Canonical Cosmological Perturbation Theory using Geometrical Clocks

Transcription

1 Canonical Cosmological Perturbation Theory using Geometrical Clocks joint work with Adrian Herzog, Param Singh arxiv: and arxiv: International Loop Quantum Gravity Seminar Kristina Giesel, Institute for Quantum Gravity, FAU Erlangen

2 Plan of the Talk I. Motivation for canonical cosmological perturbation theory II. Brief remarks on Lagrangian framework III. Canonical formulation Framework of extended ADM phase space Gauge fixing Relation formalism and observable map IV. Construction of common gauge invariant quantities Two examples: Longitudinal and spatially flat gauge V. Summary and Outlook

3 I. Motivation Why Cosmological Perturbation Theory (CPT)? Interesting for us: Possibility to learn something about the early universe: Test (quantum) cosmological models [Wilson-Ewing, Ashtekar, Agullo, Bojowald, Singh, Mena-Marugan, ] Gauge invariant perturbations: observations Aim: Obtain gauge invariant equation of motion for perturbations CPT: Common gauges and gauge invariant variables exist

4 I. Motivation Why Cosmological Perturbation Theory (CPT)? Relational Formalism: Provides naturally a framework to construct gauge invariant quantities once reference fields (clocks) have been chosen. In particular useful for higher order perturbations theory Geometrical interpretation of gauges (dynamical observers) However, often matter reference fields are chosen: [(K.G. Thiemann),(Domagala, K.G., Kaminski, Lewandowski),(Husain,Pawlowski),(K.G., Oelmann)] As far as reduced quantization is considered matter models have advantage of simple observable algebra In LQC: scalar field and common gauge invariant quantities

5 I. Cosmological Perturbation theory Phase space formulation: Langlois: Only reduced ADM+ scalar field, only Mukhanov-Sasaki Dittrich, Tambornino: Connection formulation, lapse and shift not dynamical but gauge fixed. K.G., Hofmann, Thiemann, Winkler Non-linear Brown-Kuchar-dust clocks, not all common gauges Dapor, Lewandowski, Puchta: Different gauge invariant variables Cosmological perturbations on quantum spacetimes Elizaga Navascues, Martin de Blas, Mena Marugan: effects of quantum spacetime on perturbations Indicates: Natural clocks for each common gauge choice? Relevant for reduced quantized models in full and LQC?

6 II. Cosmological Perturbation theory One considers perturbations around a flat FLRW spacetime ds 2 = N(t) 2 dt 2 + a(t) 2 ijdx i dx j Aim: Gauge invariant EOM for perturbations N, N a, q ab Consider k=0 case only Linearized perturbations theory: Scalar-vector-tensor decomposition N = N(1 + g ),N a = µ! B,a q+ S ab,n,n a a q ab =2a 2 ( ab + E,<a,b> + F (a,b) htt (q ab,p ab ) N,N a q ab = {q ab,h can } ṗ ab =, {p,e,b ab,h can } Restrict discussion to scalar sector here ab ) At linear order perturbed C(q, p) =0 Einstein C a (q, equations p) =0 decouple, however complicated Poisson algebra due to projections.

7 Lagrangian Formalism One considers linear perturbations and linearized diffeom.: x 0µ = x µ + µ Gravitational sector: µ =(, ~ ) T =(, ˆ,a + a?) T 0 0 = + 1 N,t B 0 = B = + H N ˆ N a 2 + ˆ,t E 0 = E + ˆ Matter sector: ' 0 = ' + 1 N ',t minimally coupled scalar field

8 Gauge invariance: Lagrangian Formalism General idea: Choose among the gauge variant 5 scalar dof,b,,e,' two out of them to construct linearized gauge invariant extensions of the remaining three. Particular choice of such a pair: Common gauge in cosmology Two examples: Bardeen potentials and Mukhanov-Sasaki variable := v := ' 1 3 E + Ha2 N 2 (B Ė) := + 1 N ' H 1 3 E d dt a 2 (B Ė) N gauge invariant extension of lapse and some spatial metric perturbations (longitudinal gauge, B=0, E=0) gauge invariant extension of scalar field (spatially flat gauge =0,E =0)

9 III. Aim: Do the same on phase space Here we follow the following strategy: Step 1: Express Bardeen potentials, Mukhanov-Sasaki variable and their dynamics in ADM phase space + scalar field Step 2: Use the relational formalism and choose appropriate geometrical clocks from which Bardeen potentials and Mukhanov- Sasaki variables are obtained naturally from the observable map, relation to gauge choice, physical dof. Step 3: Analyze the dynamics of the observables and check that standard results can be reproduced Step 4: Use results to go further: geometric interpretation of gauges, higher order perturbations, non-linear clocks (work in progress)

10 Why extended phase space? Realize: Bardeen potential (lapse perturbation) is the gauge invariant extension of Explicit form of, suggests that B is among the 'natural' clocks In reduced ADM phase space we have: M ' R g µ! q ab,n,n a Reduced ADM: (q ab,p ab ) N,N a =0, a =0 Einstein s eqns: q ab = {q ab,h can } ṗ ab = {p ab,h can } Constraints: Can. Hamiltonian: C(q, p) =0 C a (q, p) =0 H can := Z d 3 x(nc + N a C a ) C, C a acts trivially on lapse and shift, construction of, B as clock?

11 Extension to full ADM Phase space [Pons, Garcia] CPT: Mostly discussed in Lagrangian framework Phase space: M ' R g µ! q ab,n,n a Full ADM: Einstein s eqns: (q ab,p ab ), (N, ), (N a, a ) q ab = {q ab,h can } ṗ ab = {p ab,h can } Ṅ = {N,H can } = = {,Hcan } 0 Ṅ a = {N a,h can } = a a = { a,h can } 0 Constraints: C(q, p) =0,C a (q, p) =0, =0, a =0 Can. Hamiltonian: H can = R d 3 x(nc + N a C a + + a a ) Extended Symmetry generator: G b, ~ b = C[b]+ ~ C[b]+ (ḃ + ba N,a N a b,a )+ a (ḃa + q ab (bn,b Nb,b ) N a b b,a + b a N b,a) reduces to primary Hamiltonian for b = N,b a = N a

12 Remarks on Gauge Fixing Reduced ADM phase space: Choose gauge fixing conditions: G µ 0 of the form G µ = F µ (q, p) f µ (x, t) Stability of the gauge fixing conditions: dg µ dt = {G µ,h can } dg µ dt = {G µ,h can } 0! 0 fixes lapse and shift if G µ is appropriately chosen, no further cond. Full ADM phase space: Choose gauge fixing conditions: G µ 0 of the form G µ = F µ (q, p, N,,N a, a ) f µ (x, t) Stability here: since lapse and shift are no Lagrange multipliers, further stability fixes, a extended phase space allows to choose lapse & shift dof in G µ!

13 Relational Formalism: Observables Start with constrained theory Choose clocks [Rovelli, Dittrich] {T I } s.t. {T I,C J } I J, {G I,C J } T I satisfying: (q A,P A ), {C I }, I label set {T I } s.t. {T I,C J } I J, {G I,C J } I J Gauge invariant extensions (observables) associated with f: O C f,t ( ) = 1 P n=0 (weak Abelianization, linear in clock momentum) 1 n! ( T I ) n P {f,c I } (n) = 1 n=0 ( 1) n n! (G I ) n {f,c I } (n) Dirac observables: {O f,c I } ' 0 Geometrical clocks Choose clocks from metric dof Algebra of observables: {O f,o g } ' O {f,g} Gauge invariant dynamics: do f d = {O f,h phys } = O {f,h}

14 Observables map on full ADM Phase space Extended observable map G b, ~ b = C[b]+ ~ C[b]+ (ḃ + ba N,a N a b,a )+ a (ḃa + q ab (bn,b Nb,b ) N a b b,a + b a N b,a) One can show [Pons, Shepley, Sundermeyer] G b, ~ b ' C[b]+ ~ C[b]+ [ḃ]+ a[ḃa ] Weak abelianization has to be performed wrt primary and secondary G µ = µ T µ, Gµ := (G µ, Ġµ ), Cµ := (C µ, µ ) Extended observable formula 1X Z Of,T G 1 = d 3 x 1 d 3 x n n! Gµ (x 1 ) G µ (x n ){f, C µ (x 1 )}, }, C µ (x n )} For n=0 reduces to usual observable map (q ab,p ab ) O C f,t

15 IV. Use following Strategy: Step 1: Express Bardeen potentials, Mukhanov-Sasaki variable and their dynamics in full ADM phase space + scalar field Consider linearized phase space: N = N, N a = B,a + S a, a = 1 N p, a =,a + ˆ a =(p B ),a +(p S ) a q ab =2a 2 ( ab +(E,<ab> + F (a,b) htt p ab =2 P a 2(p ', ' ab )) ab +(p E ),<ab> +(p F ) (a,b) (p h) TT ab ) Derive Hamilton's equations for linearized perturbations and use them to construct analogues of Bardeen potentials and Mukhanov-Sasakivariable

16 Gauge transformations on extended ADM Behavior under linearized gauge transformations: (geometry) G 0 b, ~ b G 0 b, ~ b = 1 N b,t G 0 B = b, ~ b = H N b ˆb G 0 E = ˆb b, ~ b N A b + ˆb,t Conjugate momenta: G 0 b, ~ bp = 1 4 H N + apple 8! N H p b N b + ˆb A H G 0 b, ~ bp E = G 0 b, ~ bp =0 G 0 b, ~ bp B =0 N 4A H b ˆb expected transformation behavior similar to Lagrange framework

17 Step 1: Longitudinal gauge Natural observables: Bardeen Potentials, LM := 1 3 E + HA N (B 2 B Ė LM! 4 H(E + p E ). Ė). used EOM for E to get result Bardeen potential = 1 3 E + P 2 (E + p E ) on phase and conjugate momentum: := p 1 6 E (E + p E) 1 4 P 2 + apple 2 Ap (E + p E ) Likewise we get: LM := + 1 N d dt Ā N (B Ė) EOM yields directly to = Phase space form of Bardeen potentials gives us hint what clocks to choose

18 Step 2: Choice of geometrical clocks Realize: Natural choice of geometrical clock for given gauge in CPT Longitudinal/Newtonian gauge: (E=0, B=0 + stability) Observables via map Linear Perturbations: O G f,t O G f,t = O G f,t O G f,t Explicitly on linearized phase space: Z O f,t [ ] = f + Z f + apple d 3 y Z d 3 y g µ! q ab,n,n a Ġ µ (y){f, µ (y)} + G µ (y){f, Cµ (y)} (q ab,p ab ) N,N a {f,c (z)} d 3 z B h µ(z,y) Ġµ (y){f, (z)} G µ (y) + d 3 w d 3 v B (w, v){ġ (v),c (z)} {f, (w)} Z Z q ab = {q ab,h can } ṗ ab = {p ab,h can } C(q, p) =0 C a (q, Z p) =0 Question: What kind of gauge fixing and hence clocks needs to be chosen such that observables H can are := the d 3 Bardeen x(nc + potentials? N a C a ).

19 Choice of geometrical clocks I Realize: Natural choice of geometrical clock for given gauge in CPT Longitudinal/Newtonian gauge: (E=0, B=0 + stability) T 0 =2 Pa(E + P E ) T a = ab (E,b + F,b )! ˆT = E Stability of the clocks: T 0 0, ˆT 0, T 0 0, ˆT 0, is equivalent to B 0, E 0, p E 0,. Results for observables: (q ab,p ab ) N,N a O G,T =, OG,T =, OG,T =, OG p,t = 1 N q ab = {q ab,h can } ṗ ab = {p ab,h can } correct form of Bardeen potentials, others vanish O G ',T = ' (gi), O G ',T = (gi) ' correct form of gauge invariant matter dof

20 Step 3: Dynamics of observables +3H +(2 Ḣ + H 2 ) = 4 GA T (gi). We are interested in EOMs of physical degrees of freedom. Two possibilities to compute EOM for observables: (more in outlook) 1. Derive Poisson algebra for observables: {O f,o g } ' O {f,g} Dirac bracket is quite complicated due to non-commuting clocks { T 0 (x), T 0 (y)} =0 { T 0 (x), T a (y)} = 3 4 Z apple p A g µ! q ab,n,n a We derived already all EOMs (q ab,p ab for ) all N,N gauge a variant quantities, use these q ab = {q ab,h can } ṗ ab = {p ab,h can } +3H +(2 Ḣ C(q, + Hp) 2 ) =0 = C4 GA a (q, Z p) T (gi) =0. H can := d 3 x(nc + N a C a ) d 3 a 2. Use that observables are combinations of gauge variant quantities Expected result for Bardeen potential: remaining 2 phys. dof in tensor sector.

21 Step1: Spatially flat gauge Repeat the same for spatially flat gauge. Mukhanov-Sasaki variable LM (v) = ' ' H Mukhanov-Sasaki variable on phase space: ' 1 v = ' N ' A 3/2 H 3 E v = ' ' E E a 3 ' dv d' ( ') N H 1 3 E Again we use these to choose the natural clocks for this gauge

22 Step 2: Choice of geometrical clocks Realize: Natural choice of geometrical clock for given gauge in CPT Spatially flat gauge: ( T 0 = 2 Pa( Stability of clocks: =0,E =0 1 3 E) T 0 0, ˆT 0, T 0 0, 0, E 0, 2p + stability) g µ! q ab,n,n a Results for observables: (q ab,p ab ) N,N a q ab = {q ab,h can } ṗ ab = {p ab,h can } C(q, p) =0 C a (q, p) =0 T a = ab (E,b + F,b )! ˆT = E ˆT p E, B 4 Hp E, O G ',T = v, O G ',T = v O G,T = 2 ( apple P 2 Ap), OG p,t = 1 N O G B,T = N 2 is equivalent to, O Ha 2 p G Z B,T = ˆ H can := d 3 x(nc + N a C a ) O G p,t = +, O G p E,T = 1 P 2 3 Combinations of Bardeen potential and its momentum, others vanish

23 Step 3: Dynamics of observables +3H +(2 Ḣ + H 2 ) = 4 GA T (gi). Again two possibilities to compute EOM for observables: 1. Derive Poisson algebra for observables: {O f,o g } ' O {f,g} { T µ (x), T (y)} =0, µ, =0,, 3 Reason: Projectors to define variables 2. Using again the EOMs of the gauge variant quantities we can derive { T µ (x), T (y)} =0, µ, =0,, 3 g µ! q ab,n,n a the Mukhanov-Sasaki equation from the Hamiltonian EOM of v, v v 00 k + (q ab,p ab ) N,N k a 2 a q 00 ab = v{q ab,h can } ṗ ab = {p ab k =0,H can } a C(q, p) =0 C a (q, p) =0 Z here already formulated in H terms can := of Fourier d 3 x(nc modes. + N a C a ) remaining 2 phys. dof in tensor sector.

24 Further gauges: In our papers we also considered 3 more common gauges: 1. Uniform field gauge: ' = E = 0, 2. Synchronous gauge: = B = 0, 3. Comoving gauge: ' = B =0. We could also determine the natural clocks for these gauges and construct associated observables.

25 Outlook: Dynamics and physical Hamiltonian Reconsider dynamics: Once observables constructed natural to compute EOM at the gauge invariant level Might be complicated due to Dirac bracket involved, see also paper by Dittrich and Tambornino However, here: All gauge fixing conditions are of the form G µ =ḡ(t)f (,p,b,p B,E,p E,,p, ', ' ) Work in progress: Candidate for a physical Hamiltonian s.t. do G f,t dt WIP with Singh & Winnekens = {O G f,t,h phys } Could be framework to efficiently compute EOM

26 III. Summary and Conclusions Idea: Use observable map in relational formalism and geometrical clocks to derive gauge invariant quantities relevant for CPT First derived formulation of linearized perturbation theory in full ADM phase space Bardeen potentials and Mukhanov-Sasaki variable in phase space formulation with associated natural clocks Consistent with Lagrangian dynamics at linear order Choice of natural geometrical clocks associated to common gauges in CPT, Analyzed dynamics

27 III. Open Questions and Outlook Geometrical interpretation of clocks: In the light of non-linear clocks one would like to choose consistent clocks in each order of perturbation theory Not possible for all clocks that are used in CPT Reduced quantization with geometric clocks? Computed perturbed linearized observable algebra Matter versus geometrical clocks? Non-linear clocks: Learn more about their particular geometric properties Would also allow to perturb gauge invariant Einstein Eqn now possible to discuss in full ADM phase space