Gauge invariant variables for cosmological perturbation theory using geometrical clocks
aa r X i v : . [ g r- q c ] J un Gauge invariant variables for cosmological perturbation theory usinggeometrical clocks
Kristina Giesel , ∗ Adrian Herzog , † and Parampreet Singh ‡ Institute for Quantum Gravity, Department of Physics,FAU Erlangen – N¨urnberg, Staudtstr. 7, 91058 Erlangen, Germany and Department of Physics and Astronomy,Louisiana State University, Baton Rouge, LA 70803, U.S.A.
Using the extended ADM-phase space formulation in the canonical framework we analyzethe relationship between various gauge choices made in cosmological perturbation theoryand the choice of geometrical clocks in the relational formalism. We show that variousgauge invariant variables obtained in the conventional analysis of cosmological perturbationtheory correspond to Dirac observables tied to a specific choice of geometrical clocks. Asexamples, we show that the Bardeen potentials and the Mukhanov-Sasaki variable emergenaturally in our analysis as observables when gauge fixing conditions are determined viaclocks in the Hamiltonian framework. Similarly other gauge invariant variables for variousgauges can be systematically obtained. We demonstrate this by analyzing five common gaugechoices: longitudinal, spatially flat, uniform field, synchronous and comoving gauge. For allthese, we apply the observable map in the context of the relational formalism and obtainthe corresponding Dirac observables associated with these choices of clocks. At the linearorder, our analysis generalizes the existing results in canonical cosmological perturbationtheory twofold. On the one hand we can include also gauges that can only be analyzed inthe context of the extended ADM-phase space and furthermore, we obtain a set of naturalgauge invariant variables, namely the Dirac observables, for each considered choice of gaugeconditions. Our analysis provides insights on which clocks should be used to extract therelevant natural physical observables both at the classical and quantum level. We also discusshow to generalize our analysis in a straightforward way to higher orders in the perturbationtheory to understand gauge conditions and the construction of gauge invariant quantitiesbeyond linear order.
I. INTRODUCTION
In general relativity, only those quantities are physically observable which are invariantunder spacetime diffeomorphisms. In the formulation of general relativity as a constrainttheory, namely the ADM-phase space formulation, these invariant quantities are the Diracobservables. These observables commute with the Hamiltonian in the canonical descriptionof general relativity. If these observables do not depend explicitly on time, then they areconstants of motion. But if there is an explicit time dependence, these observables areevolving, but, then are not invariant under spacetime coordinate transformations. Topermit evolving observables one needs to go to a relational formalism where the evolutionof observables is studied with respect to some other fields [1–5]. The latter are the clocksor reference fields which capture the dynamics of other fields in the spacetime without ∗ [email protected] † [email protected] ‡ [email protected] using spacetime coordinates. These clocks can be matter fields, such as scalar fields ordust, or can be chosen from metric variables. Given a choice of clocks, the relationalformalism can be used to systematically construct Dirac observables in the canonicaldescription of gravity [6–10]. For instance this has been used to derive a reduced phasespace, that is the phase space of the gauge invariant quantities, for general relativity in[11–13] that was taking as the starting point for quantization. At the classical level sucha reduced phase space approach was considered in order to formulate general relativisticperturbation theory in [14]. An application of this to linearized cosmological perturbationtheory in the ADM-phase space can be found in [15], whereas a generalization to scalar-tensor theories has been presented in [16]. The reduced phase space of LTB spacetimeshas been derived in [17]. The work in [18, 19] follows more the conventional approachin cosmological perturbation theory, that is first considering linearized perturbations andafterwards constructing quantities invariant under linearized gauge transformations, butbased on Ashtekar-variables.In the canonical framework, there have been earlier works on cosmological perturbationtheory, most notably by Langlois [20]. Using the ADM-phase space, where the lapsefunction and the shift vector are treated as Lagrange multipliers, Langlois found a phasespace formulation of the Mukhanov-Sasaki variable and its evolution in the canonicalsetting. However, in the usual ADM-phase space formulation, the lapse function and theshift vector are not treated as phase space variables. Often this is called the reducedADM-phase space. As a result, it is not possible to understand various gauges and gaugeinvariant variables such as the Bardeen potentials in this setting. In other words, one cannot generalize Langlois’ work to various other gauge invariant quantities in the ADM-phasespace based on gauge fixing conditions involving lapse and shift variables. To accomplishthis, which is one of the main objectives of our work, we have to go beyond the (reduced)ADM-phase space and use the extended ADM-phase space studied in pioneering worksby Pons, Salisbury, Sundermeyer and others [21–24]. An advantage of this phase space,as discussed in [25] and demonstrated here, is that it is well suited to capture the fullscope of the relational formalism. While in the companion work [25], we formulatedcosmological perturbation theory in the setting of the extended phase space, in this workwe will apply the relational formalism to it. In particular, we will aim to understandthe relationship between gauge fixing conditions and gauge invariant quantities, used inconventional cosmological perturbation theory, in the setting of the relational formalismin the extended phase space. In comparison with Langlois’ earlier results, the novelty withthe current approach based on the extended phase space is that we are able to providea phase space formulation of various gauges and find their associated gauge invariantvariables using reference clock fields even for those cases where lapse and shift variablesare involved in the gauge fixing conditions. Furthermore, next to the results of Langlois [20]there is more recent work in the canonical formalism [26] that also considers second orderperturbations. Canonical cosmological perturbation theory has been also formulated inAshtekar variables in [27], but also here by treating lapse and shift as Lagrange multipliers.As a result, also in this work understanding of the issue of gauge invariant variables forvarious gauge choices is restricted to the reduced ADM-phase space.The issue of gauge fixing and gauge invariant quantities is a subtle one in cosmologicalperturbation theory. To relate the goals of our analysis in comparison to the conventionaltreatment, let us revisit it briefly. In the conventional analysis of cosmological pertur-bations, in order to extract any meaningful predictions, one must handle the problem ofgauge freedom in perturbations. This arbitrariness arising due to the freedom in perform-ing the space-time coordinate transformations, called diffeomorphisms, affects the scalarand vector metric perturbations, as well as the matter perturbations. There are two waysto deal with this this gauge freedom. One can either fix a gauge from the very onsetand work entirely in it, or one can construct gauge invariant variables. In the method ofgauge fixing, one imposes conditions on metric and/or matter variables to eliminate theunderlying freedom due to spacetime diffeomorphisms. An example where this freedomcan be completely eliminated is the longitudinal or the Newtonian gauge in which thelongitudinal part of the spatial metric perturbations is put to zero, and as a result thephysically relevant scalar metric perturbations are the ones in the lapse and the trace ofthe spatial metric. However, in general some residual freedom can remain even after gaugefixing. This happens for example for the synchronous gauge where the perturbations inthe shift vector vanish, yet the freedom under diffeomorphisms can not be fully eliminated.In the method of gauge invariant variables, one finds the right combination of metricand/or matter perturbations which are invariant under diffeomorphisms up to correctionsthat are of higher order than the considered order in perturbation theory. There are numer-ous ways to construct such gauge invariant variables. As examples, the Bardeen potentialsΦ and Ψ are obtained using the gauge transformation properties of linear perturbationsin the lapse and the trace of the spatial metric perturbation respectively. Similarly, theMukhanov-Sasaki variable v is constructed using the transformation properties of the lin-ear perturbations in the scalar field under gauge transformation. The Bardeen potentialsare in fact linear gauge invariant extensions of the lapse and the trace of the spatial met-ric perturbations. Similarly, the Mukhanov-Sasaki variable v is a linear gauge invariantextension of the scalar field perturbation.Though, in conventional treatments there is an emphasis on the usage of gauge invariantvariables, it is to be noted, and as is stressed by Bardeen himself [28], that the latterapproach is not more beneficial than the former to extract physical predictions about therelevant metric and matter perturbations. As is well known, this is because the physicalrelevance of gauge invariant variables is tied to the gauge fixing conditions. As an example,it is only in the longitudinal gauge that the Bardeen potentials can be naturally relatedto the metric perturbations, namely the perturbations in the lapse and the trace of thespatial metric. In any other gauge, the physical relevance of Bardeen potentials is ratherunnatural. This can easily seen for the case of the spatially flat gauge in which the gaugeinvariant extension of the trace of the perturbed metric vanishes. Similarly, it is only inthe spatially flat gauge that the Mukhanov-Sasaki variable can be naturally interpretedas the gauge invariant variable corresponding to the perturbations in scalar field.Thus, there is a subtle relationship between the gauge fixing conditions and the gaugeinvariant variables. While working with the latter, the gauge fixing conditions may not beapparent but they are critical to extract any physically relevant prediction for cosmologicalperturbations. For every gauge fixing condition, there is a natural set of physically relevantgauge invariant variables. Here it is important to note that such a correspondence isfurther tied to the order of perturbation theory in which the gauge invariant variables areconstructed.In the relational formalism, the choice of gauge fixing is tied to a particular choice ofclocks, see for instance also the early work in [29, 30] for a discussion on clocks in thecontext of general relativity and its quantization. As discussed earlier, these clocks canbe chosen using geometrical or matter degrees of freedom. In this manuscript, we choosecomponents of the perturbed metric as well as combinations of perturbed metric and mat-ter degrees of freedom as the clocks. The linearized gauge fixing conditions δG µ = 0 definea hypersurface and involve the perturbations in the clock variables, δT µ . This allows usto express the linearized gauge fixing conditions in terms of the linearized perturbationsof the clock variables. Thus, for various choices of gauge fixing conditions, we can iden-tify corresponding clocks whose stability conditions are equivalent to the stability of thegauge fixing condition. Once a particular choice of clock variables is made, the relationalformalism can be applied to explicitly construct Dirac observables associated with thoseclocks. These Dirac observables turn out to be the gauge invariant variables usually con-structed from combinations of the metric and matter perturbations. And as discussedabove, these gauge invariant variables are exactly the ones which are physically tied tothe corresponding gauge fixing conditions. As an example, the gauge invariant variablessuch as the Bardeen potentials emerge naturally as the Dirac observables for the clockscorresponding to the longitudinal gauge fixing conditions.We carry out this analysis for five commonly used gauge conditions in cosmologicalperturbation theory. The first three gauge conditions are the ones where the longitudinalpart of spatial metric perturbations vanishes. These are the longitudinal or the Newto-nian, the spatially flat and the uniform field gauges. The other two gauge conditions havevanishing perturbation in the shift vector. The gauges considered in this category are thesynchronous and the comoving gauges. In each of these gauges, we apply the relationalformalism in the extended phase space to identify consistent clocks and find the Diracobservables which are the gauge invariant variables naturally relevant for these particulargauges. It is to be noted that to achieve this task for various common gauges in cosmo-logical perturbation theory, it is necessary to have an extended phase space formulation.Without it, one is severely restricted in the above objective and can not even under-stand for example the Bardeen potentials in the relational formalism. As noted earlier,in particular one faces this limitation if one works with Langlois’ formulation of canonicalcosmological perturbation theory.Our analysis using the relational formalism based on clocks not only provides an al-ternate path to understand various gauge fixing conditions and gauge invariant variables,it further comes with some advantages not available in the conventional approach. Thefirst of which distinguishes it from the conventional procedure in the classical cosmologicalperturbation theory is its straightforward applicability to any higher order in perturba-tions. In the conventional framework, going beyond the linear order requires finding newgauge invariant quantities afresh at each order. Where as in relational formalism, oncea choice of non-linear clocks is made, one can compute the gauge invariant quantities forany order. Thus, providing a systematic procedure to obtain gauge invariant quantitiesat linear and higher orders. The second important advantage is that this formalism canprovide useful insights on how to extract predictions for the cosmological perturbationsin the canonical quantization framework. Given this formalism the path to understandclocks and gauge invariant quantities in quantum cosmological spacetimes becomes muchclearer. Further, the formalism can guide us on how to choose the right clocks relevantfor extracting analogs of Bardeen potentials and Mukhanov-Sasaki variable in a quantumcosmological perturbation theory.This manuscript is organized as follows. In section II we start with an overview of thecanonical cosmological perturbation theory in the extended phase space. Here we presenta brief introduction in the relational observable formalism for general relativity on the ex-tended phase space. For details of this discussion, the reader is referred to our companionarticle [25]. We divide section II in two parts. The first part focuses on various details ofthe cosmological perturbation theory in the extended ADM-phase space, and the secondpart deals with the relational formalism and construction of observables in the extendedADM-phase space. In section III we apply the relational formalism to understand how tochoose clocks corresponding to various gauge conditions in the cosmological perturbationtheory. The formalism is then used to derive Dirac observables in each of the cases whichyield the relevant gauge invariant quantities. We study the longitudinal, spatially flat,uniform field, synchronous and comoving gauges in detail and obtain geometric clocksand gauge invariant quantities for each of them. At the end of this section for part ofthe gauges we also discuss possible modifications of the gauge fixing conditions that arisenaturally in the relational formalism. The results from various gauges are summarizedin tables I and II. We conclude with a summary and discussion of open issues in section IV.For the benefit of the reader, in the following we provide a list of our main notation andconventions that will be used throughout the article. We use the Einstein sum convention.The spacetime metric signature is chosen to be ( − , , , { q, p } = 1. And, the Legendre map is denoted by LM and thecorresponding inverse Legendre map by LM ∗ .The following tables summarize various symbols used in the manuscript (see also tableIII in section III). General notations
Notation Meaning a, b, c, ... = 1 , , µ, ν, ρ, ... = 0 , ..., ∂ a f, f ,a , ∂f∂x a Partial derivatives D a f, f | a Spatial covariant derivatives x µ , y µ , ... Spacetime coordinates x j , y j , ... Spatial coordinates g µν Spacetime metric q ab Induced metric on Σ, spatial metricΓ abc
Spatial Christoffel symbols P ab Conjugate momentum to q ab N, N a Lapse function and shift vector fieldΠ , Π a Momenta of lapse and shift, primary constraints λ, λ a Lagrange multipliers associated to ˙ N , ˙ N a R (3) dabc Spatial Riemann tensor [ D a , D b ] ω c = R (3) dabc ω d K ab Extrinsic curvatureTr( T ) = q ab T ab Trace of the tensor TT Tab = T ab − q ab Tr( T ) Traceless part of TT
Notation Meaning a Cosmological scale factor A = a The squared scale factor H = ˙ aa Hubble parameter¯ N Background lapse, ¯ N = √ A : conformal time, ¯ N = 1: proper time˜ H = LM ( H ) Hubble parameter in phase space˜ P = − √ A ˜ H ¯ N ∝ momentum of A , ¯ P ab = ˜ P δ ab ρ, p Energy-density and pressure¯ ϕ, ¯ π ϕ Background scalar field and its momentum
Cosmological perturbation theory
Notation Meaning T ( n ) ≡ n ! δ n T n ’th perturbation of
Tφ, B, ψ, E, p ψ , p E Scalar perturbations S a , F b , p cF Transversal vector perturbations h T Tab , p cdhT T
Transversal traceless tensor perturbations δN, δN a Lapse and shift perturbations δN = ¯ N φ , δN a = B ,a + S a δq ab Spatial metric perturbation δq ab = 2 A ( ψδ ab + E ,
Observables in canonical general relativity
Notation Meaning T µ Clock fields G µ = τ µ − T µ Gauge fixing constraints O f,T [ τ ] Observable of f A µν ( x, y ) := { T µ ( x ) , C ν ( y ) } , matrix for weak abelianization B := A − ˜ C µ weakly abelianized constraints for reduced ADM( ˜Π µ , ˜˜ C ν ) weakly abelianized constraints for full ADM-phase space {· , ·} ∗ Dirac bracket with respect to constraint set ( G µ , ˜ C ν ) II. REVIEW OF CANONICAL COSMOLOGICAL PERTURBATION THEORYIN EXTENDED PHASE SPACE IN THE RELATIONAL FORMALISM
The goal of this section is to provide an overview of cosmological perturbation theory inthe canonical setting in the extended phase space. While canonical cosmological perturba-tion theory goes back to the work of Langlois [20], its generalization to the extended phasespace has been recently introduced in the companion article [25], to which the reader isreferred to for various details. A short review is presented in this section which is dividedinto two parts. The first one includes a discussion of canonical cosmological perturbationtheory in extended phase space, whereas the second part focuses on the relational formal-ism and how it can be used to formulate cosmological perturbation theory in terms of socalled Dirac observables.
A. Canonical cosmological perturbations in extended ADM phase space
In cosmology literature, cosmological perturbation theory is often discussed in theLagrangian framework. In this case one chooses a given background solution such asfor instance spatially flat FLRW spacetime and considers perturbations of all ten metriccomponents including particularly the g and the g a components that are parametrizedby the lapse function and the shift vector in the canonical framework. If in addition, wehave matter degrees of freedom as for instance a scalar field, one considers perturbationsof these degrees of freedom in a similar manner. At the level of linearized perturbationtheory one chooses a certain subset among the perturbed metric and matter variables andadds to them specific combinations of the remaining metric and matter degrees of freedomsuch that the final resulting quantities are invariant under linearized diffeomorphisms. Bythis we mean that the quantities constructed as above are invariant under diffeomorphismsup to corrections that are second order and higher. At linear order, prominent examplesof such gauge invariant quantities are the Bardeen potentials and the Mukhanov-Sasakivariable. Given these gauge invariant quantities, that will also be called observables in thefollowing, one can derive their corresponding equations of motion and hence obtain thelinearized Einstein equations expressed exclusively in terms of gauge invariant objects.If we want to carry the framework over to the Hamiltonian formalism, the first observa-tion we make is that for the usual ADM phase space, only the spatial metric componentsand their momenta are treated as elementary phase space variables whereas the lapseand shift degrees of freedom take the role of Langrage multipliers and are therefore phasespace independent quantities. The reason that such a different treatment of the two setsof variables is possible is that one obtains the reduced ADM phase space by working onthe constraint hypersurface that is obtained from the primary constraints. This reducesthe 10 metric degrees of freedom by four to six independent degrees of freedom that areencoded in the six components of the spatial ADM metric q ab , whereas the lapse functionand the shift vector become arbitrary Lagrange multipliers. However, as discussed in de-tail in [25] if our goal is to find the canonical counter-parts of for instance the Bardeenpotentials then it is necessary to treat all of the metric variables on an equal footing alsoin the Hamiltonian framework of general relativity. Therefore, in [25] as well as in thisarticle, we consider the full or the so called extended ADM phase space for which theprimary constraints have not yet been reduced. As a consequence, the lapse and shift takethe role of elementary phase space variables likewise to the spatial metric components inthe extended phase space.In the next subsection we will summarize the results of linearized canonical cosmolog-ical perturbation theory in extended ADM-phase space, that has been analyzed in thereview [25] to which we refer for more details. Afterwards we will briefly introduce therelational formalism and the way conventional cosmological perturbation theory can beembedded into it. For the latter step we will build on the seminal work of Pons, Salisbury,Sundermeyer, and others [21–24].
1. Canonical cosmological perturbation theory in extended ADM-phase space
Throughout this article we consider linear perturbations around k = 0 FLRWcosmological spacetimes. Hence, we will start our presentation with summarizing thenecessary equations for the background solutions in the canonical framework. This willalso serve to fix the notation for the background quantities. Again the derivation of theseresults as well as a more elaborate discussion on this topic can be found in [25]. For the spatially flat FLRW spacetime, the line element is given by:d s = g µν dx µ dx ν = − ¯ N ( t )d t + a ( t ) δ ab d x a d x b . (2.1)In our notation greek indices are spacetime indices and run from 0 to 4, while latin indiceslabel the spatial coordinates and run from 1 to 3 only. In the following we will denotebackground quantities with a bar on the top. The background metric components can bewritten as: ¯ q ab = A ( t ) δ ab ¯ N = ¯ N ( t ) ¯ N a = 0 , (2.2)where we chose A := a as our elementary configuration variable. This choice is convenientto write various formulae in canonical perturbation theory in a simpler form. We denotethe corresponding momenta of above quantities by P ab , Π and Π a respectively. Note, thatin order to keep a certain freedom for the time parametrization of the background solutionswe do not completely specify the lapse function ( ¯ N ) of the background here. In case onechooses ¯ N = 1, the background evolution will be measured in cosmic time, while for thechoice ¯ N = √ A = a , the line element is parameterized by conformal time. Performing aLegendre transformation we realize that it is singular which is reflected in the fact thatthe momenta associated with lapse and shift vanish, as can be seen below: P ab =: ˜ P δ ab = − A ¯ N ˙ AA δ ab , Π = 0 , Π a = 0 . (2.3)Using ˜ P we can introduce ˜ H the analogue of the Hubble parameter on phase space, thatis, ˜ H := − ¯ N ˜ P √ A . (2.4)The definition of ˜ H follows from the requirement that ˜ H when pulled back to the tangentbundle should agree with conventional Hubble parameter which in the Lagrangian pictureis the relative velocity of the scale factor H = ˙ aa . Thus we require LM ∗ ˜ H = H . Using ˙ aa =
12 ˙ AA and the equation of motion for A in (2.7) we get exactly the expression in (2.4).We denote the canonically conjugate momentum of A as P A which is related to ˜ P by P A ( t ) = 3 ˜ P ( t ), where the factor of 3 is due to the trace of δ ab being equal to 3.The momenta of lapse and shift, Π = 0 and Π a = 0, are primary constraints and areassumed to be satisfied for the background solution. A stability analysis of the primaryconstraints yields for flat FLRW cosmologies the following secondary constraints:¯ C = − √ A ˜ P + κA / ρ = 0 , and , ¯ C a = 0 . (2.5)Here ¯ C is the background Hamiltonian constraint and ¯ C a the background spatial diffeo-morphism constraints. That the latter trivially vanish for FLRW spacetimes is expectedbecause it is linear in the spatial derivatives of momenta and configuration variables,and these derivatives need to vanish in order to be consistent with the homogeneity andisotropy symmetries assumed for FLRW solutions. Thus, the symmetry reduced ADMHamiltonian for flat FLRW cosmologies reads: H = Z d x (cid:0) ¯ N ¯ C + ¯ N a ¯ C a + ¯ λ Π + ¯ λ a Π a (cid:1) = Z d x ¯ N ¯ C . (2.6)Here in the last step we used the fact that for the background solution the primaryconstraints are satisfied and that the shift vector identically vanishes.The Hamiltonian equations of motion can then be derived by computing the Poissonbracket of the canonically conjugate phase space variables A and 3 ˜ P . An alternative butequivalent way to obtain these equations of motion is to specialize the full relativisticequations of motions for q ab and P ab to the case of flat FLRW cosmologies. In both caseswe end up with the following equations for ˙ A and ˙˜ P :˙ A = − ¯ N √ A ˜ P , ˙˜ P = ¯ N
14 ˜ P √ A + κ √ Ap ! . (2.7)Using ˜ H these equations can be written as:˙ A = 2 ˜ H A, ˙˜ P = −
12 ˜ H ˜ P + κ N √ Ap . (2.8)As the matter content we introduce a minimally coupled Klein-Gordon scalar field with anarbitrary potential V ( ¯ ϕ ) whose phase space variables we denote by ¯ ϕ and ¯ π ϕ . In order thatthese variables comply with the symmetries of FLRW spacetime, both ¯ ϕ and ¯ π ϕ do onlydepend on the temporal coordinate. The corresponding contribution to the Hamiltonianconstraint is given by: ¯ C ϕ = κ λ ϕ λ ϕ ¯ π ϕ p det(¯ q ) + V ( ϕ ) ! . (2.9)Here the coupling constant λ ϕ should not be confused with the Lagrange multiplier λ associated with the primary constraints. Also, det(¯ q ) = A with ¯ q ab = A δ ab . The total0Hamiltonian constraint becomes ¯ C tot = ¯ C geo + ¯ C ϕ . The resulting Hamiltonian equationsfor the matter variables are:˙¯ ϕ = ¯ N λ ϕ A / ¯ π ϕ , ˙¯ π ϕ = − ¯ N A / λ ϕ
12 d V d ϕ ( ¯ ϕ ) . (2.10)As usual for cosmological models we introduce the associated energy density and pressurefor the matter content that has for the scalar field the following form: ρ = 12 (cid:18) λ ϕ A ¯ π ϕ + 1 λ ϕ V ( ¯ ϕ ) (cid:19) p = 12 (cid:18) λ ϕ A ¯ π ϕ − λ ϕ V ( ¯ ϕ ) (cid:19) . (2.11)Let us note here that in our notation the potential V ( ϕ ) is twice the usual value of thepotential due to on overall factor of that we chose for the scalar field action. For example,in the case of the usual quadratic inflationary potential, V ( ϕ ) above will be m ϕ and not m ϕ .Given these we can rewrite the Hamiltonian equations of the scalar field as first orderdifferential equations for ρ and p :˙ ρ = − H ( ρ + p ) ˙ p = − H ( ρ + p ) − ¯ NA / ¯ π ϕ d V d ϕ ( ¯ ϕ ) , (2.12)where we have used the definition of ρ , p and the equations of motion for the backgroundscalar field in (2.10). Next, we want to consider perturbations around the spatially flat FLRW solutionthat was discussed in the last subsection. Since we work in extended phase space we willconsider independent perturbations of all 10 metric degrees of freedom q ab , N, N a andtheir conjugate momenta P ab , Π , Π a . In addition to the gravitational sector, we also haveto introduce the perturbations of the minimally coupled scalar field. We obtain: q ab = ¯ q ab + δq ab , P ab = ¯ P ab + δP ab , N = ¯ N + δN, N a = ¯ N a + δN a ,ϕ = ¯ ϕ + δϕ, π ϕ = ¯ π ϕ + δπ ϕ , Π = ¯Π + δ Π , Π a = ¯Π a + δ Π a . (2.13)Considering the explicit form of the flat FLRW solution discussed above these phase spacevariables simplify to: q ab ( ~x, t ) = A ( t ) δ ab + δq ab ( ~x, t ) , P ab ( ~x, t ) = ˜ P ( t ) δ ab + δP ab ( ~x, t ) ,N ( ~x, t ) = ¯ N ( t ) + δN ( ~x, t ) , Π( ~x, t ) = δ Π( ~x, t ) ,N a ( ~x, t ) = δN a ( ~x, t ) , Π a ( ~x, t ) = δ Π a ( ~x, t ) ,ϕ ( ~x, t ) = ¯ ϕ ( t ) + δϕ ( ~x, t ) , π ϕ ( ~x, t ) = ¯ π ϕ ( t ) + δπ ϕ ( ~x, t ) . (2.14)The linearized Einstein equations for a generic background have been derived in our com-panion paper [25]. Moreover, these equations have been specialized to the case of the choiceof a flat FLRW cosmological background spacetime. Therefore, we will only present thefinal results here and refer the reader to [25] for a more detailed presentation. The equa-tions of motion for the perturbation of the spatial metric and its momentum are given1by: δ ˙ q ab = 2 ˜ H Aδ ab δN ¯ N − H (cid:18) δ ca δ db − δ ab δ cd (cid:19) δq cd − H A ˜ P (cid:18) δ ac δ bd − δ ab δ cd (cid:19) δP cd + 2 δN ( a,b ) , (2.15)and δ ˙ P ab = 14 1 √ A ˜ P δ ab δN + 1 √ A (cid:16) ∂ a ∂ b − δ ab ∆ (cid:17) δN − ¯ N (cid:20) A / ˜ P (cid:18) δ ac δ bd − δ ab δ cd (cid:19) δq cd + 1 √ A (cid:18) δ ac δ bd − δ ab δ cd (cid:19) δR (3) cd + 1 √ A ˜ P (cid:18) δ ac δ bd − δ ab δ cd (cid:19) δP cd (cid:21) + ˜ P (cid:16) δN c,c δ ab − δN ( a,b ) (cid:17) + κ N √ A (cid:20) pδ ab δN ¯ N − (cid:18) pδ ac δ bd + 12 ρδ ab δ cd (cid:19) δq cd A + P δ ab (cid:21) . (2.16)Here we have introduced P and E as the perturbations of the energy-density and pressurerestricted to those terms that contain perturbations of the scalar field and its momentum.These are given by: P := λ ϕ A ¯ π ϕ δπ ϕ − λ ϕ d V d ϕ ( ¯ ϕ ) δϕ, (2.17)and E := λ ϕ A ¯ π ϕ δπ ϕ + 12 λ ϕ d V d ϕ ( ¯ ϕ ) δϕ. (2.18)Note, that these expressions only contain the perturbations of scalar field and its momen-tum, and the terms involving perturbations of the geometry are not included.The perturbed secondary constraints turn out to be [25]: δC = −
14 ˜ P √ A δ ab δq ab − √ A ˜ P δ ab δP ab − √ A (cid:16) ∂ a ∂ b − δ ab ∆ (cid:17) δq ab + κ − √ A pδ ab δq ab + E ! , (2.19)and δC a = − Aδ ab δP bc,c − P (cid:18) δ ba ∂ c − δ bc ∂ a (cid:19) δq bc + κ ¯ π ϕ δϕ ,a . (2.20)It is to be mentioned that the above equations agree with the work of Langlois in [20] (seeequations (19) and (20) in [20]). For a comparison, the following differences in notationmust be considered: κ = 2 κ Langlois , V = V Langlois , ϕ is denoted as φ , A = e α , ¯ q ab , ¯ P ab are denoted by γ ij and π ij respectively, π α = 6 A ˜ P and the respective Poisson bracket of γ ij with π ij does not involve κ . The equations of motion for lapse and shift and theirmomenta turn out to be: δ ˙ N = δλ, δ ˙Π = − δC, δ ˙ N a = δλ a , δ ˙Π a = − δC a . (2.21)2These differential equations for lapse and shift involve the perturbations of the Lagrangemultipliers λ and λ a associated with the primary constraints. Expressing these in termsof background quantities and perturbations we obtain: λ = ¯ λ + δλ = ˙¯ N + δλ,λ a = ¯ λ a + δλ a = δλ a . (2.22)Finally, the equations of motion for the perturbation of scalar field and its momentum areof the form: δ ˙ ϕ = δN λ ϕ A / ¯ π ϕ + ¯ N λ ϕ A / (cid:18) δπ ϕ − A ¯ π ϕ δ ab δq ab (cid:19) ,δ ˙ π ϕ = − δN A / λ ϕ
12 d V d ϕ ( ¯ ϕ ) + ¯ N A / λ ϕ (cid:18) A ∆ δϕ − A d V d ϕ ( ¯ ϕ ) δ ab δq ab −
12 d V d ϕ ( ¯ ϕ ) δϕ (cid:19) + ¯ π ϕ δN a,a . (2.23)This concludes our discussion on the equations of motion at the level of linear cosmologicalperturbation theory. One of the main advantages of a decomposition of a generic tensor into scalar,vector and tensor parts is that in linear cosmological perturbation theory the correspond-ing equations of motion completely decouple and thus can be analyzed independently. Asa consequence, the task of finding solutions for the linearized Einstein equations presentedin the next section simplifies. Again a detailed introduction of how such projectors ontothe scalar, vector and tensor components can be defined can be found in our companionpaper [25]. In this work we will just list the results that are needed for our analysis.We want to define projectors for k=0 FLRW spacetimes that decompose a given sym-metric tensor of rank 2 into its scalar, vector and tensor part. For this purpose we definethe following differential operators: D a = 1 A ∂ a , ∆ = 1 A ∆ , ( M − ) ab = A ∆ − (cid:18) δ ab −
14 ∆ − ∂ a ∂ b (cid:19) , (2.24)where ∆ − stands again for the Green’s function of the Poisson equation, with the asso-ciated integral kernel of ∆ − denoted by G ( x, y ), that is:∆ − f ( x ) = Z ¯Σ d y G ( x, y ) f ( y ) . (2.25)These operators allow us to define a Helmholtz decomposition of a vector into its longitu-dinal (scalar) and transversal parts:( ˆ P S V ) = A ∆ − ∂ a V a , ( ˆ P ⊥ V ) a = V a − ∂ a ( ˆ P S V ) . (2.26)3Further, if we introduce a projector that projects a second rank tensor onto its trace givenby ( ˆ P Tr T ) = 13 A δ ab T ab , (2.27)we can formulate the following projectors:( ˆ P L T ) a := (cid:0) M − (cid:1) ba (cid:20) ∂ c T bc − ∂ b ( ˆ P Tr T ) (cid:21) , ( ˆ P TT T ) ab = T ab − ( ˆ P T r T ) q ab − ∂ , ( ˆ P LS T ) = 34 ∆ − ∂ T ab , ( ˆ P LT T ) a = ∆ − (cid:16) δ ba ∂ c − ∂ a δ bc (cid:17) T bc − ∂ a ( ˆ P LS T ) , (2.28)where L, T T, LS, LT are abbreviations for longitudinal, transverse-traceless, longitudinal-scalar and longitudinal-traceless respectively. Here T
43 ∆ (cid:18) ψ −
13 ∆ E (cid:19) δ ab − (cid:18) ψ −
13 ∆ E (cid:19) ,
12 ∆ h T Tab . (2.36)It is also useful to introduce the spatial energy momentum perturbation: δ ˜ T := 1 A / ( P − ρ + p ) ψ ) . (2.37)Using the above equation together with (2.36) and (2.16), it is straightforward to obtainthe equations of motion for p ψ and p E :˙ p ψ = 16 ¯ N A ˜ H ∆ (cid:18) φ + ψ −
13 ∆ E (cid:19) + (cid:18) −
12 ˜ H + κ N ˜ H p (cid:19) (cid:18) p ψ − ψ (cid:19) − κ N ˜ H δ ˜ T − (cid:18)
12 ˜ H + κ N ˜ H p (cid:19) φ + 16 ∆ B, ˙ p E = −
14 ¯ N A ˜ H (cid:18) φ + ψ −
13 ∆ E (cid:19) + (cid:18)
52 ˜ H + κ N ˜ H p (cid:19) ( E + p E ) − B . (2.38)The equations of motion for the vector and tensor perturbations can be derived anal-ogously by applying the corresponding projectors onto δq ab and δP ab . For the vectorperturbations, these turn out to be:dd t (cid:20) δ ab F b p aF (cid:21) = (cid:20) − H − H ̥ ̥ (cid:21) (cid:20) δ ab F b p aF (cid:21) + (cid:20) S a − S a (cid:21) . (2.39)5And, for the tensor perturbations, we obtain:dd t (cid:20) δ ac δ bd h T Tcd p abh TT (cid:21) = (cid:20) − H − H ̥ + ̟ ∆ ̥ (cid:21) (cid:20) δ ac δ bd h T Tcd p abh TT (cid:21) . (2.40)Here, we have introduced ̥ := 5 ˜ H κ N p ˜ H and ̟ := − ¯ N A ˜ H . (2.41)Next, we want to discuss the equations of motion of the decomposed quantities thatare associated with the perturbed lapse and shift as well as their conjugate momenta. Thedynamics of the lapse and shift perturbations are related to yet undetermined functionswhich are the perturbations of the Lagrange-multipliers λ = ¯ λ + δλ , λ a = ¯ λ a + δλ a .Using the decomposition of perturbation in Lagrange multiplier δλ a in terms of scalar andtransversal parts: δλ a = δ ˆ λ ,a + δλ a ⊥ , we obtain,˙ φ = − ˙¯ N ¯ N φ + δλ ¯ N , ˙ B = δ ˆ λ, ˙ S a = δλ a ⊥ . (2.42)Using the projected quantities, and (2.19) and (2.20), the linearized secondary con-straints become: δC = − √ A ˜ P ( ψ + 4 p ψ ) + 4 √ A ∆ (cid:18) ψ −
13 ∆ E (cid:19) + κA / ( − p ψ + E ) δC a = − A ˜ P (cid:18) p ψ − ψ + 23 ∆( E + p E ) (cid:19) ,a − A ˜ P ∆( F a + p bF δ ab ) + κ ¯ π ϕ δϕ ,a . (2.43)These perturbations of the secondary constraints stabilize the perturbations of the primaryconstraints δ Π, δ Π a : δ ˙Π = − δC, δ ˙Π a = − δC a . (2.44)Let us note that the form of the perturbed Hamiltonian constraint δC agrees with onein for example [15] (equation (19)), where all perturbed quantities were constructed withdust clocks. However, there are differences in notation. There the authors use ¯ N = √ A , φ = 0, Ξ denotes the scalar field, and ψ corresponds to ψ + ∆ E . Note that due to dustclocks, the perturbed quantities have a different interpretation because one starts froma different model, that is gravity with a minimally coupled scalar field plus the coupledBrown-Kuchaˇr dust.For the reason that the perturbed spatial diffeomorphism constraint is a covector itcan be decomposed into a scalar part δ ˆ C and a transversal part δC ⊥ a yielding: δ ˆ C = − A ˜ P (cid:18) p ψ − ψ + 23 ∆( E + p E ) (cid:19) + κ ¯ π ϕ δϕ,δC ⊥ a = 2 A ˜ P ( F a + p bF δ ab ) . (2.45)Finally, let us discuss the Hamiltonian equations of the scalar matter field perturbations.Because δϕ is already a scalar and δπ ϕ is a scalar density no further decomposition ofthese quantities has to be performed. However, since also geometrical degrees of freedom6are involved in the matter equations of motion, of course these equations need also to berewritten in terms of the decomposed quantities. We get, δ ˙ ϕ = ¯ N λ ϕ A / ¯ π ϕ (cid:18) φ − ψ + δπ ϕ ¯ π ϕ (cid:19) ,δ ˙ π ϕ = ¯ N A / λ ϕ (cid:20) −
12 d V d ϕ ( ¯ ϕ )( φ + 3 ψ ) + 1 A ∆ δϕ −
12 d V d ϕ ( ¯ ϕ ) δϕ (cid:21) . (2.46)After having presented all necessary equations of motion that will be relevant for ouranalysis later, in the next session we give a brief introduction to the relational formalismand how it can be used to construct gauge invariant quantities in general relativity. B. Relational formalism and Dirac observable in extended ADM-phase space
In the last subsection we discussed the way linear cosmological perturbation theorycan be formulated in terms of appropriate variables in extended ADM phase space suchthat we can analyze the scalar, vector and tensor sector independently because theircorresponding equations of motion fully decouple. However, all of the scalar and vectorvariables introduced above share the common feature that they are not invariant undergauge transformation, that means invariance under arbitrary coordinate transformationsalso called diffeomorphisms or also called gauge transformations in the context of generalrelativity. As we analyze linear cosmological perturbation theory this invariance is requiredorder by order and thus in our case we want to construct quantities that are invariant underlinearized gauge transformations. We denote the resulting gauge invariant quantity as thegauge invariant extension of the corresponding perturbed quantity. Note that the tensorprojections are already invariant under these transformations up to linear order.In the Lagrangian framework the strategy one adopts is as follows. For the scalar aswell as the vector projections one chooses a certain subset of the variables and considersthe way members of these subsets transform under linearized gauge transformations. Thenone adds to these chosen variables specific combinations of the remaining variables suchthat the final sum of the original chosen variables together with these specific combinationslead to a quantity that is invariant under linearized gauge transformations.Now we aim at doing the same construction in the Hamiltonian picture. A crucialconcept that enters the above construction is that one considers two sets of variables bothnot invariant under gauge transformations. Then one combines them in an appropriateway so as to obtain a quantity that is invariant under (linearized) gauge transformations.This construction can be naturally embedded into the so called relational formalismwhere gauge invariant extension of a given variable of the first of those sets is definedwith respect to so called reference fields, that play the role of the second set of variablesmentioned above. In order to explain how this embedding can be actually performedin the first instance we will briefly summarize how (linearized) diffeomorphisms can beimplemented on the extended ADM-phase space following the seminal work of Pons,Salisbury and Sundermeyer et al [21–24]. Afterwards, we will discuss how this can beused to define a so called observable map that maps each phase space variable to itsgauge invariant extension once a set of reference fields which are often also called clockfields has been chosen. A more detailed introduction to these topics can be found in[25] and references therein. Here we are particularly interested in understanding how agiven choice of clock fields is related to a certain choice of a gauge fixing condition at7the Lagrangian level. Formulated in a different way we want to address the followingquestion: What are the appropriate clock or reference fields that we must choose in orderto obtain in the Hamiltonian framework gauge invariant extensions that correspond forexample to the Bardeen potentials and the Mukhanov-Sasaki variable?
Before discussing how diffeomorphism are implemented on the extended ADM-phasespace, let us briefly recall how this can be done on the reduced ADM-phase spacewhere the primary constraints Π ≈ a ≈ q ab , P ab ). Under diffeomorphism q ab , P ab willtransform accordingly and in the Hamiltonian formulation these transformations can beformulated by means of the spatial diffeomorphism constraint C a and the Hamiltonianconstraint C respectively. The first one C a generates spatial diffeomorphisms within thespatial hypersurfaces one obtains from the standard ADM 3+1 decomposition of the fourdimensional spacetime. The second one generates diffeomorphism orthogonal to thesehypersurfaces. Note that for both generators this is only true on-shell, that is whenthe equations of motion as well as the constraints are satisfied. In order to introduce anotation for the action of these generators on the reduced ADM phase space, let us definethe smeared versions of the constraints given by: C [ b ] := Z d xC ( x ) b ( x ) and C a [ b a ] := Z d xC a ( x ) b a ( x ) , (2.47)where we call b and b a smearing or test functions respectively. Then the action of thegenerators on a generic phase space function f can then be written as: δ ~b f := { f, C a [ b a ] } and δ b f := { f, C [ b ] } (2.48)If we restrict to the gravitational sector for the moment, then all functions f on thereduced ADM-phase space can be understood as functions of the elementary variables q ab , P ab . Consequently if we know the action of the generators C [ b ] and C a [ b a ] on them,we can easily compute their action on a generic f . One obtains on-shell for the spatialdiffeomorphisms: δ ~b q ab = κ ( L ~b q ) ab and δ ~b P ab = κ ( L ~b P ) ab (2.49)and δ b q ab = κ ( L ~b q ) ab and δ b P ab = κ ( L ~b P ) ab , (2.50)which is exactly the expected transformation behavior. In the case of the orthogonaldiffeomorphisms generated by C [ b ] the discussion is slightly more complicated, see forinstance [32] for a pedagogical presentation. If we consider that in the ADM 3+1 split thespacetime metric g µν can be expressed as g µν = q µν − n µ n ν where n µ is the co-normal vectorof a spatial hypersurface and choose a frame in which n t = − b and n a = 0, correspondingto a normal vector n µ of the form n t = b , n a = − b a /b , then on-shell the action of C [ b ]onto the ADM metric q µν can be expressed as δ b q µν = κ ( L bn q ) µν , (2.51)8where we used that in this frame q tt = 0 and q at = 0. A similar but slightly longercalculation shown for instance in [32] shows that also for the ADM momenta P ab one getson-shell the expected transformation behavior under orthogonal diffeomorphisms, that canbe expressed as: δ b P µν = κ ( L bn P ) µν . (2.52)Note, that for the momenta on-shell involves also the constraints and hence this identityholds only on the constraint hypersurface of the Hamiltonian constraint.As it will be convenient for our later calculation we consider the combination of thespatial and orthogonal diffeomorphisms and define the following generator: G b,~b = 1 κ ( C [ b ] + C a [ b a ]) , (2.53)Note, that if we add matter as for instance a scalar field that we will need later, we canstill work with the generator G b,~b in which then C and C a denote the total spatial andHamiltonian constraint. The total constraints consist of the sum of the geometric andmatter contributions, that is C = C geo + C matter and similarly for C a = C geo a + C matter a .At first we will restrict our discussion to the gravitational sector only and consider ageneralization later when needed.This finishes our discussion on the reduced ADM-phase space. Now, if we aim at goingover the extended ADM-phase space, then we realize that since the generator G b,~b is alinear combination of C and C a only, it trivially commutes with lapse and shift and theirconjugate momenta. As a result, it certainly does not generate diffeomorphisms for thelapse and shift degrees of freedom. Consequently, if we instead work on the extendedADM-phase space where the primary constraints have not yet been reduced and thus wecan treat ( N, Π) and ( N a , Π a ) as full phase space variables, we need a modified generator,that we denote by G ′ b,~b , that also generates diffeomorphisms for these set of variables. Sucha generator has been derived by Garcia, Pons, Salisbury, and Shepley in Refs. [21, 22], towhich we refer the reader for more details. A brief summary of the relevant details for thefollowing discussion can be found in our companion paper [25]. Here we will just list theresults needed for the present analysis.In the extended phase space, the modified generator G ′ b,~b takes the following form: G ′ b,~b = 1 κ (cid:16) C [ b ] + C a [ b a ] + Π[˙ b + b a N ,a − N a b ,a ] + Π a [˙ b a + q ab ( bN ,b − N b ,b ) − N a b b,a + b a N b,a ] (cid:17) . (2.54)Naively, one could expect that the generator G b,~b needs to be extended by terms involvingthe primary constraints because these have a non-trivial action on lapse and shift vari-ables. However, the particular form of the phase space dependent smearing functions forthe primary constraints comes from the requirement that one wants to match the im-plementation of the diffeomorphisms on the extended ADM-phase space with the one inLagrangian framework, see [22]. Note, that the fact that these extra terms in G ′ b,~b are pro-portional to the primary constraints Π µ with µ = 0 , · · · , := Π, has the effect thaton fields other than lapse and shift the action of G ′ b,~b reduces to the gauge transformationgenerated by G b,~b in (2.53).The transformation of phase space variables under the modified generator G ′ b,~b can becomputed in a straightforward way. The configuration variables in the gravitational sector9transform according to [25]: δ G ′ b,~b N = b ,t − N a b ,a + b a N ,a ,δ G ′ b,~b N a = q ab ( bN ,b − N b ,b ) + b a,t + b b N a,b − N b b a,b ,δ G ′ b,~b q ab = ˙ q ab | N = b,N a = b a , (2.55)where we have introduced the following notation δ G ′ b,~b f := { f, G ′ b,~b } . (2.56)Similarly, for the corresponding conjugate momenta we obtain: δ G ′ b,~b Π = (Π a q ab b ) ,b + Π a q ab b ,b + (Π b a ) ,a ,δ G ′ b,~b Π a = Π b ,a + (Π a b b ) ,b + Π b b b,a ,δ G ′ b,~b P ab = ˙ P ab (cid:12)(cid:12)(cid:12) N = b,N a = b a + q c ( a q b ) d Π c ( bN ,d − N b ,d ) . (2.57)As can be seen from the above equation the primary constraint hypersurface Π ≈ , Π a ≈ G ′ b,~b . This is a necessary requirement for the gener-ator as otherwise G ′ b,~b would map out of the physical sector. Considering the action onthe momenta P ab , we realize that δ G ′ b,~b P ab contains an additional term proportional toΠ a . However, on the physical sector, where all constraints are satisfied, this term clearlyvanishes.Given that we have a generator of diffeomorphisms on the extended ADM-phase spaceavailable, we can use it to analyze how the relevant variables for cosmological perturbationstheory transform under linearized diffeomorphisms. As far as the background is consideredwe assume that we choose fixed coordinates. For linear perturbations we restrict to thecase of infinitesimal diffeomorphisms. This means that the change of the tensor fieldscaused by diffeomorphisms is of the order of the perturbations ǫ . The gauge descriptors b and ~b will also be of the same order ǫ . Now considering a generic tensor field Q onthe extended ADM-phase space, we obtain the following transformation behavior underlinearized diffeomorphisms generated by G ′ b,~b : Q ′ = Q + { Q, G ′ b,~b } = ¯ Q + δQ + { Q, G ′ b,~b } + O ( ǫ ) . (2.58)Here the bar on the Poisson bracket denotes that it is computed on the backgroundspacetime. To make contact to our former notation introduced above, we define δ G ′ Q := Q − ¯ Q as the perturbation of the variable Q . Then we obtain, δ G ′ b,~b Q = { Q, G ′ b,~b } + O ( ǫ ) . (2.59)Note that the above transformation does not affect the background geometry since it doesnot contain any terms of the order ǫ . That is, ¯ Q is unaffected by the above diffeomor-phism. In order to find the change in the linear perturbation δQ (1) , we consider only thoseterms which are of ǫ order in the above equation. However, it is to be noted that in generalthe action of diffeomorphism generator results in terms of order ǫ and higher. Thus, thechange in first order perturbations can be written as,( δ G ′ b,~b δQ ) (1) = { Q, G ′ b,~b } . (2.60)0In the following we drop the superscript above the parenthesis for brevity. Since we needthe transformation behavior of the earlier introduced projected quantities it is convenientto decompose ~b into its scalar and transversal part: ~b = ˆ b ,a + b a ⊥ . Looking at the results inequations (2.55) and (2.57) we can rewrite these taking the decomposition of the spatialdescriptor b a into account. This leads to the following form of the transformations: δ G ′ b,~b N = b ,t , δ G ′ b,~b N a = − ¯ NA b ,a + ˆ b ,a,t + b a ⊥ ,t ,δ G ′ b,~b q ab = 2 A " ˜ H ¯ N b + 13 ∆ˆ b ! δ ab + ˆ b ,
14 ˜ H ¯ N − κ N ˜ H p ! b + 16 ∆ (cid:18) ¯ NA ˜ H b + ˆ b (cid:19)! δ ab − (cid:18) ¯ N A ˜ H b + ˆ b (cid:19) ,
14 ˜ H ¯ N + κ N ˜ H p ! b + 16 ∆ (cid:18) ¯ NA ˜ H b + ˆ b (cid:19) , p E → p E − ¯ N A ˜ H b − ˆ b, (2.65)1where we used in the first line the result in (2.61), namely that δ G ′ b,~b Π = 0 and δ G ′ b,~b Π a = 0because the primary constraints are satisfied for the background solution.Further, the vector variables transform under linearized diffeomorphisms in the follow-ing way: S a → S a + b a ⊥ ,t , p S a → p S a ,F a → F a + b ⊥ a , p aF → p aF − b a ⊥ . (2.66)The tensor sector encoded in h T Tab and p abT T consists of already gauge invariant variablesand hence no further analysis is needed here. In the Lagrangian framework usually aninfinitesimal diffeomorphism is parametrized as x µ → x µ + ǫ µ where x µ denotes spacetimecoordinates. In order to compare our results here with the literature (e.g. [31]) we haveexpressed ǫ µ in terms of the descriptors b and b a . We have ǫ µ = bn µ + X µa b a and for anadaptive frame we can use X µa = δ µa and n µ = N − (1 , − N a ). In our companion paper [25] we discussed in detail how Dirac observables can beconstructed in the context of the relational formalism. The techniques developed in thisformalism by several authors [4, 5, 7–9, 23, 24] allow us to construct observables, whichare gauge invariant extensions, for generic phase space functions. The general startingpoint is a Hamiltonian system with constraints for which general relativity formulatedin ADM variables is a prominent example. A detailed discussion of the construction ofobservables for constrained systems and in particular its application to general relativitywas presented in the review [25]. Here we only summarize the main points for our analysisand refer to [25] as well as the original literature [4, 5, 7–9, 23, 24] for further details.The idea of the relational formalism is to construct observables for a constrained systemby means of formulating values that a chosen set of field variables can take in a relationalmanner. By this we mean that in the framework of general relativity it is not a gaugeinvariant statement to say that the metric takes a certain value at a spacetime coordinate x µ . However, what can be formulated in a gauge invariant way is the values that the metrictakes if other so called reference or clock fields take a certain value. A familiar exampleis finding the volume of a definite finite region in space. If we compute the volume andthen apply a coordinate transformation, the actual number the volume takes changes andthus this number is not invariant under diffeomorphisms. But if we are able to definethe volume of that region relative to another field, for instance we define the region asthe part of space where some matter density is non-vanishing, then we can formulate adiffeomorphism invariant expression for the volume of this region.Hence, the main idea behind this formalism is to provide a framework in which observ-ables for gauge variant phase space functions can be constructed by defining their valuesas well as their dynamics relative to other reference/clock fields that are themselves gaugevariant fields on phase space. For general relativity such reference fields correspond to thechoice of physical spatial and temporal coordinates.It was shown earlier that once a set of clock fields has been chosen, one can define anobservable map that maps a given phase space function onto its gauge invariant extension[7–9, 23, 24]. Of course the actual form of this map crucially depends on the chosenclock fields as well as the constraints the system under consideration possesses. We now2summarize how such an observable map can be constructed for the extended ADM-phasespace.Let us start by recalling that on the extended ADM-phase space we have for eachspacetime point eight first class constraints: four primary and four secondary ones, thatwe write in the following compact notation as:Π µ = (Π , Π a ) and C µ = ( C, C a ) µ = 0 , · · · , . (2.67)As discussed above, the quantity G ′ b,~b in (2.54) generates diffeomophisms for all geometricas as well as matter degrees of freedom on extended phase space. Our aim is to constructquantities, called observables, denoted by O that are invariant under diffeomorphisms,that is general coordinate transformations. At the canonical level this condition can beformulated by requiring that such quantities need to at least weakly Poisson commute withthe generator G ′ b,~b , that is { O, G ′ b,~b } ≈
0. Here weakly means equality on the constraintshypersurface.Next we discuss the role of the clock fields in this formalism. The general strategy oneembarks on is that for each constraint in the system one introduces a clock that has tosatisfy certain conditions that we will discuss below. Now in the case of general relativityin the reduced ADM-phase space we have for each spacetime point four constraints C µ .Hence, we have 4 × ∞ many constraints and a possible choice for clocks would be fourscalar fields because this have exactly 4 × ∞ many degrees of freedom. In order that thechosen fields can be used as clock fields they have to satisfy the following condition:det ( { T µ ( x ) , C ν ( y ) } ) = 0 ∀ µ, ν = 0 , · · · , . (2.68)Here x and y denote local coordinates on the spatial manifold Σ. The above condition onclocks needs to hold in some local neighborhood of x and y . Whether or not the clockscan be used globally depends on whether the above condition is globally satisfied.If the clocks satisfy the condition (2.68), it ensures that the matrix A µν ( x, y ) defined as: A µν ( x, y ) := { T µ ( x ) , C ν ( y ) } , (2.69)is invertible and clocks locally parameterize the gauge orbits. This allows us to use theinverse of A denoted by B := A − in order to construct an equivalent set of first classconstraints given by: ˜ C µ ( x ) := Z d y B νµ ( y, x ) C ν ( y ) . (2.70)It turns out that the clock fields are weakly canonically conjugate to this equivalent setof constraints. That is, { T µ ( x ) ˜ C ν ( y ) } ≈ δ µν δ ( x, y ). Further, the mutually Poisson bracketbetween the constraints of the equivalent set ˜ C µ yields terms that are at least quadraticin the constraints. The above procedure of choosing constraints ˜ C µ is called weak abelian-ization because the associated Hamiltonian vector fields χ µ := {· , ˜ C µ } are weakly abelian.Given a set of clock fields T µ satisfying the assumptions above, we can define a set ofgauge fixing constraints: G µ = τ µ − T µ , (2.71)where τ µ is a generic function of spacetime coordinates and can be interpreted as thevalue that the clock T µ takes. Since τ µ is phase space independent also, − G µ is weaklycanonically conjugate to ˜ C µ if T µ is.3Now it has been proven in [7] that the following formal power series O f,T [ τ ] = f + ∞ X n =1 n ! Z d x ... Z d x n G µ ( x ) ...G µ n ( x n ) { ... { f, ˜ C µ ( x ) } , ... ˜ C µ n ( x n ) } , (2.72)is gauge invariant, where we have suppressed the µ -label for the clocks T µ and their values τ µ to keep our notation compact and simple. Gauge invariance means, that at least weaklythe observable commutes with all constraints C µ relevant in the reduced ADM-phase space,that means {O f,T [ τ ] , C µ } ≈
0. The map f → O f,T [ τ ] returns the value of a generic phasespace function f at those values where the clock fields T µ take the values τ µ . This canbe also seen by rewriting the formula for observables in a slightly different but equivalentway in terms of the gauge flow induced by the constraints denoted by α ˜ G b,~b : O f,T [ τ ] ≈ α ˜ G b,~b ( f ) (cid:12)(cid:12)(cid:12) b = G ,b a = G a . (2.73)An important point, also relevant for the above formal power series is the following. Firstthe gauge flow acting on f is computed for general phase space independent descriptors b,~b and only afterwards these descriptors are identified with the phase space dependentgauge fixing conditions. The observable formula in (2.72) is sufficient for all phase spacefunctions other than lapse and shift, and moreover if the clock fields also do not dependon lapse and shift either.Now let us briefly discuss how this formula can be generalized in case we want to havegauge invariant extensions of variables that involve lapse and shift degrees of freedom asshown in [21–24]. Still we assume that the clock fields do not depend on lapse and shiftvariables which is sufficient for the cases discussed in this article. In case we drop also thisassumption one can still define an observable map but the weak abelianization gets morecomplicated in general.In the reduced ADM-phase space where lapse and shift are Lagrange multipliers, thestability of a gauge fixing condition G µ , that is ˙ G µ ≈
0, involves lapse and shift. Con-sequently, the stability requirement of G µ just fixes the Lagrange multipliers N and N a .In the case of the extended ADM-phase space, where N, N a are elementary phase spacevariables, the situation is different. The lapse and shift are dynamical variables in theextended phase space, and as a result the stability condition for gauge fixing results insecondary gauge fixing constraints. To be precise, we get G (2) µ ( x ) := ˙ G µ ( x ) = [ ˙ τ ν − ˙ T µ ]( x )= [ ∂ t τ ν − ∂ t T µ ]( x ) − κ − Z Σ d y A µν ( x, y ) N ν ( y ) . (2.74)A consequence of the secondary gauge fixing constraints is that weak abelianization is re-quired for secondary as well as the primary constraints, modifying the observable formula.In particular, the descriptors in the diffeomorphism generator G ′ b,~b are replaced by G µ ,and the time derivatives of descriptors by G (2) µ . As a side remark note that given theeight primary and secondary constraints, one could naively think that we will need eightinstead of four clock fields in the extended phase space. However, since the generator G ′ b,~b is not an independent linear combination of these eight constraints and in particularinvolves four descriptors. Hence, a choice of only four clock fields is natural.4In the extended phase space, the procedure to obtain the weakly abelianized constraintsgoes as follows. We need to compute the matrix elements A IJ := −{ G I , C J } betweengauge fixing constraints G I := ( G µ , G (2) µ ), and primary and secondary constraints C I :=( C µ , Π µ ) for I = 1 , .., . . The negative sign in the definition of the matrix elements isbecause of the relative sign difference between the gauge fixing constraints and the clocks.Using our notation A µν = { T µ , C ν } , and the identity ˙ T µ = ∂ t T µ + R d x A µν ( · , x ) N ν ( x ),we find A IJ = − (cid:20) A µν { ˙ T µ , C ν } A µν (cid:21) . (2.75)Its inverse matrix denoted by B := A − can be in the following way [23]: B IJ = ( A − ) IJ = (cid:20) B µν S µν B µν (cid:21) , (2.76)where S µν ( x, y ) = − Z d z Z d v B µρ ( x, z ) B σν ( v, y ) { ˙ T ρ ( z ) , C σ ( v ) } . (2.77)Given B IJ we can construct the set of weakly abelianized constraints as:˜ C I ( x ) = Z d y B JI ( y, x ) C J ( y ) . (2.78)Let us define the equivalent abelian set of constraints by ˜ C I ( x ) =: ( ˜˜ C µ , ˜Π µ ). Then ˜Π µ and˜˜ C µ expressed in terms of the original constraints are given by:˜Π µ ( x ) = Z d y B νµ ( y, x )Π ν ( y ) , ˜˜ C µ ( x ) = Z d y B νµ ( y, x ) (cid:20) C ν ( y ) − Z d z Z d v B σρ ( v, z ) { ˙ T ρ ( z ) , C ν ( y ) } Π σ ( v ) (cid:21) . (2.79)Also in the case of the extended phase space we aim at writing down a formal power seriesformula for the observable map. This final formula crucially simplifies if we take a resultproven in [23] into account, namely that up to second order in the gauge generators, wecan replace the rather complicated generator G ′ b,~b in (2.54) by a simpler gauge generator.Explicitly, we have [23]: G ′ b,~b + O ( C ) = ˜ G ′ ξ,~ξ =: κ − (cid:16) ˜Π µ [ ˙ ξ µ ] + ˜˜ C µ [ ξ µ ] (cid:17) , (2.80)where ξ µ = R d x A µν ( · , x ) b ν ( x ). As a result we can construct observables using ˜ G ′ ξ,~ξ : O f,T [ τ ] ≈ α ˜ G ′ ξ,~ξ ( f ) (cid:12)(cid:12)(cid:12)(cid:12) ξ µ = G µ , ˙ ξ µ = G (2) µ . (2.81)As before we can rewrite this formula in a formal power series that for the extended phasespace has the following form: O f,T [ τ ] = f + ∞ X n =1 n ! Z d y ... Z d y n G I ( y ) ... G I n ( y n ) { ... { f, ˜ C I ( y ) . } , ... ˜ C I n ( y n ) } . (2.82)5Here we used the definitions of G I = ( G µ , G (2) µ ) and ˜ C I ( x ) = ( ˜˜ C µ , ˜Π µ ). This observableformula is the required generalization needed for the extended ADM-phase space tohave the same techniques available for constructing gauge invariant quantities as in thereduced ADM-phase space. Since we aim at applying these techniques in the contextof cosmological perturbation theory, in the next subsection we will briefly discuss howobservables can be computed perturbatively. For the reason that we aim at constructing observables associated with the scalar, vectorand tensor perturbations, the question arises how the observable map discussed in the lastsubsection needs to be modified if we consider instead of a generic phase space function f its corresponding projection ˆ P f . We discuss this in the case that our f will be chosenamong the linear perturbations of the geometry or matter sector. Note that the action ofthe projectors ˆ P only affects the function f in the observable formula. This can be seenfrom the observable formula in (2.82). The projectors ˆ P involve background quantitiesas well as derivative operators with respect to the variables the function f depends on.However, the constraints as well as the gauge fixing conditions are evaluated at a different,independent variable and thus the action of ˆ P on them becomes trivial. Furthermore, theiterated Poisson bracket involved in the observable formula is evaluated with respect tothe phase space of the linear perturbations. As discussed earlier for perturbations arounda flat FLRW background the relevant projectors depend only on differential operatorsand background quantities. In other words as far as the phase space of the perturbationsis considered those projectors are phase space independent. Given this, we have for allprojectors ˆ P considered in our further computations that the following relation holds: O ˆ P f,T [ τ ] = ˆ P O f,T [ τ ] . (2.83)In the following we want to use the observable map in the framework of linear cosmologicalperturbation theory and we aim at showing that the conventional gauge invariant quan-tities such as the Mukhanov-Sasaki variable or the Bardeen potentials can be obtainednaturally from the application of the observable map with an appropriate choice of clockvariables. In order to do so, we need to discuss how perturbations of these observablescan be formulated. At first we consider only phase space functions that are independentof lapse and shift. In that case we can use the observable formula shown in (2.72). Itsfirst order perturbation leads to: δ O f,T [ τ ] = δf + Z d y δG µ ( y ) O { f, ˜ C µ ( y ) } ,T [ τ ]+ ∞ X n =1 n ! Z d y ... Z d y n ¯ G µ ( y ) ... ¯ G µ n ( y n ) { ... { f, ˜ C µ ( y ) } , ..., ˜ C µ n ( y n ) } (1) . (2.84)As discussed above, the functions τ µ involved therein should be in the range of the clocks T µ , that is there exists a gauge such that T µ = τ µ leading to the gauge fixing conditions G µ = τ µ − T µ . For the background solution we assume that these gauge fixing conditions6are satisfied, that is ¯ G µ = 0 and thus ¯ T µ = ¯ τ µ . As a consequence, the formula for thelinear perturbations of the observables can be written as: δ O f,T [ τ ] = δf + Z d yδG µ ( y ) { f, ˜ C µ ( y ) } ≈ δf + Z d y Z d z δG µ ( y ) ¯ B νµ ( z, y ) { f, C ν ( z ) } . (2.85)Note that as expected in the gauge δG µ = 0 the gauge variant quantities δf coincide withtheir corresponding observables δ O f,T [ τ ].Generalizing to the case that f can also depend on lapse and shift, we need to considerperturbations of the observable formula in (2.82). Assuming again that ¯ G µ = 0 and¯ G (2) µ = 0, we get, δ O f,T [ τ ] = δf + Z d y (cid:20) δ ˙ G µ ( y ) { f, ˜Π µ ( y ) } + δT µ ( y ) { f, ˜˜ C µ ( y ) } (cid:21) ≈ δf + Z d y Z d z ¯ B νµ ( z, y ) h δ ˙ G µ ( y ) { f, Π ν ( z ) } − δG µ ( y ) (cid:16) { f, C ν ( z ) } + Z d w Z d v ¯ B ρσ ( w, v ) { ˙ T σ ( v ) , C ν ( z ) } { f, Π ρ ( w ) } (cid:19)(cid:21) . (2.86)In section III, we will extensively use these observable formulae to obtain gauge invariantvariables natural to various gauge choices. For these gauges, we will have to choose δτ µ = 0, that is use δG µ = − δT µ ≈
0. The stability of this gauge fixing constraint and ofsubsequent conditions will play a central role in our analysis.
Both of the formulae for the linearized observables derived above show that weneed to compute the Poisson bracket of the quantity f , whose observable we want toconstruct, with the constraints and then evaluated it on the background solution, whichin our case is the flat FLRW spacetime. As discussed in detail in [25] (in appendix A),these Poisson brackets can be related to certain Poisson brackets on the linearized phasespace. Explicitly, we have for two generic phase space functions f, g : { f ( x ) , g ( y ) } = { δf ( x ) , δg ( y ) } δ , (2.87)where on the right-hand side the Poisson bracket {· , ·} δ denotes the Poisson bracket ofthe linearized phase space.To compute various observables, we need several Poisson brackets of the linearizedcosmological perturbations with the linearized constraints. Since the tensor perturbationsare already gauge invariant we here focus on the Poisson brackets involving scalar andvector perturbations only.To calculate Poisson brackets among the decomposed perturbations, we use the factthat one can pull the phase space independent projectors ˆ P I out of the Poisson brackets.This leads to { ˆ P I ( δq )( x ) , ˆ P J ( δP )( y ) } = ˆ P abI ( x ) ˆ P Jcd ( y ) { δq ab ( x ) , δP cd ( y ) } = κ ˆ P ( ab ) I ( x ) ˆ P Jab ( y ) δ ( x, y )(2.88)7where I, J label different projectors introduced above. For the elementary phase spacevariables in the extended ADM phase space we get: { φ ( x ) , p φ ( y ) } = κδ ( x, y ) , { B ( x ) , p B ( y ) } = − κG ( x, y ) , { S a ( x ) , p S b ( y ) } = κδ ab δ ( x, y ) + 2 κG ( x, y ) , { ψ ( x ) , p ψ ( y ) } = κ A ˜ P δ ( x, y ) , { E ( x ) , p E ( y ) } = κ A ˜ P Z d zG ( x, z ) G ( z, y ) , { F a ( x ) , p bF ( y ) } = κ A ˜ P (cid:20) − δ ba G ( x, y ) + Z d zG ( x, z ) ∂ G ( z, y ) ∂z a ∂z b (cid:21) . (2.89)All of the remaining Poisson brackets in the scalar-vector sector vanish. Consideringthe form of the secondary constraints in terms of the projected variables in (2.43) we cancalculate the necessary Poisson brackets of the perturbations and the linearized constraints(see also [25] for more details). The results are given by:1 κ { ψ ( x ) , δC ( y ) } = ˜ H ¯ N δ ( x, y ) , κ { ψ ( x ) , δC a ( y ) } = 13 ∂∂x a δ ( x, y ) , κ { E ( x ) , δC ( y ) } = 0 , κ { E ( x ) , δC a ( y ) } = ∂G∂x a ( x, y ) , (2.90)and1 κ { p ψ ( x ) , δC ( y ) } = − ˜ H N + ¯ N A ˜ H ∆ − κ NA / ˜ H F − ϕ ! δ ( x, y ) , κ { p ψ ( x ) , δC a ( y ) } = 16 ∂∂x a δ ( x, y ) , κ { p E ( x ) , δC ( y ) } = − ¯ N A ˜ H δ ( x, y ) , κ { p E ( x ) , δC a ( y ) } = − ∂G∂x a ( x, y ) . (2.91)For the vector perturbations we find the non-vanishing Poisson brackets as:1 κ { F a ( x ) , δC ( y ) } = 0 , κ { F a ( x ) , δC b ( y ) } = (cid:18) δ ab δ ( x, y ) − ∂ G ( x, y ) ∂y a ∂y b (cid:19) , κ { p aF ( x ) , δC ( y ) } = 0 , κ { p aF ( x ) , δC b ( y ) } = − (cid:18) δ ab δ ( x, y ) − ∂ G ( x, y ) ∂y a ∂y b (cid:19) . (2.92)We will use these results frequently in the next section where the explicit form of theobservables is derived and discussed. III. GAUGE CHOICES AND CORRESPONDING CLOCKS
The goal of this section is to construct geometrical clocks and the associated observablesfor metric and momentum perturbations at the linear order for various gauges used incosmological perturbation theory. The observables turn out to be gauge invariant variablesspecific to the choice of a particular gauge condition. The main idea in this construction isthe following. The hypersurface defined by the linearized gauge fixing constraint δG µ = 0involves the perturbations δT µ , that is δG µ = δτ µ − δT µ . Hence, a choice of perturbed8clocks δT µ can be directly related to a family of a gauge fixings parametrized by δτ µ . Inparticular, the first order observable δ O f,T [ τ ] equals δf on δG µ = 0. Each gauge conditionused in cosmological perturbation theory is tied to a specific choice of clocks which in ourframework naturally leads to a set of observables or gauge invariant variables. Usingour framework, we can systematically find clock fields that yield the Bardeen potentials,Mukhanov-Sasaki variable etc.In the context of the relational formalism the choice of a clock corresponds to the choiceof a reference field that on one hand is used to construct gauge invariant quantities and onthe other hand defines a notion of physical time or spatial coordinates respectively. By thiswe mean that the evolution of such gauge invariant quantities is not defined with respect tocoordinate time, but with respect to the values that the temporal reference field takes, thatis τ in our notation. As discussed in [14, 24], for a certain type of coordinate gauge fixingconditions, one can show that the gauge invariant evolution of the observables and thegauge-fixed evolution can be mapped into each other under an appropriate identificationand both formulations yield equivalent results. Exactly, by means of this property weare able to rederive the gauge invariant observables as well as their equations of motioncommon in linearized cosmological perturbation theory in the context of the relationalformalism.Now, in case we apply perturbation theory, we have gauge fixing conditions for thebackground solution ¯ G µ = ¯ τ µ − ¯ T µ and of course these conditions need to be consistentwith the equations of motion of the background clocks ¯ T µ . The same is true at the linearorder where a choice of δG µ = δτ µ − δT µ can be only considered if such a choice does notcontradict the equations of motion of δT µ .In order to reproduce the gauges common in cosmological perturbation theory, we willoften choose components of the perturbed metric as clocks whose corresponding compo-nents vanish on the background such as for instance linear perturbations of the shift vector.Thus, in order to ensure consistency with the background solutions for some gauges wehave to choose ¯ τ µ = 0. However, in general this would still allow us to work with a genericvalue of δτ µ and would yield a family of gauge fixings represented by δG µ = δτ µ − δT µ .Depending on the gauge under consideration, in certain cases we will also need to choose δτ µ = 0 to reproduce the gauge invariant observables conventionally constructed for cos-mological perturbation theory. An example of this is the case of the longitudinal gauge,for which in the Lagrangian framework the longitudinal part of the spatial metric pertur-bation E and the perturbation in shift vector B are set to zero. As explained before, theobservable formula, when applied to the clocks T µ , maps them onto the values τ µ . Car-rying this over to the perturbative case, we can access gauges like the longitudinal gaugewith clocks chosen from the linear perturbations of the metric and its momenta simply bysetting δτ µ = 0. In such a case we have δG µ = − δT µ and the construction formula forthe observables further simplifies to: δ O f,T [ τ ] (cid:12)(cid:12)(cid:12) τ =0 = δf − Z d yδT µ ( y ) { f, ˜ C µ ( y ) }≈ δf − Z d y Z d z δT µ ( y ) ¯ B νµ ( z, y ) { f, C ν ( z ) } . (3.1)Note that as expected in the gauge δG µ = 0, that is δT µ = 0 if δτ µ = 0 has been chosen,the gauge variant quantities δf coincide with their corresponding observables δ O f,T [ τ ].As has been shown in [14, 15], if one can identify non-linear clocks, that is clocks forfull general relativity, then one can first construct manifestly gauge invariant quantities9and afterwards consider the linearized perturbations around the FLRW solution. In thiscase, the non-linear temporal clock determines the temporal clock at any order. As aresult, one has a well-defined notion of physical time for the background and for arbitraryorders in perturbation theory. From this perspective, we would interpret a clock for whichwe have ¯ τ µ = 0, to be not natural because with such a choice we are not able to definea notion of physical time for the background. As a consequence, it might be impossibleto find non-linear clocks that reproduce such clocks in the background as well as at thelinear order. Moreover, it may seem that if we need to choose ¯ τ µ = 0 as well as δτ µ = 0in order to reproduce a common gauge in cosmology, the interpretation of the evolution ofthe gauge invariant observables in the context of the relational formalism is problematic.However, it turns out that even in these cases the observable map opens the possibility toobtain these gauge invariant quantities very systematically and a consistent and judiciousnotion of evolution can be formulated. As far as the derivation of their evolution equa-tions is considered the relational formalism is very useful because it technically simplifiesto explicitly derive these evolution equations. We will discuss these important points ina separate investigation in more detail. For this article, we focus on different gauges anddiscuss the precise relationship between different gauge fixing conditions in linear cosmo-logical perturbation theory, clocks and observables. For some gauges, a generalization ispossible allowing us to identify the clock in the relational formalism with the physicaltime. These will be discussed in section III F.We will consider the following five common gauges, which are discussed in cosmologicalperturbation theory (see e.g. [28, 31, 33]). Though we will focus our discussion on scalarperturbations, the clocks and the observables for the transversal vector components arealso found in the process. These gauges are:1. Longitudinal gauge: E = B = 0,2. Spatially flat gauge: ψ = E = 0,3. Uniform field gauge: δϕ = E = 0,4. Synchronous gauge: φ = B = 0,5. Comoving gauge: δϕ = B = 0 .Above gauges can be divided into two broad categories. The first three of these sharea common feature that the longitudinal part of the spatial metric perturbation vanishes.This is the isotropic threading of spacetime. There is no residual freedom in the spatialcoordinates. In contrast, the latter two gauges correspond to a time slicing such that theshift perturbations vanish. There is a residual freedom associated with the shift in spatialcoordinates in these cases. It turns out that the construction of the geometrical clocksand observables is much more straightforward and less involved in the first three gaugesthan the last two.In the following we begin with the longitudinal gauge where we demonstrate the entiremethod to appropriate clocks and construct the respective observables in detail. Thecalculations for the other gauges in subsequent subsections follow the same strategy, albeitfor some subtleties and technical issues related in particular to the vanishing of the shiftperturbation. For this reason, discussion of other gauges, except for latter issues, is shorterin presentation than the longitudinal gauge.0 A. Longitudinal Gauge
The longitudinal or the Newtonian gauge amounts to an isotropic threading and ashear free slicing. Using its definition in the Lagrangian picture, we impose the followingin the constraint framework. Note that we will use the weak equality symbol ≈ to denoteequalities on the gauge fixing constraint hypersurface. This gauge is identified as: B ≈ , E ≈ . (3.2)In the Lagrangian formulation the corresponding velocities have also to be set equal tozero for stability reasons. In the Hamiltonian picture, the stability analysis of the gaugefixing condition will yield a further condition among the perturbed quantities involvingscalar perturbations of the momenta. The stability conditions are,˙ B ≈ , ˙ E ≈ . (3.3)Using the Hamilton’s equations (2.35) and (2.42), we can write˙ B = δ ˆ λ ≈ E = − H ( E + p E ) + B ≈ − H p E ≈ . (3.5)(3.4) does not require further stability conditions because the perturbation δ ˆ λ gets fixed.The same is not true for (3.5). We further need to impose ˙ p E ≈
0. The stability of p E ≈ φ ≈ − ψ . (3.6)Using (2.42), the above equation results in fixing the perturbation of the Lagrange multi-plier as δλ ≈ ˙¯ N φ + 2 ¯ N ˜ H ( ψ − p ψ ) . (3.7)No further conditions are required for the stability of the longitudinal gauge.In order to consider this gauge in the framework of geometrical clocks, we need tofind the gauge descriptors consistent with the longitudinal gauge. At the linear order inperturbations, we need the first order transformation properties of perturbations underthe action of the gauge operator G ′ b,~b . Using these properties of metric perturbations andtheir momenta, given by (2.64) and (2.65), the following relations must hold: b ! = 4 ˜ H A ¯ N ( E + p E ) , ˆ b ! = − E . (3.8)Using these relations we can obtain the longitudinal gauge by an appropriate choice ofgauge descriptors, even if one started with an arbitrary gauge. Note that the aboverelations do not imply that the gauge descriptors are functions of the phase space variables.The above identification is to be made after the gauge transformation with phase spaceindependent descriptors has been calculated.1
1. Geometrical clocks
The linearized gauge fixing constraints δG µ = δτ µ − δT µ ≈ δT µ . For the background we have chosen ¯ T µ ! = τ µ and the latter choice needs ofcourse to be consistent with the background solution. The range of values allowed for δτ µ of course depends on the specific choice of clocks ¯ T µ and their linear perturbationsrespectively. For the longitudinal gauge the components of E and B are set to zero inthe Lagrangian framework. As explained before the observable formula when applied tothe clocks T µ maps them onto the values τ µ . Carrying this over to the perturbativecase, we can access gauges like the longitudinal gauge with clocks chosen from the linearperturbations of the metric and its momenta simply by setting τ µ = 0 and δτ µ = 0. Thefirst choice is necessary because the background metric components associated with E and B just vanish. For the choice regarding the perturbations we get δG µ = − δT µ and hencethe linearized gauge fixing conditions δG µ constitute the hypersurface defined by δT µ ≈ δT µ such that δT µ ≈ δT and δ ˆ T , wherethe latter is the scalar projection of δT a , such that the following conditions: δT ≈ , δ ˆ T ≈ , δ ˙ T ≈ , δ ˙ˆ T ≈ , (3.9)are equivalent to the longitudinal gauge on the phase space, identified by B ≈ , E ≈ , p E ≈ , φ ≈ − ψ . (3.10)To find the appropriate clocks consistent with longitudinal gauge, we recall that for a phasespace function f assumed to be independent of lapse and shift, the first order observableformula reads [25]: δ O f,T [ τ ] = δf + { f, ˜ C µ [ b µ ] } (cid:12)(cid:12)(cid:12) b µ = − δT µ , (3.11)where ˜ C µ denote weakly abelianized constraints:˜ C µ ( x ) = Z d y B νµ ( y, x ) C ν ( y ) . (3.12)Here, as noted earlier, the matrix B is the inverse of matrix A defined as A µν ( x, y ) := { T µ ( x ) , C ν ( y ) } . (3.13)And, the second term in the observable formula (3.11) corresponds to the change of f under an infinitesimal diffeomorphism with descriptors b µ = − δT µ evaluated after thePoisson bracket has been calculated.Using the identification in (3.8) we are led to the following perturbed clocks for thelongitudinal gauge: δT = − H A ¯ N ( E + p E ) = 2 √ A ˜ P ( E + p E ) , δ ˆ T ! = E , (3.14)where ˜ P = P A / P A as the conjugate momentum of A .2Note that the gauge descriptors ˆ b and b do not determine δT a ⊥ . However, the latter canbe fixed knowing δ ˆ T . As δ ˆ T is the longitudinal scalar part of δq ab we choose δT a ⊥ to bethe longitudinal transversal part thereof such that: δT a = ¯ q ab ( ˆ P L δq ) b = δ ab ( E ,b + F b ) . (3.15)The consistency of above clocks can be verified using δT µ ≈ δ ˙ T µ ≈
0. It is easilyseen that δ ˆ T ≈ ⇒ E ≈ , δT ≈ ⇒ p E ≈ ,δ ˙ˆ T ≈ ⇒ B ≈ δ ˙ T ≈ ⇒ ψ = − φ . (3.16)Stabilization of above conditions fix the Lagrange multipliers δλ and δ ˆ λ consistent withthe longitudinal gauge and result in (3.4) and (3.7).On the other hand, the Lagrange multiplier δλ a ⊥ gets fixed as follows. The condition δ ˙ T a ⊥ = δ ab ˙ F b ≈ p aF ≈
14 ˜ H S a . Demanding its further stabilization yields δλ a ⊥ +3 ˜ H S a ≈
0. We note that this condition arises just by using the equations of motion for p aF and ˙ S a = δλ a ⊥ , and does not involve conditions from other clocks or gauge descriptorscorresponding to the scalar perturbations of the longitudinal gauge. This is not surprisingonce we recall that the clock δ ˙ T a ⊥ is introduced without using the gauge descriptors ˆ b and b which identify the longitudinal gauge. It will be useful to note that above constraint on δλ a ⊥ and S a arises when ever the clock δ ˙ T a ⊥ is the longitudinal scalar part of the perturbationin the spatial metric. Thus, this will be true for the spatially flat and uniform field gaugesas well.An important contribution to the observable map are the abelianized constraints thatinvolve the inverse of the matrix A µν . As explained in section II, the requirement that theinverse of the matrix exists is a condition on the clock fields that we can choose. At the levelof linear perturbation theory we choose clock fields defined on the linearized phase spaceand hence the condition on the existence of the inverse of the matrix A µν has to be fulfilledon that phase space. Since a Poisson bracket between two linear functions on the linearizedphase space yields a resulting expression that depends on the background quantities only,we will denote the matrix relevant for our further analysis with ¯ A µν (see also section IIfor more details ). Using the Poisson brackets of perturbations of metric componentsand their momenta, with perturbations of the Hamiltonian and spatial diffeomorphismconstraints, it is straightforward to verify that¯ A µν ( x, y ) = { δT µ ( x ) , δC ν ( y ) } = κδ µν δ ( x, y ) . (3.17)As a result, at the level of the linearized phase space the matrix A µν can be triviallyinverted. However, there is a subtlety with the clocks in the longitudinal gauge. It turnsout that the above choice of clocks is non-commuting. One can check that though { δT ( x ) , δT ( y ) } = 0 (3.18) Note, that for some calculations, as for instance derivation of the equations of motion and the Diracbracket of the observables, one also needs the perturbation δ A µν of this matrix. However, this perturba-tion will not be relevant for the work discussed in this paper. { δT ( x ) , δT a ( y ) } = − κ √ A Z d z G ( x, z ) ∂G ( z, y ) ∂y a , (3.19)where G ( x, y ) is is the integral kernel of the Green’s function of the Poisson equation:∆ − f ( x ) = Z ¯Σ d y G ( x, y ) f ( y ) . (3.20)The non-commuting clocks suggest that the linearized observable algebra will be non-standard and may have a complicated form, an aspect relevant once the quantizationof such an observable algebra is wished to be achieved. Nevertheless, we will see belowthat the Bardeen potentials follow very naturally from the longitudinal clocks as gaugeinvariant extensions of ψ and φ .
2. Observables
Now we use the clocks corresponding to longitudinal gauge to construct first orderobservables of the scalar, vector and tensor perturbations. We use the notation O (1) δf,T [ τ ] = δ O f,T [ τ ] (3.21)for the first order gauge invariant extension of a perturbation δf . As discussed above forthe longitudinal gauge we need to choose τ µ = 0 and δτ µ = 0. Therefore, we will eitherwrite this out explicitly or will suppress the τ µ dependency of the observables in whatfollows for simplicity where possible. Let us first calculate the observables for perturbations δf other than lapse and shift. In this case we can use the following perturbed observableformula [25]: δ O f,T [ τ ] (cid:12)(cid:12)(cid:12) τ =0 = δf − Z d yδT µ ( y ) { f, ˜ C µ ( y ) }≈ δf − Z d y Z d z δT µ ( y ) ¯ B νµ ( z, y ) { f, C ν ( z ) } (3.22)with ¯ B µν ( x, y ) = κ − δ µν δ ( x, y ) (3.23)to derive the following formula for observables constructed with the longitudinal clocks: O (1) δf,T ≈ δf + 4 ˜ H Aκ ¯ N Z d y ( E +Σ)( y ) { f, C ( y ) }− κ Z d y δ ab ( E ,b + F b )( y ) { f, C a ( y ) } . (3.24)Note, that in the above observable formula we neglected terms proportional to the sec-ondary constraints. In the following expressions, we will write an equal sign = insteadof the weak equality ≈ to distinguish between the gauge fixing constraint hypersurfaceand the hypersurface where the linearized secondary constraints are fulfilled. Using theexpressions for the Poisson brackets of the perturbations with the linearized secondary4constraints (2.90) and (2.91) that have been derived in [25], it is straightforward to calcu-late the first order observables of the scalar-vector-tensor perturbations. The observablecorresponding to ψ turns out to be one of the Bardeen’s potential: O (1) ψ ( x ) ,T = ψ ( x ) + 4 ˜ H A ¯ N Z d y ( E + p E )( y ) ˜ H ¯ N δ ( x, y ) − Z d y δ ab ( E ,b + F b )( y ) 13 ∂∂x a δ ( x, y )= ψ ( x ) + 4 ˜ H A ¯ N ( E + p E )( x ) −
13 ∆ E ( x )=: Ψ( x ) . (3.25)Similarly, we can compute: O (1) p ψ ( x ) ,T = p ψ ( x ) + 4 ˜ H A ¯ N Z d y ( E + p E )( y ) −
14 ˜ H ¯ N + 16 ¯ N ˜ H A ∆ − κ N ˜ H p ! δ ( x, y ) − Z d y δ ab ( E ,b + F b )( y ) 16 ∂∂x a δ ( x, y )= " p ψ −
16 ∆ E + 23 ∆( E + p E ) − ˜ H A ¯ N + κA p ! ( E + p E ) ( x )=: Υ( x ) (3.26)which is the gauge invariant extension of the momentum p ψ usually not considered inthe Lagrangian framework but which also appears in the equation of motion for Ψ in thecanonical framework of gauge invariant cosmological perturbation theory (see [25]). Ifwe apply the linearized observable map to the clock degrees of freedom that are for thelongitudinal gauge encoded in E and p E these quantities are mapped to zero for the reasonthat we have chosen δτ µ = 0. That this is indeed the case can be seen below: O (1) E ( x ) ,T = E ( x ) − Z d y δ ab ( E ,b + F b )( y ) ∂G∂x a ( x, y )= E ( x ) − Z d yE ( y )∆ G ( x, y )= 0 , (3.27)and O (1) p E ( x ) ,T = p E ( x ) + 4 ˜ H A ¯ N Z d y ( E + p E )( y ) (cid:18) − ¯ N H A (cid:19) δ ( x, y )+ Z d y δ ab ( E ,b + F b )( y ) ∂G∂x a ( x, y )= 0 . (3.28)Here we have used ∂ a F a = 0 as well as the symmetry of the Green’s function and its partialderivatives. For the transversal vector perturbations one can derive using the backgroundPoisson brackets in (2.92), the following: O (1) F a ( x ) ,T = F a ( x ) − Z d y ( E ,b + F b )( y ) (cid:18) δ ba δ ( x, y ) − ∂ G ( x, y ) ∂y a ∂y b (cid:19) = 0 (3.29)5and O (1) p aF ( x ) ,T = p aF ( x ) + Z d y ( E ,b + F b )( y ) (cid:18) δ ab δ ( x, y ) − ∂ G ( x, y ) ∂y a ∂y b (cid:19) = p aF ( x ) + δ ab F b ( x ) =: ν a ( x ) . (3.30)Note that the fact that the gauge invariant extension of F a just vanishes is again consistentwith δT a = δ ab ( E ,b + F b ) being clock variables. For the scalar field perturbation and itsmomentum, we obtain, O (1) δϕ ( x ) ,T = δϕ ( x ) + 4 ˜ H Aκ ¯ N Z d y ( E + p E )( y ) { ϕ ( x ) , C ( y ) }− κ Z d yδ ab ( E ,b + F b )( y ) { ϕ ( x ) , C a ( y ) } = δϕ ( x ) + 4 ˜ H λ ϕ κ √ A ¯ π ϕ ( E + p E ) =: δϕ ( gi ) ( x ) . (3.31)Similarly for δπ ϕ we get: O (1) δπ ϕ ( x ) ,T = δπ ϕ ( x ) − H A / ¯ N λ ϕ d V d ϕ ( ¯ ϕ )( E + p E )( x ) − ¯ π ϕ ∆ E ( x ) =: δπ ( gi ) ϕ ( x ) . (3.32)Finally, let us compute the observables corresponding to lapse and shift perturbations.For these we have to use the more involved perturbed observable formula [25]: O (1) δN µ ( x ) ,T = δN µ ( x ) − Z d yδ ˙ T ν ( y ) { N µ ( x ) , ˜Π ν ( y ) } − Z d yδT ν ( y ) { N µ ( x ) , ˜˜ C ν ( y ) } = δN µ ( x ) − Z d yδ ˙ T ν ( y ) κ ¯ B µν ( x, y ) − Z d yδT ν ( y ) { N µ ( x ) , ˜˜ C ν ( y ) } , (3.33)where we once again neglected terms proportional to the primary and secondary con-straints. Here ˜Π µ ( x ) = Z d y B νµ ( y, x )Π ν ( y ) (3.34)and˜˜ C µ ( x ) = Z d y B νµ ( y, x ) (cid:20) C ν ( y ) − Z d z Z d v B σρ ( v, z ) { ˙ T ρ ( z ) , C ν ( y ) } Π σ ( v ) (cid:21) . (3.35)Using the latter we can compute { N µ ( x ) , ˜˜ C ν ( y ) } = − κ Z d z Z d v ¯ B µσ ( x, v ) ¯ B ρν ( z, y ) { ˙ T σ ( v ) , C ρ ( z ) } = − κ Z d z Z d v ¯ B µσ ( x, v ) ¯ B ρν ( z, y ) { δ ˙ T σ ( v ) , δC ρ ( z ) } . (3.36)The time derivatives in the above formulas are evaluated on shell, i.e. they have to bereplaced by the respective Hamilton’s equations of motion. These are given by, δ ˙ T = dd t [2 √ A ˜ P ( E + p E )] = ¯ N (cid:18) φ + ψ −
13 ∆ E (cid:19) + 4 ˜ H A ¯ N ( E + p E ) ,δ ˙ T a = ˙ E ,a + δ ab ˙ F b = − H [( E + p E ) ,a + ν a ] + δN a . (3.37)6Using these results, it is straightforward to obtain { N ( x ) , ˜˜ C ( y ) } = 0 , { N ( x ) , ˜˜ C a ( y ) } = 0 , { N a ( x ) , ˜˜ C ( y ) } = − ¯ NA ∂∂x a δ ( x, y ) , { N a ( x ) , ˜˜ C b ( y ) } = 0 . (3.38)This yields the following expressions for the observables corresponding to the lapse andshift perturbations: O (1) δN,T = − ¯ N ψ −
13 ∆ E + 4 ˜ H A ¯ N ( E + p E ) ! = − ¯ N Ψ , (3.39)and O (1) δN a ,T = 4 ˜ H ν a . (3.40)Using the definitions δN = ¯ N φ and δN a = B ,a + S a we find, O (1) φ,T =: Φ =: O − ψ,T = − Ψ , O (1) B,T = 0 , O (1) S a ,T = 4 ˜ H ν a . (3.41)For their conjugate momenta we obtain, O (1) p φ ,T = 1¯ N δ Π , O (1) p B ,T = ˆ δ Π , O (1) p Sa ,T = δ Π a ⊥ , (3.42)where we used that the projected primary constraints Poisson commute with the generator G ′ . The result for the observables associated to S a given by O (1) S a ,T is also consistent withour former result for the observable O (1) p aF ,T if we take the relation p aF =
14 ˜ H S a into account.As expected, we find that the first order observable associated to φ is the negative ofthe first order observable of ψ . This reflects the well known result that the two Bardeenpotentials Ψ and Φ coincide up to a minus sign for a scalar field as matter content, seefor instance [25, 31].In summary, we find that the geometrical clocks corresponding to the longitudinalgauge lead to the gauge invariant quantities Ψ, Υ, δϕ ( gi ) and δπ ( gi ) ϕ for the scalar degreesof freedom and in the case of the vector degrees of freedom we get an observable propor-tional to ν a and a corresponding momentum observable. In addition, taking into accounttensor perturbations, we can similarly write an observable corresponding to h T Tab , anda corresponding momentum observable. For the configuration variables, we get in totalseven gauge invariant variables: 3 scalars (Φ, Ψ, O (1) δϕ,T ), 2 vector components ( O (1) S a ,T ), and2 tensor components ( O (1) h TTab ,T ). However, only three of these should be independent whichamount to three physical degrees of freedom in the gravity plus scalar field configurationspace.This can be understood by noting that in our analysis in the extended phase space,imposing gauge fixing constraint in terms of clocks, δG µ = δT µ ≈ δλ µ . It should be noted that this isa general argument which applies to all the gauges, if no residual freedom is left after gaugefixing. Let us contrast the situation in the conventional picture in the reduced ADM-phasespace. There the gauge fixing constraints δG µ ≈ O (1) h TTab ,T . Thus, we find that using stability of geometrical clocks we recover the correctcounting of degrees of freedom via the observable formulae. In the next subsection, wewill see that in the case of the spatially flat gauge, the independent degrees of freedom arecaptured by the Mukhanov-Sasaki variable and O (1) h TTab ,T .The example of longitudinal gauge shows that in our framework the resulting gaugeinvariant quantities are obtained in a straightforward and systematic way. The conven-tional formalism for perturbations which results in identification of Bardeen potentialsframework can therefore be naturally embedded into the observable formalism in the caseof the longitudinal clocks. Let us summarize all the observables: O (1) φ,T = − Ψ , O (1) B,T = 0 , O (1) S a ,T = 4 ˜ H ν a , O (1) ψ,T = Ψ , O (1) E,T = 0 , O (1) F a ,T = 0 , O (1) p ψ ,T = Υ , O (1) p E ,T = 0 , O (1) p aF ,T = ν a , O (1) δϕ,T = δϕ ( gi ) , O (1) δπ ϕ ,T = δπ ( gi ) ϕ . (3.43)For the momenta involving the decomposed primary constraints we get: O (1) p φ ,T = 1¯ N δ Π , O (1) p B ,T = ˆ δ Π , O (1) p Sa ,T = δ Π a ⊥ . (3.44)As expected all elementary variable that are involved in the clock fields, for the longitu-dinal gauge these are E, p E , B, F a are mapped onto zero by the observable map which iscompletely consistent with the requirement δT ≈ δT a ≈ B. Spatially Flat Gauge
We now present the case of the spatially flat gauge. The gauge corresponds to anisotropic threading with a flat slicing. In this gauge the perturbation in intrinsic curvaturedue to scalar perturbations vanish. It is defined as, ψ ≈ , E ≈ . (3.45)8It is easily seen that (2.36) then implies δR (3) ab = 0 in the absence of tensor perturbations.Using Hamilton’s equations for ψ and E , the stability of spatially flat gauge conditionsamounts to ˙ ψ ≈ ˜ H φ + 2 ˜ H p ψ + 13 ∆ B ! ≈ E ≈ − H Σ + B ! ≈ , (3.46)which imply φ ≈ − p ψ −
43 ∆ p E , and B ≈ H p E . (3.47)The stability check of the above two equations do not yield any new constraints, but fixthe perturbations of the Lagrange multipliers δλ and δ ˆ λ .Following the analogous strategy used for the longitudinal gauge to identify the gaugedescriptors, we find that the spatially flat gauge can be implemented by choosing thefollowing descriptors in the gauge generator G ′ b,~b : b ! = − ¯ N ˜ H (cid:18) ψ −
13 ∆ E (cid:19) and ˆ b ! = − E . (3.48)
1. Geometrical clocks
To choose the perturbed clocks δT µ corresponding to the spatially flat gauge, let usrecall that the gauge constraint hypersurface is given via δG µ = δτ µ − δT µ ≈
0. Forthe background metric, the metric components defining the spatially flat gauge are zero,which implies that ¯ τ µ = ¯ T µ = 0. Setting δτ µ = 0, we can identify the perturbed clockssuch that δT µ ≈ δT = ¯ N ˜ H (cid:18) ψ −
13 ∆ E (cid:19) , δT a ! = δ ab ( E ,b + F b ) . (3.49)Here δT a is composed of their derivative of δ ˆ T determined by ˆ b , and δT a ⊥ chosen as thelongitudinal transversal part of the perturbation in spatial metric. This choice againyields ¯ A µν ( x, y ) = κδ µν δ ( x, y ). Since the gauge fixing constraints δT ≈ δ ˆ T ≈ ψ ≈ E ≈
0, the stabilization of perturbed clocks, δT and δ ˆ T yield (3.47). The stability conditions of these clocks also fix the Lagrangemultipliers δλ and δ ˆ λ . The stabilization of δT a ⊥ = δ ab F b ≈ p aF ≈
14 ˜ H S a . And,the stability of the latter gives δλ a ⊥ + 3 ˜ H S a ≈
0, as in the case of the longitudinal gauge.Note that the clocks corresponding to the spatially flat gauge only involve metric per-turbations and do not depend on the momentum perturbations. As a result, this set ofclocks commute in contrast to the non-commuting clocks for the longitudinal gauge. Fur-ther, the algebra of observables corresponding to the non clock degrees of freedom willbe the respectively simpler standard algebra of configuration and momentum degrees offreedom.9
2. Observables
The first order observables of the perturbations can be constructed in the same wayas the case of longitudinal gauge using the Poisson brackets of metric and momentumperturbations with those of the linearized constraints and the time derivatives of theperturbations in clocks. In these calculations it is useful to note that (3.23) holds for thespatially flat gauge too. After some straight forward computations we find the followingform of the first order observables for the clocks corresponding to spatially flat gauge: O (1) φ,T = − − (cid:18)
12 + κ ˜ P Ap (cid:19) Ψ , O (1) B,T = ¯ N ˜ H A Ψ , O (1) S a ,T = 4 ˜ H ν a , O (1) ψ,T = 0 , O (1) E,T = 0 , O (1) F a ,T = 0 , O (1) p ψ ,T = Υ + α Ψ , O (1) p E ,T = 1˜ P Ψ , O (1) p aF ,T = ν a , O (1) δϕ,T = v, O (1) δπ ϕ ,T = π v . (3.50)Here α := (cid:18)
14 + κ P Ap (cid:19) −
23 1˜ P ∆ , (3.51) v is the Mukhanov-Sasaki variable v := δϕ − λ ϕ A / ˜ H ¯ N ¯ π ϕ (cid:18) ψ −
13 ∆ E (cid:19) (3.52)and π v is the corresponding gauge invariant momentum observable: π v := δπ ϕ − ¯ π ϕ ∆ E + 12 A / λ ϕ d V d ϕ ( ¯ ϕ ) ¯ N ˜ H (cid:18) ψ −
13 ∆ E (cid:19) . (3.53)For the momenta involving the degrees of freedom of the primary constraints we obtain: O (1) p φ ,T = 1¯ N δ Π , O (1) p B ,T = ˆ δ Π , O (1) p Sa ,T = δ Π a ⊥ . (3.54)It turns out that the gauge invariant observables which naturally emerges for the choiceof clocks consistent with spatially flat gauge are not the Bardeen potentials. As discussedin section I, the natural gauge for the Bardeen potentials is the longitudinal gauge and ifany other gauge is chosen, none of the Dirac observables is equal to the Bardeen poten-tials. Instead the Mukhanov-Sasaki variable v occurs as the first order gauge invariantextension of the scalar field perturbation δϕ . This is expected, because the Mukhanov-Sasaki variable v coincides with the scalar field perturbation in the spatially flat gauge.Finally, let us note that as in the case of longitudinal gauge, including the observables fortensor perturbations, there are 7 (configuration) gauge invariant variables for the metricand scalar field perturbations. Of these, only three are independent which include theMukhanov-Sasaki variable v and O (1) h TTab ,T . Thus, the degrees of freedom in metric andscalar field sector turn out to be three. As discussed in the case of the longitudinal gauge,this can be shown in the extended phase space using stability of geometrical clocks in ourframework.0 C. Uniform Field Gauge
In the presence of the scalar field, as in our analysis, we can choose the gauge suchthat perturbations in the scalar field vanish. The scalar field is thus homogeneous in thiscase. In the observable formalism the scalar field perturbation will serve as a clock, withrespect to which evolution of other metric and matter variables can be studied. The metricperturbation ψ captures the curvature perturbation, which itself turns out to be gaugeinvariant [34]. In order to fix the gauge, another condition is required, such as the isotropicthreading which requires vanishing of the longitudinal scalar part of the perturbation inthe spatial metric. The uniform field gauge requires: δϕ ≈ E ≈ . (3.55)The stability of the gauge conditions yields: φ ≈ ψ − δπ ϕ ¯ π ϕ B ≈ H p E . (3.56)As before, the stability requirements fix the Lagrange multipliers δλ and δ ˆ λ .Using the transformation properties of δϕ (2.62) and metric perturbations (2.64) underthe action of the gauge generator G ′ b,~b , we obtain the following gauge descriptors for theuniform field gauge: b ! = − A / δϕλ ϕ ¯ π ϕ and ˆ b ! = − E . (3.57)
1. Geometrical Clocks
As before, we choose perturbed clocks such that we can reproduce the common results inlinearized cosmological perturbation theory. For the background gauge fixing condition weare allowed to choose a non-trivial and time-dependent ¯ τ because the temporal descriptorinvolve the scalar field linearly. For the spatial part we still need to choose ¯ τ a = 0since the associated linearized descriptor is completed determined by E whose associatedbackground quantity vanishes. Now at the linearized level it turns out that a choice of δτ µ ≈ δG µ = − δT µ ≈ δτ µ = 0. With the choice of setting δτ µ = 0, we can identify using the gauge descriptorsthe following perturbed clocks for the uniform field gauge: δT = A / λ ϕ ¯ π ϕ δϕ, δT a ! = δ ab ( E ,b + F b ) . (3.58)The clocks satisfy ¯ A µν ( x, y ) = κδ µν δ ( x, y ) which simplifies the calculations of the observ-ables. It is straightforward to check that as in the case of the spatially flat gauge, theseclocks do commute. It is easy to verify that the stability of the above clocks yield condi-tions which are consistent with the uniform field gauge and the related stability conditions.Note, that δT a is the same for all choices of clocks that were yet considered for the isotropicthreading of spacetime. Hence, stability of δT a ⊥ ≈ p af ≈
14 ˜ H S a as in the case of thelongitudinal and spatially flat gauges.1
2. Observables
The observables can be constructed using the linearized observable formulas in (3.1)and (3.33). Inserting the uniform field gauge clocks into the respective formulas, thefollowing first order observables can be derived: O (1) φ,T = − (3 κ + ς ) v − π v ¯ π ϕ , O (1) B,T = ¯ N ˜ H A (Ψ + κ v ) , O (1) S a ,T = 4 ˜ H ν a , O (1) ψ,T = − κ v, O (1) E,T = 0 , O (1) F a ,T = 0 , O (1) p ψ ,T = Υ + α (Ψ + κ v ) , O (1) p E ,T = 1˜ P (Ψ + κ v ) , O (1) p aF ,T = ν a , O (1) δϕ,T = 0 , O (1) δπ ϕ ,T = π v + ¯ π ϕ ςv . (3.59)Here we have defined, κ := A / λ ϕ ¯ π ϕ ˜ H ¯ N (3.60)and ς := 12 A λ ϕ ¯ π ϕ d V d ϕ ( ¯ ϕ ) . (3.61)Apart from the above observables, we also obtain the following related to primary con-straints, O (1) p φ ,T = 1¯ N δ Π , O (1) p B ,T = ˆ δ Π , O (1) p Sa ,T = δ Π a ⊥ . (3.62)Note, that the combination Ψ + κ v appearing in the linearized observables can berelated to δϕ ( gi ) : δϕ ( gi ) = 1 κ (Ψ + κ v ) , δπ ( gi ) ϕ = π v − ς κ ¯ π ϕ Ψ . (3.63)Further, the first order observable for ψ is proportional to the Mukhanov-Sasaki variable v . This is, however, not surprising because instead of interpreting v as a gauge invariantextension of the scalar field involving the geometric perturbations ψ and E we can analo-gously interpret it as a gauge invariant extension of ψ involving δϕ and E . Similar to thelongitudinal and spatially flat gauge, we will have three in independent physical degreesof freedom, one in the scalar and two in the tensor sector. D. Synchronous Gauge
In the cosmological perturbation theory, synchronous gauge has been studied exten-sively, see for e.g. [35]. The underlying idea is to use the gauge freedom of the theory toset the temporal-temporal and temporal-spatial components of the metric perturbation δg tt , δg ti equal to zero. In terms of the ADM variables this gauge is equivalent to choosingvanishing lapse and shift perturbations, that is δN µ = 0. For the scalar perturbations thisgauge requires: φ ≈ , B ≈ . (3.64)2The stability of these gauge conditions directly translate to the conditions on the per-turbations of the Lagrange multipliers, via (2.42). No other conditions on any otherperturbations arise. Using (2.64), we find the gauge descriptors for the synchronous gaugeas: b ( x, t ) ! = − t Z d t ′ ¯ N ( t ′ ) φ ( x, t ′ ) − c ( x ) , ˆ b ( x, t ) ! = − t Z d t ′ ¯ NA ( t ′ ) t ′ Z d t ′′ ¯ N ( t ′′ ) φ ( x, t ′′ ) + c ( x ) − t Z d t ′ B ( x, t ′ ) − c ( x ) . (3.65)In comparison to the longitudinal and spatially flat gauges, the gauge descriptors for thesynchronous gauge have more non-trivial expressions. First, they involve time integralswhich are quite non-trivial to deal with in passage to the Hamiltonian formulation. Second,there are two arbitrary functions c ( x ) and c ( x ) which are constant in time and hencecan be any constants of motion. These arbitrary ‘constants’ are the non-physical gaugemodes in the synchronous gauge (see for e.g. [31]). Finally, the gauge descriptors involvelapse and shift perturbations. Thus, the resulting clocks will depend on the latter.
1. Geometrical clocks
Let us consider the clocks for the synchronous gauge using the identification of thegauge descriptors. For the background metric, B vanishes and in agreement with this wechoose ¯ τ a = 0. For the temporal gauge fixing condition we obtain ¯ T = ¯ N because wehave N = ¯ N + δN = ¯ N + ¯ N φ . Thus we realize that we can choose a non-vanishing timedependent τ for the synchronous gauge and therefore also in this gauge we can potentiallydefine a notion of physical time for the background solution. Similar to the uniform fieldgauge also here in principle a generalized perturbed gauge fixing condition with δτ µ = 0can be formulated as also discussed in section III F. However, for being able to reproducethe results of the Lagrangian framework, we chose again δτ µ = 0 and write the linearizedgauge fixing constraint as δG µ = − δT µ ≈
0. As a result, the perturbed clocks turn out tobe: δT ! = ˆ I ¯ N φ + c ,δ ˆ T ! = ˆ I ¯ NA h ˆ I ¯ N φ + c i + ˆ IB + c . (3.66)Since the descriptors involve a time integral the same will be true for the correspondingclocks. For this purpose we denoted this time integral as ˆ I which is defined such that itsatisfies dd t (cid:16) ˆ If (cid:17) ( t ) := dd t t Z d t ′ f ( t ′ ) = f ( t ) (3.67)where f is an arbitrary phase space function and time derivatives of f are expressed viathe Poisson bracket of f and the perturbed Hamiltonian δH (2) on the linearized phasespace. Note, that ˆ I and Poisson brackets do not commute in general and also ˆ I will ingeneral be non unique and may only exist on a subset of the phase space.3For the reason that δ ˆ T is the scalar part of the shift perturbation, we choose δT a ⊥ asthe transverse part of the latter: δT a ⊥ ! = ˆ IS a + c , (3.68)where like c and c , c is a function of spatial coordinates which is a constant in time.Using the on-shell relations of the perturbations in the metric and the Lagrange mul-tiplier we obtain:[ ˆ I ¯ N φ ]( x, t ) = t Z d t ′ t ′ Z d t ′′ δλ ( x, t ′′ ) , [ ˆ IB ]( x, t ) = t Z d t ′ t ′ Z d t ′′ δ ˆ λ ( x, t ′′ ) (3.69)and [ ˆ IS a ]( x, t ) = t Z d t ′ t ′ Z d t ′′ δλ a ⊥ ( x, t ′′ ) . (3.70)For the stability of these clocks, we need δ ˙ T = ¯ N φ ≈ , δ ˙ˆ T = ¯ NA h ˆ I ¯ N φ + c i + B ≈ , (3.71)and δ ˙ T a ⊥ = S a ≈ . (3.72)The first condition yields φ ≈
0. Noting that δT ≈ c ≈
0, from the secondcondition we obtain B ≈
0. And, the third condition yields S a ≈
0. The stability of theclocks immediately yield: δλ ≈ δ ˆ λ ≈ δλ a ⊥ ≈
0. Let us note that these are notthe most general clocks for the synchronous gauge. The reason is that we have obtainedthem by fixing c ≈ c ≈ c ≈
0, the latter two getting fixed using δ ˆ T ≈ δT a ⊥ ≈
0. Other choices of c , c and c are possible by defining, σ µ := τ µ − ¯ T µ , as a resultof which we obtain δT µ ≈ σ µ . Following the above analysis, one is then led to relationsbetween components of σ µ and c , c and c . The constant functions c and c are thusdetermined by choice of σ µ . In the following we will consider the choices where constants c , c and c are all vanishing.
2. Observables
The clocks we have found for the synchronous gauge, by construction satisfy: { T, G ′ b,~b } , = b { ˆ T , G ′ b,~b } = ˆ b . (3.73)Using these the observable formula can be solved for the descriptors quite easily. The firstorder observable formula becomes: O (1) f,T ( x ) = δf ( x ) + { f ( x ) , G ′ b,~b } (cid:12)(cid:12)(cid:12) b µ = − δT µ . (3.74)Thus the first order observable of a scalar perturbation is just the expression of its in-finitesimal transformation behavior with b → − δT and ˆ b → − δ ˆ T . This results in thefollowing first order observables:4 O (1) φ,T = 0 , O (1) B,T = 0 , O (1) ψ,T = ψ − ˜ H ¯ N ˆ I ¯ N φ −
13 ∆ ˆ I (cid:18) ¯ NA ˆ I ¯ N φ + B (cid:19) , O (1) E,T = E − ˆ I (cid:18) ¯ NA ˆ I ¯ N φ + B (cid:19) , O (1) p E ,T = p E + ¯ N A ˜ H ˆ I ¯ N φ + ˆ I (cid:18) ¯ NA ˆ I ¯ N φ + B (cid:19) , O (1) p ψ ,T = p ψ + ˜ H N + κ N p ˜ H ! ˆ I ¯ N φ −
16 ∆ (cid:18) ¯ NA ˜ H ˆ I ¯ N φ + ˆ I (cid:18) ¯ NA ˆ I ¯ N φ + B (cid:19)(cid:19) , O (1) F a ,T = F a − ˆ IS a , O (1) p aF ,T = p aF + ˆ IS a , O (1) S a ,T = 0 , O (1) δϕ,T = δϕ − λ φ A / ¯ π φ ˆ I ¯ N φ, O (1) δπ ϕ ,T = π ϕ − ¯ π ϕ ∆ ˆ I (cid:18) ¯ NA ˆ I ¯ N φ + B (cid:19) + 12 A / λ ϕ d V d ϕ ˆ I ¯ N φ . (3.75)And for the observables corresponding to momenta of lapse and shift perturbations weget, O (1) p φ ,T = 1¯ N δ Π , O (1) p B ,T = ˆ δ Π , O (1) p Sa ,T = δ Π a ⊥ . (3.76)Though we have found the observables corresponding to the synchronous gauge, we mustnote that the operator ˆ I might not be unique. This is to be contrasted with the resultsfor the previous gauges where no such ambiguity exists. E. Comoving gauge
In this gauge the slicing is chosen such that the scalar field perturbations vanish. Inthe presence of fluids this translates to comoving slicing in which the time slices areorthogonal to the fluid velocity. In particular, the fluid velocity perturbation must matchthe perturbation in the shift. For the case of the scalar field, we obtain, δϕ ≈ , B ≈ . (3.77)As in the case of the uniform field gauge, the scalar field is homogeneous. Though the timeslicing is fixed, there is a residual freedom in the choice of origin of spatial coordinates. Aswe will see, this will get reflected in the presence of an arbitrary constant in the geometricalclocks corresponding to scalar perturbations. Stability of the comoving gauge conditionsrequire: δ ˙ ϕ ≈ , and ˙ B ≈ . (3.78)The second condition fixes the perturbation, δ ˆ λ ≈
0. Whereas the first condition, usingHamilton’s equations, results in φ − ψ − δπ ϕ ¯ π ϕ ≈ . (3.79)5The stability of the above condition results in fixing the perturbation δλ . Using thetransformation properties of δϕ and B we can find the gauge descriptors as before. Thedifference in contrast to previous gauges is that the gauge descriptor b takes a simple formas in the longitudinal gauge, whereas the gauge descriptor ˆ b involves time integrals as inthe synchronous gauge. These are given by b ( x, t ) ! = − A / δϕλ ϕ ¯ π ϕ , (3.80)and ˆ b ( x, t ) ! = − Z d t ′ ¯ N ( t ′ ) A / ( t ′ ) λ ϕ ¯ π ϕ ( t ′ ) − Z d t ′′ B ( x, t ′′ ) − c ( x ) . (3.81)As in the case of the synchronous gauge, the latter gauge descriptor involves an integrationin time which creates some ambiguity in the Hamiltonian formulation. The presence of thefunction c ( x ) represents a residual gauge freedom corresponding to a shift of the spatialcoordinates.
1. Geometrical Clocks
As in the other gauges, we want to to choose the clocks such that we can reproduce thegauge fixing constraints used in cosmology. Considering the background quantities in thecomoving gauge, we realize that we need to choose ¯ τ a = 0 since the metric component cor-responding to the shift perturbation is zero. However, as far as the temporal backgroundclock is considered we have the freedom to choose ¯ τ = 0 and again this in principle allowsto define a notion of physical time for the background solution. At the linearized levelwe choose δτ a = 0 and also here a non-vanishing δτ is consistent with the linearizedequations of motion. But, likewise to the other gauges already discussed, for reproducingthe exact gauge fixing conditions used in cosmological perturbation theory, we considerthe specific choice of δτ = 0. The generalized gauge fixing condition with δτ = 0 willbe analyzed in more detail in section III F. With δτ µ set to vanish, the perturbed clocksusing the expressions of b and ˆ b are: δT = A / δϕλ ϕ ¯ π ϕ , δ ˆ T ! = ˆ I ¯ N A / δϕλ ϕ ¯ π ϕ + ˆ IB + c . (3.82)Since δ ˆ T is proportional to the shift perturbation, it leads us to identify δT a ⊥ ! = ˆ IS a + c , (3.83)where c , like c , is an arbitrary constant in time which depends on spatial coordinates.Note that in contrast to the uniform field gauge, where also δϕ ≈ δ ˆ T contains shiftperturbation. This is what results in the presence of c for the scalar perturbation whichreflects the residual freedom mentioned earlier. The stability of δT ≈ δλ . On the other hand, stability of δ ˆ T and δT a ⊥ fix δλ a , consistent with the gaugeconditions for the comoving gauge.6
2. Observables
To find the observables we use the general formulas as noted earlier, (3.1) and (3.33),and also (3.74). Using the clocks corresponding to the comoving gauge, we obtain: O (1) φ,T = − (3 κ + ς ) v − π v ¯ π ϕ , O (1) B,T = 0 , O (1) S a ,T = 0 , O (1) ψ,T = ψ − ˜ H A / δϕ ¯ N λ ϕ ¯ π ϕ − ∆3 ˆ I ( βδϕ + B ) , O (1) E,T = E − ˆ I ( βδϕ + B ) , O (1) F a ,T = F a − ˆ IS a , O (1) p E ,T = p E + βδϕ H + ˆ I ( βδϕ + B ) , O (1) p ψ ,T = p ψ + A / δϕ λ ϕ ¯ π ϕ ˜ H ¯ N + κ ¯ N p H ! − ∆6 (cid:18) β ˜ H δϕ + ˆ I ( βδϕ + B ) (cid:19) , O (1) p aF ,T = p aF + ˆ IS a , O (1) δϕ,T = 0 , O (1) δπ ϕ ,T = δπ ϕ − ˙ π ϕ ˙ ϕ δϕ − ¯ π ϕ ∆ ˆ I (¯ π ϕ βδϕ + B ) , (3.84)where β := ¯ N A / λ ϕ ¯ π ϕ . (3.85)And as for all other gauges, we also have O (1) p φ ,T = 1¯ N δ Π , O (1) p B ,T = ˆ δ Π , O (1) p Sa ,T = δ Π a ⊥ . (3.86)Unlike the case of the uniform field gauge, some of the observables consist of the ˆ I operator.These observables, as in the case of the synchronous gauge, have a certain non-uniquenessassociated with the ˆ I . F. Generalized gauge fixing constraints and modified gauges
As mentioned earlier for the uniform field, the synchronous and the comoving gauge, theequation of motions are consistent with choosing temporal functions ¯ τ and δτ that do notvanish. This corresponds to gauge fixing conditions ¯ G µ = ¯ τ µ − ¯ T µ and δG µ = δτ µ − δT µ .We showed in the previous subsections that a choice of δτ = 0, that is δG µ = − δT µ ≈ δτ seems to be problematic since ingeneral the observable map of a function f returns the value of f at those values where theclocks take the values τ µ . As far as the spatial clocks are considered we can set δτ a to zerobecause the physical evolution of the observables is defined with respect to the temporalclock only. More generally, if we choose τ a = τ a ( x j ) such that it is time independent,this corresponds to an induced slicing for which the shift vector vanishes on the constrainthypersurface where the gauge-fixing conditions are satisfied. In this subsection we want toanalyze generalized gauges for the uniform field, the synchronous and the comoving gauge.7For this purpose we choose the simple generalization of a temporal function τ ( t ) = 0corresponding in linearized cosmological perturbation theory to ¯ τ ( t ) = 0 and δτ ( t ) = t .In the linearized theory the latter choice corresponds to δG = t − δT ≈ t for δτ is the most simple choice for a coordinate gaugefixing constraints as discussed for instance in [14, 24]. As usual in the relational formalismwith such generalized gauge fixing constraints we would construct a τ -dependent familyof observables and hence more general observables than conventionally used in the contextof cosmological perturbation theory. By introducing a non-vanishing function δτ ( t ) wealso modify the gauge and hence gauge fix the clock fields differently. As a consequencealso the stability conditions of the clocks become modified and involve additional terms.Nevertheless, these modified conditions still merge into our former results presented aboveif we again consider the choice of δτ = 0.In the following we focus on the way uniform field, synchronous and comoving gaugesare modified when the gauge fixing constraint is δG = t − δT ≈
0. As a result, thestability requirement of the temporal gauge fixing constraint leads to modified conditionson the geometric and matter perturbations respectively. However, as far as the spatialgauge fixing condition is considered our former results still apply because we still choose δτ a = 0, that is δG a = − δT a . Note that for the uniform field and the comoving gauge,the perturbed temporal clock is the same and hence we do not need to discuss these twocases separately since the results are identical. The latter gauges will be discussed afterthe case of the synchronous gauge that we will start with.Let us consider a generalization of the gauge fixing constraint which led to the syn-chronous gauge in the above analysis. The perturbed temporal clock in the synchronousgauge was determined by the perturbation in the lapse. Requiring δG = t − δT ≈ b is given by b ( x, t ) ! = t − t Z d t ′ ¯ N ( t ′ ) φ ( x, t ′ ) − c ( x ) , (3.87)which leads to the following perturbed temporal clock δT = ˆ I ¯ N φ . Reinserting this backinto the gauge fixing constraint yields, δT = ˆ I ¯ N φ ≈ t . Here as in the case of the synchronous gauge we have chosen the constant c to be vanishing.The spatial clocks and corresponding gauge descriptors remain unchanged. The stabilityof the above gauge fixing constraint gives: δ ˙ T = ¯ N φ ≈ . Using (2.64), we easily see that the stability of above equation yields δλ = 0 exactly as inthe case of the perturbation in temporal clock for the synchronous gauge. Recall that one ofthe conditions for the synchronous gauge, φ ≈
0, is equivalent to the stability of the gaugefixing: δT ≈
0. The stability of the generalized gauges requires that ¯
N φ ≈
1. Thus, if theperturbed temporal clock δT is required to be linear in time and the perturbed spatialclock takes vanishing value then the synchronous gauge condition for the perturbationsmodifies to: φ ≈ N , B ≈ . (3.88)8Similarly, we can consider the generalization of the gauge fixing constraint leading tothe uniform field and the comoving gauges. For both the cases, the perturbation in thetemporal clock is the same. The gauge descriptor corresponding to δG = t − δT ≈ b ! = t − A / δϕλ ϕ ¯ π ϕ . (3.89)The associated perturbed temporal clock is chosen to have the form δT = A / δϕ/λ ϕ ¯ π ϕ leading to the following gauge fixing constraints δT = A / δϕλ ϕ ¯ π ϕ ≈ t . (3.90)The stability of the above constraint yields a differential equation relating time derivativesof background quantities with δ ˙ ϕ : δ ˙ T = A / δϕλ ϕ ¯ π ϕ ! · δϕ + A / δϕλ ϕ ¯ π ϕ δ ˙ ϕ ≈ δϕ ≈ ( t + t ′ δϕ ) λ ϕ ¯ π ϕ /A / , where t ′ δϕ is a constantof integration which we set to zero. We then need to consider the stability of the aboveequation with respect to the equations of motion for perturbed variables. This resultsin determining the value of δλ in terms of background and perturbation variables, as itwas the case for the uniform field and comoving gauges. Thus we obtain a consistentperturbed temporal clock that can be gauge fixed to be linear in time. The associatedmodified uniform field gauge in this case is given by: δϕ ≈ t λ ϕ ¯ π ϕ A / , E ≈ δϕ ≈ t λ ϕ ¯ π ϕ A / , B ≈ . (3.93)This concludes our discussion of the generalized gauge fixing constraints that lead tomodified uniform field, synchronous and comoving gauges for which in the relationalformalism a notion of physical time can be defined via these geometrical clocks. Likewiseto the case of the unmodified gauges, we could apply the observable map now and obtaina family of Dirac observables parametrized by t , where this parameter is interpreted asphysical time. For the choice of t = 0 these observables coincide with the observablesconstructed in the former subsections.Finally, let us summarize all the results obtained so far in this section for various gaugesin our formalism of reference clocks in tables I and II. In table I we summarize the resultsfor longitudinal, spatially flat and uniform field gauges all which involve isotropic thread-ing. Table II summarizes the synchronous and comoving gauges which have vanishingperturbations of the shift. Table III summarizes various symbols and key equations intables I and II.9 Variable Longitudinal Spatially flat Uniform field δT P √ A ( E + p E ) ¯ N ˜ H ( ψ − ∆ E ) κ ¯ N ˜ H δϕδT a δ ab ( E ,b + F b ) δ ab ( E ,b + F b ) δ ab ( E ,b + F b ) φ − Ψ − − (cid:16) + κ ˜ P Ap (cid:17) Ψ − (3 κ + ς ) v − π v ¯ π ϕ B ¯ N A ˜ H Ψ ¯ N A ˜ H (Ψ + κ v ) S a H ν a H ν a H ν a ψ Ψ 0 − κ vE F a p ψ Υ Υ + α Ψ Υ + α (Ψ + κ v ) p E P Ψ P (Ψ + κ v ) p aF ν a ν a ν a δϕ v + κ Ψ v δπ ϕ π v − ¯ π ϕ ς κ Ψ π v π v + ¯ π ϕ ςv Table I: Summary of geometrical clocks and linearized observables corresponding to various metricperturbations and their momenta for the gauges where the longitudinal part of the spatial metricperturbation vanishes. Various symbols are summarized in table III.
IV. CONCLUSIONS AND OUTLOOK
The main objective of our manuscript was to apply the relational formalism in theextended phase space to linearized cosmological perturbation theory and to understandthe relationship between the choice of clocks, gauge fixing conditions and the associatedgauge invariant quantities. Our manuscript, which is a companion article to the review [25],extends the results already present in the literature for the reason that the considerationof the extended phase space opens a window to a larger class of gauge fixing conditionsthat can not be dealt with if only the reduced ADM-phase space is considered where lapseand shift are treated as Lagrange multipliers. Let us note that in a seminal work, Langloisformulated the canonical description using ADM variables and derived the analogue ofthe Mukhanov-Sasaki variable in phase space [20]. However, in this study lapse and shiftwere treated as Lagrange multipliers and therefore the analysis had in built restrictions.As an example, it is impossible to obtain the Bardeen potentials in Langlois’ analysisbecause one of the Bardeen potentials is tied to perturbations in the lapse. As we haveshown this restriction can be avoided by formulating linear canonical perturbation theoryin the extended phase space. This generalization that is strongly based on earlier workby Pons, Salisbury, Sundermeyer et al [21–24] was started in the review [25]. There, asa preparation for the work in this article, the phase space analogues of gauge invariantquantities such as for instance the Bardeen potential and the Mukhanov-Sasaki variablewere constructed. These results were used in our work as the explicit form of these gaugeinvariant quantities defined on the extended ADM-phase space were taken as the guidingprinciple for the choice of clocks.0
Variable Synchronous Comoving δT ˆ I ¯ N φ A / δϕλ ϕ ¯ π ϕ δ ˆ T ˆ I ¯ NA h ˆ I ¯ N φ i + ˆ IB ˆ I ¯ NA / δϕλ ϕ ¯ π ϕ + ˆ IBδT a ⊥ ˆ IS a ˆ IS a φ − (3 κ + ς ) v − π v ¯ π ϕ B S a ψ ψ − ˜ H ¯ N ˆ I ¯ N φ − ∆ ˆ I (cid:16) ¯ NA ˆ I ¯ N φ + B (cid:17) ψ − ˜ H A / δϕ ¯ Nλ ϕ ¯ π ϕ − ∆3 ˆ I ( βδϕ + B ) E E − ˆ I (cid:16) ¯ NA ˆ I ¯ Nφ + B (cid:17) E − ˆ I ( βδϕ + B ) F a F a − ˆ IS a F a − ˆ IS a p ψ p ψ + (cid:16) ˜ H ¯ N + κ Np ˜ H (cid:17) ˆ I ¯ N φ − ∆ (cid:16) ¯ NA ˜ H ˆ I ¯ Nφ + ˆ I (cid:16) ¯ NA ˆ I ¯ N φ + B (cid:17)(cid:17) p ψ + A / δϕ λ ϕ ¯ π ϕ (cid:16) ˜ H ¯ N + κ ¯ Np H (cid:17) − ∆6 (cid:16) β ˜ H + ˆ I ( βδϕ + B ) (cid:17) p E p E + ¯ N A ˜ H ˆ I ¯ N φ + ˆ I (cid:16) ¯ NA ˆ I ¯ N φ + B (cid:17) p E + βδϕ H + ˆ I ( βδϕ + B ) p aF p aF + ˆ IS a p aF + ˆ IS a δϕ δϕ − λ φ A / ¯ π φ ˆ I ¯ Nφ δπ ϕ δπ ϕ − ¯ π ϕ ∆ ˆ I (cid:16) ¯ NA ˆ I ¯ N φ + B (cid:17) + A / λ ϕ d V d ϕ ˆ I ¯ Nφ δπ ϕ − ˙ π ϕ ˙ ϕ δϕ − ¯ π ϕ ∆ ˆ I (¯ π ϕ βδϕ + B )Table II: Summary of geometrical clocks and first order observables for the gauge choices corre-sponding to vanishing perturbation in shift. Unlike table I, we have split the perturbation in clockscorresponding to shift since the clock components are different. For definition of symbols, see tableIII. A difference of our analysis to the conventional approach in cosmological perturbationtheory lies in the context of the relational formalism. The latter has been earlier appliedto study linearized cosmological perturbations [14, 15, 18, 19], but the applications havebeen limited to the reduced ADM-phase space or its corresponding extension in terms of1
Symbol Relation to background and perturbation variables Equation α + κ P Ap −
23 1˜ P ∆ (3.51) β ¯ NA / λ ϕ ¯ π ϕ (3.85) κ A / λ ϕ ¯ π ϕ ˜ H ¯ N (3.60) ν a p aF ( x ) + δ ab F b ( x ) (3.30)Ψ ψ ( x ) + H A ¯ N ( E + p E )( x ) − ∆ E ( x ) (3.25) π v δπ ϕ − ¯ π ϕ ∆ E + A / λ ϕ d V d ϕ ( ¯ ϕ ) ¯ N ˜ H (cid:0) ψ − ∆ E (cid:1) (3.53) ς A λ ϕ ¯ π ϕ d V d ϕ ( ¯ ϕ ) (3.61)Υ p ψ + ∆ E + ∆ p E − (cid:16) ˜ H A ¯ N + κA p (cid:17) ( E + p E ) (3.26) v δϕ − λ ϕ A / ˜ H ¯ N ¯ π ϕ (cid:0) ψ − ∆ E (cid:1) (3.52)Table III: Definitions of various symbols used in tables I and II. Ashtekar variables respectively. Our manuscript provides the first application of the rela-tional formalism to cosmological perturbation theory in the extended ADM-phase space.As a result, clocks and Dirac observables can be understood even for lapse and shift vari-ables which are treated dynamically on the same footing as all remaining phase spacevariables.In the relational formalism every gauge fixing conditions is determined by a choice ofclocks. Therefore in this manuscript, we have taken the approach where we have chosenlinearized clocks which yield the commonly used gauge conditions in the linear cosmologicalperturbation theory. Five gauge fixing constraints were considered: the longitudinal gauge,spatially flat gauge, the uniform field gauge, the synchronous gauge and the comovinggauge. In the first three cases, the gauge freedom is completely fixed, whereas in thelatter two cases there is residual freedom in the shift in the spatial coordinates. For eachof these gauge choices, we identified the clocks constructed from the metric and in somecases metric and matter perturbations at the linear order. The associated observables tothese clocks correspond to the gauge invariant variables which are naturally tied in theirphysical interpretation to the same gauge fixing conditions.For the longitudinal gauge fixing constraint, the geometrical clocks we chose result inthe Bardeen potentials as the associated and independent Dirac observables understood asthe gauge invariant extensions of the lapse and the trace of the spatial metric perturbation.For the spatially flat gauge, the Mukhanov-Sasaki variable is the naturally associated Diracobservable corresponding to the scalar field perturbation. Similarly, for the other gaugeswe find a natural set of gauge invariant quantities using geometric clocks and constructtheir associated Dirac observables.The connection between the gauge fixing conditions and gauge invariant variables iswell known in the conventional treatment of cosmological perturbation theory. Our anal-ysis bring out this relationship from the perspective of the relational formalism in thecanonical perturbation theory. As emphasized earlier in section I, each gauge invariantquantity in cosmological perturbation theory has a direct relationship with metric or mat-ter perturbations only in a specific gauge. This becomes transparent in our procedure interms of Dirac observables. As an example, even though the Bardeen potentials are gauge2invariant quantities, irrespective of the chosen gauge, they only appear as the natural
Dirac observables when the clocks are chosen to be consistent with the longitudinal gauge– their natural gauge as far as their physical interpretation is addressed. In any othergauge, they do not appear as natural Dirac observables and thus their physical interpreta-tion is lost. The same is true for the relationship between other gauges and gauge invariantquantities. We note that this conclusion is expected from the conventional treatment ofcosmological perturbation theory where the naturalness of gauge invariant quantities vis-`a-vis gauge fixing conditions is well known, but has been borne out for the first time inthe language of Dirac observables. Thus, our canonical analysis make apparent the subtleand self-consistent relationship between the choice of clocks, gauge fixing conditions andDirac observables.It is useful to point out a subtlety in comparing our work based on the extendedADM formulation to extract gauge invariant quantities with the conventional frameworkbased on the Lagrangian approach. Let us recall following the early work in cosmologicalperturbation theory by Sachs [36], and Stewart and Walker [37] (further developed byBruni, Dunsby, Ellis and Sonego [38–40]) that one starts by introducing of two spacetimemanifolds M and M where M defines the physical spacetime and M the backgroundspacetime. The latter can be understood as a fiducial manfiold in this construction. Theperturbations are then defined via a point identification map between M and M , andthe choice of a specific point identification map can be understood as a choice of gauge.The gauge invariant quantities are then those whose values do not depend on the pointidentification map and this kind of gauge invariance was called gauge of second kind bySachs due to the fact that it occurs in addition to the usual coordinate freedom presentin general relativity which we will refer to as gauge invariance of the first kind.Now following the relational formalism the idea is the following. First let us neglectthe aspects of perturbation theory. Then one would like to choose reference fields (clocks)that enter the observable map which allow to solve the constraints present in the canonicaltheory of general relativity. That is we construct gauge invariant quantities with respect tothe gauge invariance of the first kind. As a result one obtains a reduced phase space withonly physical degrees of freedom and a physical Hamiltonian that generates the dynamicsof the observables. This dynamics is unconstrained unlike the Lagrangian approach wherethe constraints of general relativity are still present among the 10 Einstein’s equations.If we aim at formulating a perturbative setting of this reduced canonical theory, thesecond kind of gauge transformation has to be considered as well. In this case the referencefields (clocks) serve a twofold purposes because on the one hand they are used to constructDirac observables and on the other hand in the above language also as a point identificationmap that eliminates the second kind of gauge freedom.Thus, to compare the complexity of the two approaches it is necessary to introduce thedemand of gauge invariance of the first kind in the conventional Lagrangian based approachto cosmological perturbation theory. An aspect where this issue becomes immediatelyrelevant is for instance when one derives the gauge invariant dynamics of for instance theBardeen potential and the Mukhanov-Sasaki variable in linear perturbation theory. Incontrast to the conventional way, this dynamics can be obtained in a very straightforwardand efficient way in our formalism [41]. This occurs thanks to the implementation of thegauge invariance of the first kind in our formalism which allows one to derive the dynamicsof these gauge invariant quantities purely at the gauge invariant level without the needto go back to the gauge variant form of the Einstein equations and derive from it theassociated dynamics of the observables.3An important question in cosmological perturbation theory is what are the gauge in-variant quantities when we go beyond linear order. In the conventional analysis, this is anon-trivial problem as one has to construct these quantities order by order in perturbationtheory, with the results obtained at lower order not being generalizable to higher orders.This task becomes easier in the relational formalism. The reason for this is that one canconstruct manifestly gauge invariant quantities already at the non-linear level, that is fullgeneral relativity, as it has for instance be done in [14–16]. This yields the gauge invariantEinstein’s equations at the full non-linear level. Perturbations of these equations involveby construction only quantities that are manifestly gauge invariant and hence are invari-ant under coordinate transformations up to arbitrary high orders. Thus, even in linearperturbation theory around a flat FLRW background we consider the linearity as far asthe perturbations on phase space are considered, but have manifestly gauge invariance forthe diffeomorphisms. As has been shown in the case of dust matter model [15], if onetruncates this manifestly gauge invariant quantities at linear order, one can reproduce theresults in linearized cosmological perturbation theory, where one linearizes the Einsteinequation first and afterwards construct linearized gauge invariant quantities.We should note that non-linear gauge invariant quantities have earlier been studiedin the covariant formulation [42] and further developed in [43, 44]. The latter method isbased on constructing gauge invariant quantities by introducing flow lines associated withfundamental observers. Using the Stewart-Walker lemma such quantities are automaticallygauge invariant if the corresponding quantities vanish in the background spacetime. Oneobtains full non-linear gauge invariant quantities and as shown by Langlois and Vernizzione can recover the usual gauge invariant quantities at linear and second order [43, 44]. Asin the case of dust model [15], one can first construct non-linear gauge invariant quantitiesand then apply perturbation theory which results in a formulation of perturbation theoryat the gauge invariant level. The key difference of this approach in comparison to ouranalysis is that the formalism by Langlois and Vernizzi considers only the gauge invarianceof the second kind but not the first kind as can been seen for instance from the factthat the constraints are still part of the theory. Furthermore, the introduction of flowlines associated with fundamental observers can be understood in the relational formalismas the choice of an idealized observer, that is one that causes no backreaction. In ourformalism observers are the reference fields (clocks) which are dynamically coupled to thesystem. Thus, backreaction will be inherently included in our approach. This is evidentvia the imprint of the chosen clocks in the equations of motion of the observables usuallyvia their energy and momentum densities. In order to compare the two approaches andtheir complexities in studying second order perturbation theory, one would need to includegauge invariance of the first kind in the approach of Langlois and Vernizzi.We showed here that the common gauge invariant variables in linear cosmologicalperturbation theory can be systematically constructed if we apply the relational formalismand the observable map to the extended phase space of linear cosmological perturbationtheory. An open question that arises from our results is whether we can find non-lineargeometrical clocks that reduce at the linear order to those we have identified here. Suchclocks have been constructed in relational formalism for the the case of dust matter inRef. [14], though they have been used to study first order cosmological perturbationtheory [15]. To generalize our current analysis to second order perturbation theory, onewill need to find non-linear clocks using purely metric perturbations or its combinationswith scalar field perturbations. For this purpose we had to generalize for instance thework in Ref. [14] to the extended ADM phase space and carefully analyze the stability of4our chosen clocks that becomes more complicated when geometrical than matter clocksare chosen. The identification of these clocks at linear order in our present analysis giveus vital insights on the nature of non-linear geometrical clocks which will be investigatedin a future work. Such an analysis would extend our results beyond the linear order andwould also in higher order have a systematic and straightforward way to construct gaugeinvariant quantities with a clear physical interpretation of such quantities.If we want to go beyond the classical theory and consider the quantization of theassociated reduced phase spaces that follow from a certain choice of geometrical clocks,then it might be the case that models where one chooses purely matter clocks as forinstance in [10–12, 14, 45–47] are of advantage. The reason for this is that in generalthe Poisson algebra of the Dirac observables that need to be considered in a reducedphase space quantization is more complicated than the standard kinematical algebra.Consequently, to find representations of the algebra, that is finding the associated quantumtheory, might be very difficult. Along with this comes the fact that the decomposed scalar,vector- and tensor gauge variant quantities in general satisfy also a more complicatedalgebra than the original ADM variables due to the projectors that are used to define suchdecomposed quantities. The matter field reference models in [10–12, 14, 45–47] are alldesigned in such a way that the Poisson algebra of the Dirac observables is as simple asthe kinematical algebra. However, a real conclusion on this point can only be drawn afterwe have identified some candidates for non-linear clocks, which will be one of our futureprojects.In the present manuscript we have focused our discussion on the classical perturba-tion theory in the canonical setting. However, one important application of our analysislies in the canonical quantization program, such as Wheeler-deWitt quantization or loopquantum gravity. For these approaches, our framework provides a natural setting to in-vestigate quantum gravitational effects on cosmological perturbations. For this purpose,the first step is to formulate a canonical formulation of classical cosmological perturbationtheory at the linear order. Our analysis accomplishes this step using the extended phasespace using ADM variables. The next step will be to incorporate quantum gravitationalconstraints in this setting and repeat the analysis with appropriate clocks at the quantumlevel. An interesting question will be to examine the role of clocks in the cosmologicalperturbation theory at the quantum level. Acknowledgements
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