer/observable)(smaller for light observer/observable)For larger V, “BB”-like vacuum noise dominates For larger V, “BB”-like vacuum noise dominates

• Quantum observers/observables are Quantum observers/observables are global global constructions.constructions.

• Finite S for eternal dS, but naturally embeds Finite S for eternal dS, but naturally embeds in larger infinite-dimensional theories. in larger infinite-dimensional theories. Similar results for eternal inflation, etc. ?? Similar results for eternal inflation, etc. ??