A global picture of quantum de Sitter space Donald Marolf May 24, 2007 Based on work w/Steve Giddings.

Perturbative gravity & dS Residual gauge symmetry when both i. spacetime has symmetries and ii. Cauchy surfaces are compact.  An opportunity to probe locality in perturbative quantum gravity!! E.g., de Sitter! Watch out for i) strong gravity ii) subtle effects on long timescale (e.g., from Hawking radiation) but keep guesses at non-pert physics on back burner.

Framework Matter QFT on dS w/ perturbative gravity Compare with perturbative QED on dS: Q 1 = E i dS i = -Q 2 Total charge vanishes! Q|  matter  0 th order: Consider any Fock state 1 st order: Gauss Law includes source i E i = . Restriction on matter states:

Matter QFT on dS w/ perturbative gravity (Moncrief, Fischer, Marsden, …Higuchi, Losic & Unruh) Similar “linearization stability constraints” in perturbative gravity! Hamiltonian constraints of GR: for any vector field , Expand in powers of l p w/ canoncial normalization of graviton. Matter QFT & free gravitons + grav. interactions 0 = H[  ] = (q dS 1/2 ) { l p -1 [ ( L  q dS ) ab  ab - ( L   dS ) ab h ab ] = (q dS 1/2 ) { l p -1 [ ( L  q dS ) ab  ab - ( L   dS ) ab h ab ] + l p 0 (T matter + free gravitons ) ab n a  b +… } + l p 0 (T matter + free gravitons ) ab n a  b +… }  A constraint for KVFs  ! 0 0 Residual gauge symmetry not broken by background. Framework

Quantum Theory If consistent, resolves Goheer-Kleban-Susskind tension between dS-invariance and finite number of states. Technical Problem: In usual Hilbert space, |  > must be the vacuum! (But familiar issue from quantum cosmology….) Requires: Q free [  ] |  matter + free gravitons > = 0 Each |  > is dS-invariant! Solution introduced by Higuchi: Renormalize the inner product! Consider |  > = dg U(g) |  > g dS For such states, define new “group averaged” product: phys := dg phys := dg g dS (Naïve norm “divided by V dS ” ) For compact groups, projects onto trivial rep. (also Landsmann, D.M.) { Fock state (seed) seeds (Not normalizeable, but like <p| ) dS-invariant! Vaccum is special case; norm finte for n > 2 free gravitons in 3+1

Results dS: A laboratory to study locality (& more?) in pert. grav. dS: A laboratory to study locality (& more?) in pert. grav. Constraints  each state dS invariant Constraints  each state dS invariant Finite # of pert states for eternal dS (pert. theory valid everywhere) Limit ``energy’’ of seed states to avoid strong gravity. (Any Frame) Compact & finite F  finite N. S = ln N ~ ( l/l p ) (d-2)(d-1)/d < S dS Finite # of pert states for eternal dS (pert. theory valid everywhere) Limit ``energy’’ of seed states to avoid strong gravity. (Any Frame) Compact & finite F  finite N. S = ln N ~ ( l/l p ) (d-2)(d-1)/d < S dS Simple relational observables (operators): O = A(x) [ O,Q  ]=0; Finite matrix elements, but (fluctuations) 2 ~ V dS. (Boltzmann Brains) Simple relational observables (operators): O = A(x) [ O,Q  ]=0; Finite matrix elements, but (fluctuations) 2 ~ V dS. (Boltzmann Brains) Solution: cut off intermediate states! O = P O P for P a finite-dim projection; e.g. F < F 1. Restricts O to region near neck. Heavy observer/observable OK for  t ~ S dS. Solution: cut off intermediate states! O = P O P for P a finite-dim projection; e.g. F < F 1. Restricts O to region near neck. Heavy observer/observable OK for  t ~ S dS. Proto-local physics over volumes ~ exp(S dS ) Other global projections assoc. w/ non-repeating events should work too. Proto-local physics over volumes ~ exp(S dS ) Other global projections assoc. w/ non-repeating events should work too. Picture looks rather different from “hot box…” Picture looks rather different from “hot box…” neck Consider F = q T ab n a n b ~ ~

Finite # of states? (Eternal dS) Conjecture for non-eternal dS: e S dS states enough for “locally dS” observer.   acceleration.   acceleration. too much   collapse! too much   collapse! As. dS in past and future if small “Energy.” As. dS in past and future if small “Energy.” At 0 th order in l p, consider F = q T ab n a n b neck Safe for F < F 0 ~ l d-3 / l p d-4 ~ M BH ; Other frames? |  > and U(g) |  > group average to same |  >; no new physical states! Finite N, dS- invariant S = ln N ~ ( l/l p ) (d-2)(d-1)/d < S dS

Observables? Try O = -g A(x) x dS Finite ( H 0 ) matrix elements for appropriate A(x), |  i >. O is proto-local for appropriate A(x). Also dS-invariant to preserve H phys. But fluctuations diverge: ~ V dS (vacuum noise, BBs) Note: =  i. Control Intermediate States? O = P O P for P a finite-dim projection; e.g. F < F 1. dS UV/IR: Use “Energy” cut-off to control spacetime volume O is insensitive to details of long time dynamics, as desired. Tune F 1 to control “noise;” safe for F 1 ~ F 0. ~ ~

Example: Schwarzschild dS Schwarzschild dS has two black holes/stars/particles.  Q[  ] = M – M = 0 Solution must be `balanced’! No “one dS Black Hole” vacuum solution.

II. Why a new picture? The static Hamiltonian is unphysical. A “boost” sym of dS Q[  ] = (q dS 1/2 ) (T matter + free gravitons ) ab n a  b  = H s R - H s L  But Q[  ] |  > = 0 |  > = dE f(E) |E L =E>|E R =E> Perfect correlations…  R = Tr L  is diagonal in E R.  R = Tr L  is diagonal in E R. [ H s R,  R ] = 0 H s R generates trivial time evolution: Static Region

A “boost” sym of dS [ H s R,  R ] = 0 H s R generates trivial time evolution: Eigenstates of H s R also unphysical |E R = 0> ~ |0> Rindler UV divergent: no role in low energy effective theory II. Why a new picture? The static Hamiltonian is unphysical.  Static Region

Observables? Try O = -g A(x) x dS Finite ( H 0 ) matrix elements for appropriate A(x), |  i >. Proto-local for appropriate A(x) Expand in modes. Each mode falls off like e -(d-1)t/2 l. Each mode gives finite integral for A ~  3,  4, etc. For |  i > of finite F, finite # of terms contribute. Free fields: Conformal case: maps to finite  t in ESU F maps to energy Large conformal weight & finite F  finite integrals! Also dS-invariant to preserve H phys.

But fluctuations diverge! Recall: |0> is an attractor…. = dx 1 dx 2 = dx 1 dx 2 ~ dx 1 dx 2 ~ dx 1 dx 2 ~ const(V dS ) Note: =  i. Control Intermediate States? O = P O P for P a finite-dim projection; e.g. F < F 1. dS UV/IR: Use “Energy” cut-off to control spacetime volume O is insensitive to details of long time dynamics, as desired. Tune F 1 to control “noise;” safe for F 1 ~ F 0. ~

Boltzmann Brains? dS thermal, vacuum quantum. In large volume, even rare fluctuations occur…. What do typical observers in dS see? Detectors or observers (or their brains) arise as vacuum/thermal fluctuations. Note: Infinity of ``Boltzmann Brains’’ outnumber `normal’ observers!!! (Albrecht, Page, etc.) I am a brain! Our story: Subtract to control matrix elements Subtract to control matrix elements Still dominate fluctuations for local questions integrated over all dS. Still dominate fluctuations for local questions integrated over all dS.  Ask different questions (non-local, finite V): O = P O P  Ask different questions (non-local, finite V): O = P O P ~ Fits with Hartle & Srednicki V

Poincare Recurrences, t ~ e S dS ? Finite N, H s : Hot Static Box Finite N, H s : Hot Static Box Global dynamics of scale factor Global dynamics of scale factor Unique neck defines zero of time, never returns. States relax to vacuum; Unique neck defines zero of time, never returns. States relax to vacuum; Relational Dynamics neck Local relational recurrences? No recurrences relative to neck. (L. Dyson, Lindesay, Kleban, Susskind) E = 0 “time-dependent background.” No issue: local observers destroyed or decay after t ~ e S dS

Summary dS symmetries are gauge  constraints!dS symmetries are gauge  constraints! H s, No “Hot Static Box” picture.H s, No “Hot Static Box” picture. Future and Past As. dS  Finite N (F dS-invariantFuture and Past As. dS  Finite N (F dS-invariant Relational dynamicsRelational dynamics “neck” gives useful t=0 states relax to vacuum, no recurrences.“neck” gives useful t=0 states relax to vacuum, no recurrences. O samples finite region R (relational, e.g., set by F 1 ). O samples finite region R (relational, e.g., set by F 1 ). For moderate R, Boltzmann brains give small noise term. Recover approx. local physics in R. For moderate R, Boltzmann brains give small noise term. Recover approx. local physics in R. ~ Vol( R ) < l (d-1) exp(S dS ), details to come!!

What limits locality in dS? Possible limits from Vacuum noise (Boltzmann Brains) V ~ exp(S dS ) Vacuum noise (Boltzmann Brains) V ~ exp(S dS ) Quantum Diffusion t ~ [ l S dS ] 1/2 Quantum Diffusion t ~ [ l S dS ] 1/2 Marker Decay/Destruction t ~ exp(S dS ) Marker Decay/Destruction t ~ exp(S dS ) Regulate & avoid eternal inflation, or Short Time Nonlocality t ~ l S dS (Arkani-Hamed) Regulate & avoid eternal inflation, or Short Time Nonlocality t ~ l S dS (Arkani-Hamed) Grav. Back-reaction t ~ l S dS (Giddings) Grav. Back-reaction t ~ l S dS (Giddings) l ln l ? l ln l ? Confusion: Durability: Need “reference marker” to select event. Other: