Local gauge-invariance at any order in cosmological perturbation theory
aa r X i v : . [ a s t r o - ph . C O ] A ug Local gauge-invariance at any order in cosmologicalperturbation theory
Chris Clarkson
Astrophysics, Cosmology & Gravity Centre, and, Department of Mathematics, and,Applied Mathematics, University of Cape Town, Rondebosch 7701, Cape Town,South AfricaE-mail: [email protected]
Abstract.
The relativistic theory of structure formation in cosmology is based mainly on linearperturbations about a homogeneous background. But we are now driven to understandthe theory of higher-order perturbations in full detail, both from observational andtheoretical points of view. An important aspect of this lies in defining gauge-invariantperturbations at any order. We present a new covariant approach to this based ona local separation of tensors into their scalar, vector and tensor parts. Such a localdecomposition necessarily requires a trio of mutually annihilating differential operatorswhich form the basis for defining gauge-invariant objects. It makes no use of non-localGreen’s functions or pre-defined gauges, and can be used to define families of scalar,vector and tensor modes at any order one chooses.
1. Introduction
A challenging problem in relativistic cosmology is perturbation theory around ahomogeneous and isotropic background when extended beyond first-order. A particulardifficulty lies in preserving general covariance under perturbations, which necessitatesconsidering general coordinate transformations order-by-order alongside a pertubativeexpansion, and trying to eliminate gauge degrees of freedom. For a tensor to be gauge-invariant (GI) at a given order, it must vanish (or be constant) at all lower orders (see [1]for a full discussion). It is widely acknowledged that using tensors which are fully gaugeinvariant is the best way to construct higher-order perturbation theory. One importantresult of Nakamura’s [2] has been to show how starting from gauge-invariant first-orderquantities in the metric approach one can generate GI objects at second- and higher-order. Thus one can define a second-order GI tensor perturbation of the metric in thePoisson gauge, for example, but this perturbation is gauge-dependent in the sense thatit depends on using the Poisson gauge to begin with; starting in a different gauge onehas a different tensor perturbation, even though it too can be GI (see [3] for details).An important restriction for these gauge-based GI quantities is that they aredefined non-locally, which requires assumptions about unknowable boundary conditions ocal gauge-invariance at any order in cosmological perturbation theory ∇ + nK involved; it is covariant, so no gauges or coordinatesare necessary to define these quantities; and finally, variables are defined such that theyare automatically scalar, vector or tensor modes.
2. The covariant approach to cosmological perturbations
In cosmology there exists physically defined observers and therefore reference frameswith which to consider physical quantities. Any observable quantity necessarily relieson such an observer being chosen. This underlies the 1+3 covariant approach to GRinitiated by Hawking, Ehlers and Ellis [4, 5]. Any observer can find, say, the uniqueCMB reference frame, or the energy frame (in which the net heat flux vanishes), anduse such a frame to define the electric and magnetic parts of the Weyl tensor which donot depend on any coordinate system, thereby preserving general covariance. That is,there are various velocity fields around which physically and invariantly define the restof the spacetime quantities; the coordinate system one uses remains irrelevant. Suchconsiderations have led to the 1+3 covariant approach to cosmological perturbationtheory in which gauge-invariance and frame-invariance are subtly separated.The 1+3 approach is a semi-tetrad conversion of the field equations of GR into aset of evolution and constraint equations which are derived from the Ricci and Bianchiidentities. The variables involved are all covariant objects (i.e., they are tensors), definedthrough projections with a physical velocity field u a and its associated spatial projectiontensor h ab = g ab + u a u b and volume element ε abc = u d η abcd [4]. All tensors may then ocal gauge-invariance at any order in cosmological perturbation theory u a is the velocity of the fluid.More usefully than this is what happens at first-order. Any of the 1+3 objects withan index must vanish in the background as otherwise it would break the symmetry, andso it must be GI by the Stewart-Walker Lemma. This important theorem states that atensor is GI provided it vanishes in the background [6]. Except in specialised situationslittle work has been carried out at higher order [7, 8] because of the difficulty in findingGI quantities. All rank-1 and -2 tensors used here are orthogonal to u a , and rank-2 tensors areprojected with h ab , symmetric and trace-free which we denote using angle bracketson indices. We define the spatial covariant derivative acting on scalars or spatial tensorsas D a X b...c = h a ′ a h b ′ b · · · h c ′ c ∇ a ′ X b ′ ...c ′ . The irreducible parts of the spatial derivative ofPSTF tensors are the divergence, curl, and distortion, defined as [9]div X b...c = D a X ab...c (1)curl X ab...c = ε de h a D d X eb...c i (2)dis X ca...b = D h c X a...b i . (3)Then, the spatial derivative of a rank- n PSTF tensor X A n = X a a ...a n may bedecomposed as (for n = 1 , , b X A n = 2 n − n + 1 div X h A n − h a n i b − nn + 1 curl X c h A n − ε ca n i b + dis X bA n . (4)Note that the divergence decreases the rank of the tensor by one, the curl preserves it,while the distortion increases it by one (and all are PSTF). Keeping this in mind onecan drop the indices on differential operators as long as it’s explicit the valance of thePSTF tensor which is being acted on.At maximal perturbative order a GI object may be considered as a field on anFLRW background. Then, a general rank-2 GI PSTF tensor can be split into a non-local scalar, vector and tensor parts X ab = S ab + V ab + T ab = D h a D b i S + D h a V b i + T ab , (5)where the scalar part S ab is curl-free, the vector part V ab is solenoidal, D a V a = 0 ⇒ D a D b V ab = D a D b D h a V b i = 0, while the tensor part is transverse, div T a = 0. A similardecomposition exists for rank-1 tensors, but we shall concern ourselves here with rank-2 tensors only: a PSTF rank-2 tensor can be formed from a rank-1 one by taking a ocal gauge-invariance at any order in cosmological perturbation theory local variables correspondingto S ab , V ab and T ab which can be found by acting with suitable combinations of differentialoperators on X ab . These are local because they do not rely on the solution to anelliptic differential equation that the usual SVT decomposition (5) implicitly relies on;consequently there is no reliance on integrals over all space and unknowable boundaryconditions. These differential operators, which act on rank-2 PSTF tensors, are: ‡S = dis D div div (6) V = dis curl div (7) T = (cid:2) dis div − D + curl (cid:3) curl . (8)These are constructed so as to preserve the rank of X ab . § Note that div T = 0,div div V = 0 and curl S = 0 as required. Then, S X ab depends only on S ab , V X ab depends only on V ab , T X ab depends only on T ab ; similarly, T X ab is a divergence-freerank-2 tensor, hence a bone fide tensor mode. These operators are mutually annihilatingwhen acting on a quantity at maximal perturbative order (i.e., acting on the background)in the sense that S V = V S = S T = T S = V T = T V = 0 . (12)
3. Gauge-invariant objects
At first-order finding gauge-invariant objects is trivial: any object with an index mustbe GI by the Stewart-Walker lemma. How do we find GI objects at second- and higher-order? The trick in going from zeroth to first lies in taking gradients of scalars: scalarsare non-zero in the background but their gradients will not be, thereby providing usefulGI objects (in addition to things like the electric and magnetic Weyl tensors which areGI anyway). At first-order, however, all of the standard 1+3 covariant quantities arenon-zero, so it has been a long standing problem as to how GI second-order objects maybe formed. Can we take derivatives of first-order quantities in a suitable way such thatthey are zero until we get to second-order? What about higher-orders?One method is to excite only a certain degree of freedom at first-order, such asscalar modes [7, 8]. Then, any variable which is a pure vector or tensor mode has to besecond-order and, hence, gauge-invariant. For example, in this case we can easily see ‡ Note that T looks a bit different from in [8]. We have substituted for the curvature in terms of curl and D so that T can be used unambiguously as a differential operator in any spacetime. § Written out in full these are: S X ab = D h a D b i D c D d X cd (9) V X ab = ε cd h a D b i D c D e X ed (10) T X ab = ε cde D h a D e D c X b i d + ε cd h a (cid:2) b i D e D c X cd − D D c X b i d − c D X b i d − D e D c D e X b i d + D e D c D b i X ed + D c D e D b i X ed + D e D b i D c X ed + D c D e D d X b i e (cid:3) (11) ocal gauge-invariance at any order in cosmological perturbation theory V E ab is also a GI vector mode at second-order, and T H ab isa GI tensor mode. What is not obvious is how to isolate the scalar modes which areinduced at second-order in a GI way. This actually illustrates the main problem: howdo we handle the general situation when all modes are excited at first?The key lies in analysing what happens in going from zeroth to first. Consider thegradient of the energy density D a ρ , which is GI at first-order. Despite appearances,D a ρ is actually a mixture of a scalar and a vector mode. We can see it contains avector mode because if we take the curl, we find curl D a ρ ∝ ω a . Similarly, to isolate thescalar part of D a ρ we have to take a divergence. The reason for this is that because ρ itself is not GI when we take its gradient, the covariant derivative is not a backgroundcovariant derivative but has first-order connection terms mixing in; the vector degree offreedom arises from this. To summarise, we must first take a derivative orthogonal tothe symmetry of the background to form a GI object; then we must differentiate againin two ways to isolate the scalar and vector part.Now let us generalise this to one order higher. Consider dis ω ab : this is a purevector degree of freedom at first-order because ω a is divergence free. Consequently, atfirst-order, we must have S dis ω ab = 0 and T dis ω ab = 0. Therefore, at second-order S dis ω ab and T dis ω ab must be GI PSTF tensors . Thinking of S , V and T as ‘orthogonaloperators’, taking derivatives orthogonal to V (since dis ω ab is a pure vector) results inGI variables at the next perturbative order. Given either of these objects we can operateonce more with S , V or T to isolate pure scalar, vector or tensor degrees of freedom.More generally, if we take any PSTF rank-2 tensor such as the shear or electricWeyl curvature, we can operate with S , V and T to form a set of 3 new PSTF objects:in the case of the shear for example we have S σ ab , V σ ab and T σ ab . At first-order thesehave an invariant meaning in terms of the SVT decomposition, but at second they donot. Now operate on each of these three with S , V and T , to form a total of 9 newrank-2 tensors. Of these, S S σ ab , V V σ ab and T T σ ab are not much use to us as they arenon-zero at first-order. However, all the mixed cases have to vanish at first-order andtherefore must be GI at second . That is, an operator to produce a GI PSTF tensor atsecond-order from any PSTF first-order one belongs to the annihilating set Π = {S V , S T , V S , V T , T S , T V } (13)After this operation, they still do not have an invariant meaning in terms of the SVTsplit because S , V and T must act on GI quantities for this to be the case (analogouslyto D a ρ not being a scalar at first-order). To extract the SVT parts of an element of Π [ σ ab ], say, we must differentiate again with S , V or T , giving a total of 6 scalars, 6vectors and 6 tensors. Explicitly, we have:scalar: S Π A σ ab S S V σ ab , S S T σ ab , S V S σ ab , S V T σ ab , S T S σ ab , S T V σ ab vector: V Π A σ ab V S V σ ab , V S T σ ab , V V S σ ab , V V T σ ab , V T S σ ab , V T V σ ab tensor: T Π A σ ab T S V σ ab , T S T σ ab , T V S σ ab , T V T σ ab , T T S σ ab , T T V σ ab , ocal gauge-invariance at any order in cosmological perturbation theory A = 1 , . . . ,
6, forming a set of 18 GI objects which now have an invariant meaningin terms of the SVT split. Similar quantities can be formed starting from any PSTFrank-2 tensor, including things like D h a D b i ρ .Let us now extend this to third-order. Again we can form objects which vanishat second-order by applying annihilating projections of S , V and T to those SVT GIvariables just defined – i.e., S V Π A , S T Π A , V S Π A , V T Π A , T S Π A , T V Π A give a totalof 6 × Π twice on a first-order PSTF tensor inorder to find all these objects. Further action with S , V or T will then split these intosets of scalars, vectors and tensors. Thus, T V S V S σ ab is a GI third-order tensor mode.We can generalise this to any order. We have seen that at second-order Π actingon a first-order PSTF tensor gives a set of 6 GI variables, while at third we have to actwith Π ⊗ Π , i.e., the outer product of Π with itself. In index notation we have Π A Π B .At perturbative order n we must act with Π n − Π ⊗ ( n − = Π ⊗ Π ⊗ · · · ⊗ Π | {z } n − times . (14)In index notation we have instead Π A Π A · · · Π A n − , giving a total of 6 n − possible GIPSTF tensors. The SVT parts can be separated with a final action of S , V and T asbefore.
4. Conclusions
We have demonstrated for the first time how to generate covariant, locally defined,GI objects at any order when perturbing away from an FLRW background. Utilisinglocal differential operators which mutually annihilate on the background we can formrank-2 tensors which represent only scalar, vector or tensor degrees of freedom, providedthey act upon a GI object at a given perturbative order. Recursively acting with theseoperators in the way described above then provides a neat way to define a large varietyof GI variables at any given order.How would such GI quantities be used in practise? In principle, for a given order n one can calculate evolution equations for Π ⊗ ( n − X ab , where X ab stands for a set ofappropriate rank-2 tensors, using standard commutation relations and the 1+3 covariantequations. Products of quantities which appear will be of perturbative order n − Π ⊗ ( n − X ab ; these will have their own evolutionequations. The final system will yield a large system of PDEs which can be solved orderby order. This is left for future work.The physical information these variables contain will be very subtle to interpret ofcourse. When considering something like T S V σ ab for example, we cannot read this asthe ‘the tensor part of the scalar part of the vector part of the shear at second-order’,because once one moves up a perturbative order the meaning of the SVT decompositiondisappears. Analogously, V σ ab is the vector part of the shear at first-order, but not ocal gauge-invariance at any order in cosmological perturbation theory also has asimple physical interpretation is implausible. Nevertheless, as higher-order perturbationtheory is developed in full generality this will be an important avenue to explore. Acknowledgments
I would like to thank Marco Bruni, George Ellis, Kenneth Hughes, Roy Maartens, BobOsano and Obinna Umeh for comments and/or discussions.
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