Second order cosmological perturbations: simplified gauge change formulas
aa r X i v : . [ g r- q c ] J a n Second order cosmologicalperturbations: simplified gauge changeformulas
Claes Uggla ∗ Department of Physics,Karlstad University, S-651 88 Karlstad, Sweden
John Wainwright † Department of Applied Mathematics,University of Waterloo,Waterloo, ON, N2L 3G1, Canada
Abstract
In this paper we present a new formulation of the change of gauge formulasin second order cosmological perturbation theory which unifies and simplifiesknown results. Our approach is based on defining new second order scalarperturbation variables by adding a multiple of the square of the correspondingfirst order variables to each second order variable. A bonus is that these newperturbation variables are of broader significance in that they also simplifythe analysis of second order scalar perturbations in the super-horizon regimein a number of ways, and lead to new conserved quantities.
Cosmological perturbation theory plays a central role in confronting theories ofthe early universe with observations. The increasing accuracy of the observations,however, has made it desirable to extend the theory from linear to second order( i.e. nonlinear) perturbations, which presents various technical challenges. For ex-ample, in applying cosmological perturbation theory at second order it is oftendesirable to use several gauges since the physical interpretation may require onegauge while the mathematical analysis may be simpler using a different gauge. The ∗ Electronic address: [email protected] † Electronic address: [email protected] See, for example, Bartolo et al (2010) [3] and Tram et al (2016) [24]. INTRODUCTION Reading andinterpreting the formulas in these papers is not easy due to their complexity, andcomparing formulas in the papers is difficult because of the lack of a standard no-tation. However, while studying the formulas in these papers we noticed that forscalar perturbations they have certain features in common that enables one to writethem in a unified and simpler form.We consider first and second order scalar perturbations of Friedmann-Lemaˆıtre(FL) universes subject to the following assumptions:i) the spatial background is flat;ii) the stress-energy tensor can be written in the form T ab = ( ρ + p ) u a u b + pδ ab ,thereby describing perfect fluids and scalar fields;iii) the linear perturbation is purely scalar.We will use the following five gauges: the Poisson (longitudinal, zero shear) gauge,the uniform curvature (spatially flat) gauge, the total matter gauge, the uniformdensity gauge, and the uniform scalar field gauge.The simplification of the gauge change formulas is accomplished by making threechoices. First, we use a common fixing of the spatial gauge freedom so that the re-maining degrees of freedom in the gauge vector fields ( r ) ξ a , r = 1 ,
2, are the temporalcomponents at first and second order. Second, we normalize background and pertur-bation variables so that they are dimensionless. In particular, as the time variablewe use the so-called e -fold time N which is defined by N = ln( a/a ), where a isthe background scale factor. The scalar N represents the number of background e -foldings from some reference time a . Third, a careful inspection of the sourceexpressions in the known gauge change formulas reveals that a number of quadraticfirst order terms can be incorporated in a systematic way into the second orderperturbation variables and the temporal gauge vector field, leaving much simplified Other papers that have been influential in developing and applying second order cosmologicalperturbation theory but do not emphasize change of gauge formulas are Bartolo et al (2004) [1] andNakamura (2007) [20]. Examples of recent papers that use change of gauge formulas at second orderare Malik (2005) [15], Christopherson et al (2011) [6], Hidalgo et al (2013) [10], Christopherson etal (2015) [7], Carrilho and Malik (2015) [5], Dias et al (2015) [8], Villa and Rampf (2016) [31] (seeequations (3.11)-(3.14) and (3.22)-(3.25) with the source terms given in equations (C.1-C.4)) andHwang et al (2017) [11]. We refer to Malik and Wands (2009) [16], section 7.5, for this terminology. See also Liddleand Lyth (2000) [13], page 343. This gauge was apparently introduced by Kodama and Sasaki(1984) [12], and called the velocity-orthogonal isotropic gauge (see page 45, case 2b). See for example, Martin and Ringeval (2006) [17].
UNIFIED FORM FOR GAUGE TRANSFORMATIONS TO SECOND ORDER f for these source-compensated second or-der perturbation variables . For the metric and matter variables we can give a unifieddefinition as follows: (2) ˆ := (2) + C (1) , (1)where the kernel represents a dimensionless metric or matter perturbation andthe coefficient C depends on the background variables, while for the temporalcomponent of the gauge vector field using e -fold time N as time coordinate wedefine (2) ˆ ξ N := (2) ξ N − (1) ξ N ∂ N ( (1) ξ N ) , (2)where we write ∂ N ≡ ∂/∂ N for brevity. In terms of these quantities, inspection ofthe known change of gauge formulas leads to the following unified form: (1) • = (1) − (1) ξ N • , (3a) (2) ˆ • = (2) ˆ − (2) ˆ ξ N • + 2 (1) ξ N • ∂ N (1) • + rem, • , (3b)where the subscript • stands for a letter describing a particular gauge choice. Wefind that the reminder term rem, • for most metric and matter variables is a simplequadratic function of the first order variables, which in the case of any scalar variableis in fact zero.The outline of the paper is as follows. In section 2, after introducing the notationthat we will use for the metric and matter variables, we present the details concerningthe unified definition (1) of the hat variables and the details concerning the unifiedform of the gauge transformation formula (3). In section 3 for each of the fivechoices of temporal gauge indicated by • we give expressions for (2) ˆ • in terms of themetric and matter perturbation variables (see equations (42)). This set of formulas,which provides an efficient unifying algorithm for calculating any gauge invariantin any of the above five gauges, is the main goal of the paper. In section 4 we useour unified scheme to give simple derivations of some important change of gaugeformulas previously presented in the cosmological literature. In section 5 we pointout that the present paper is the first of four closely connected papers. We alsocomment on how the new hatted variables result in new conserved quantities, asshown in detail in the sequel papers. In appendix A we make further comparisonsof our gauge transformation formulas in section 2 with those in Malik and Wands(2009) [16], which served as our main starting point for the present paper. To perturb a flat FL background geometry it is convenient to write the metric as ds = a (cid:0) − (1 + 2 φ ) dη + f ηi dηdx i + f ij dx i dx j (cid:1) , (4)where a is the background scale factor and η is conformal time in the background.We assume that the metric components can be expanded in powers of a perturbationparameter ǫ , i.e. as a Taylor series, for example, φ = ǫ (1) φ + ǫ φ + . . . . (5) UNIFIED FORM FOR GAUGE TRANSFORMATIONS TO SECOND ORDER f ηi = D i B + B i , (6a) f ij = (1 − ψ ) γ ij + 2 D i D j C + 2 D ( i C j ) + 2 C ij , (6b)where D i B i = 0; D i C i = 0; C ii = 0, D i C ij = 0, and where D i is the spatial co-variant derivative corresponding to the flat metric γ ij . Use of Cartesian backgroundcoordinates yields γ ij = δ ij and D i = ∂/∂x i . As regards dimensions, we make thechoice that the scale factor a is dimensionless. It then follows that the coordinates η and x i have dimensions of length since ds has dimension length . Hence φ and ψ are dimensionless while B has dimension length .We consider a stress-energy tensor of the form: T ab = ( ρ + p ) u a u b + pδ ab , (7)which encompasses perfect fluids and scalar fields. The energy density ρ , the pressure p , and the 4-velocity u a can be expanded as a Taylor series in ǫ . Perturbations ofthe energy density ρ are therefore given by ρ = ρ + ǫ (1) ρ + ǫ ρ + . . . , (8)and similarly for the pressure perturbations. We use the usual background mattervariables w and c s , and the deceleration parameter q defined according to w = p ρ , c s = p ′ ρ ′ , q = − H ′ H , (9)where ′ denotes the derivative with respect to the conformal background time vari-able η , and H = a ′ /a = aH with H the background Hubble variable. We use unitssuch that c = 1 and 8 πG = 1, where c is the speed of light and G the gravitationalconstant. It follows that H has dimension of ( length ) − and that q, w and c s aredimensionless.Since we have assumed that the spatial background is flat, the Einstein fieldequations in the background can be written as3 H = a ρ , −H ′ + H ) = a ( ρ + p ) , (10)which in conjunction with (9) yields the following relation between w and the de-celeration parameter q : 1 + q = (1 + w ) , (11)a result that we will use frequently.To define the scalar velocity perturbations we find it convenient to work withthe covariant u b , which we normalize with a conformal factor a according We use a subscript zero to denote the background value of some quantity, so that ρ and p are the background energy density and pressure. UNIFIED FORM FOR GAUGE TRANSFORMATIONS TO SECOND ORDER u b = aV b , in analogy with the conformal factor a in the metric (4). We thenexpand and decompose the spatial components of V b according to V i = ǫ (1) V i + ǫ V i + . . . , (12a) ( r ) V i = D j ( r ) V + ( r ) ˜ V i , r = 1 , , . . . , (12b)with D i ( r ) ˜ V i = 0, so that ( r ) V represents the scalar perturbations. Since the V b aredimensionless and the x i have dimension length it follows from (12) that ( r ) V hasdimension length .In the case in which the matter-energy content is provided by a minimally-coupled scalar field, we will use ϕ to denote the scalar field and define the pertur-bations according to ϕ = ϕ + ǫ (1) ϕ + ǫ ϕ + . . . , . (13)Next we turn to gauge transformations in cosmological perturbation theory. Webegin by considering an arbitrary 1-parameter family of a tensor field A ( ǫ ), whichcan be expanded in powers of ǫ , i.e. as a Taylor series:A( ǫ ) = A + ǫ (1) A + ǫ A + . . . . (14)A gauge transformation induces a change in the first and second order perturbationsof A ( ǫ ). Arguably the most geometric and straightforward approach to gauge trans-formations is the “active approach” using an exponential map described in section6 in Malik and Wands (2009) [16], and this is the approach we take as our start-ing point. First and second order gauge transformations are then represented as(equations (6.5) and (6.6), respectively, in [16]): (1) A [ ξ ] = (1) A + £ (1) ξ A , (15a) (2) A [ ξ ] = (2) A + £ (2) ξ A + £ (1) ξ (cid:0) (1) A + £ (1) ξ A (cid:1) , (15b)where (1) ξ a and (2) ξ a are independent background gauge vector fields and £ is theLie derivative (see also [4], equations (1.1)–(1.3)). Equations (15) describe how thetensor field A changes under an arbitrary gauge transformation. More importantlyfrom a physical point of view, these equations serve to define gauge invariant quan-tities in the following way. If we impose a restriction on the perturbation variablesthat determines the gauge fields uniquely, say (1) ξ a = (1) ξ a • , (2) ξ a = (2) ξ a • , then wesay that we have fixed the gauge. If we use these as the gauge fields in (15), thenthe quantities (1) A [ ξ • ] and (2) A [ ξ • ] so defined are gauge invariant quantities. Onintroducing the shorthand notation (1) A • = (1) A [ ξ • ] , (2) A • = (2) A [ ξ • ] , (16) Gauge transformations up to second order in cosmological perturbation theory can also berepresented in coordinates as follows (see e.g. Malik and Wands (2009) [16]):˜ x a = x a + ǫ (1) ξ a + ǫ (cid:16) (2) ξ a + (1) ξ a,b (1) ξ b (cid:17) . UNIFIED FORM FOR GAUGE TRANSFORMATIONS TO SECOND ORDER (1) A • = (1) A + £ (1) ξ • A , (17a) (2) A • = (2) A + £ (2) ξ • A + £ (1) ξ • (cid:0) (1) A + £ (1) ξ • A (cid:1) . (17b)We say that (1) A • and (2) A • are the first and second order gauge invariants associatedwith the tensor field A in the gauge specified by the subscript • . We list severalgauges and their identifying subscripts at the beginning of section 3.We fix the spatial gauge freedom completely by setting the metric functions C and C i in (6) to be zero order by order, which up to second order gives ( r ) C = 0 , ( r ) C i = 0 , r = 1 , . (18)The above spatial gauge fixing is arguably the essence of Hwang and Noh’s so-called “gauge ready” approach, who refer to it as the C-gauge (see for exampleNoh and Hwang (2004) [21], equation (259)). Note that this is the only way onecan algebraically completely fix the spatial gauge by using the metric componentsand matter variables for the present models (see e.g. the gauge transformationsgiven in [26]), and as a consequence all the gauges listed in the introduction andsection 3 are characterized by this condition. The only gauge that is commonlyused that does not fulfil this condition is the synchronous gauge, which is useful fortreating dust models. However, the synchronous gauge is not a fully fixed gaugeand the natural way to completely fixing this gauge, and thereby relate quantitiesto physical observables, is to relate it to the total matter gauge, which does obeythe above conditions, see Appendix B.7 in [26].As a consequence of the above spatial gauge fixing, the remaining gauge freedomis described by gauge fields to second order restricted to be of the form (1) ξ a = ( (1) ξ N , , (2) ξ a = ( (2) ξ N , D i (2) ξ + (2) ˜ ξ i ) , D i (2) ˜ ξ i = 0 , (19)where (2) ξ and (2) ˜ ξ i are determined by quadratic source terms that arise from theconditions (18), where, in particular, (2) ξ depends on (1) ξ N . As in the introductionwe are using the e -fold time defined by N = ln( a/a ) as the time coordinate insteadof the conformal time η . Note that the temporal components of the gauge fields arerelated according to ξ N = H ξ η , which follows from ∂ η = H ∂ N . (20)In this paper we will primarily use e -fold time N but we will also use conformal time η , depending on the context. Equation (20) enables one to make the transition andwe will use it frequently.We are further restricting our considerations to perturbations that are purelyscalar at linear order , i.e. the metric functions that describe vector and tensorperturbations at first order are zero: (1) B i = 0 , (1) C ij = 0 , (1) ˜ V i = 0 . (21) For a recent work using the synchronous gauge for models with dust, see e.g. Gressel andBruni (2017) [9]. See equation (B10e) and (B10f) in [26] for the transformation laws for C and C i . UNIFIED FORM FOR GAUGE TRANSFORMATIONS TO SECOND ORDER et al (2004) [1] (see page 41) argue that this restriction is reasonable onphysical grounds, since vector perturbations have decreasing amplitude and are notgenerated during inflation, while tensor perturbations are expected to be negligible.On the other hand it is well known ([1], see page 41) that even if the vector andtensor perturbations are zero at first order, they will be generated at second orderdue to the presence of source terms in the vector and tensor governing equations,since these source terms depend on the first order scalar perturbations. Thus evenif the first order restriction (21) holds we will have (2) B i = 0 , (2) C ij = 0 , (2) ˜ V i = 0 , (22)at second order. In this context, however, the second order scalar perturbationsare independent of the second order vector and tensor perturbations and hence canbe studied separately. In this paper we are choosing to consider only the scalarperturbations at second order, which physically represent density perturbations,leaving the second order vector and tensor perturbations for future work. We arethus studying second order scalar perturbations subject to the first order restric-tion (21), and they are represented by the functions ( r ) φ , ( r ) B , ( r ) ψ , ( r ) V , ( r ) ρ , ( r ) p , ( r ) ϕ ,and the remaining gauge freedom which is described by the functions ( r ) ξ N , r = 1 , (1) B, (1) ψ, (1) V, (1) ρ, (1) ϕ transformunder the remaining temporal gauge freedom. From (17a) one obtains the well-known relations: (1) B • = (1) B − H − ξ N • , (1) ψ • = (1) ψ − (1) ξ N • , (23a) (1) V • = (1) V − H − ξ N • , (1) A • = (1) A + ( ∂ N A )( (1) ξ N • ) , (23b)where A = ρ or A = ϕ . By normalizing the perturbations (apart from ψ ) thesetransformation rules can be written in the unified form given in Eq. (3a), which werepeat here: (1) • = (1) − (1) ξ N • , (24)where the kernel represents the following variables in the five different cases: = ψ, = H B, = H V, = ρ ( − ∂ N ρ ) , = ϕ ( − ∂ N ϕ ) . (25)The above normalization ensures that the variables are dimensionless (recall that B and V have dimension length and H has dimension ( length ) − ).We now consider the second order perturbation variables. The second ordergauge transformation formulas, which follow from (17b), can be written so as tohave the same leading order terms as the first order formula (24) but they alsoinclude a source term S that depends quadratically on the first order variables ina sometimes complicated way: (2) • = (2) − (2) ξ N • + S . (26) The tensor mode at second order describes gravitational waves generated by the first orderscalar ( i.e. matter) perturbations.
UNIFIED FORM FOR GAUGE TRANSFORMATIONS TO SECOND ORDER S revealsthat a number of quadratic first order terms can be incorporated into the secondorder terms in the formula (26), leaving much simplified source terms. The resultingunified formula is given by equation (3b), which we repeat here for the reader’sconvenience: (2) ˆ • = (2) ˆ − (2) ˆ ξ N • + 2 (1) ξ N • ∂ N (1) • + rem, • , (27)where stands for one of the variables in (25). The hatted variables are given byequations (1) and (2), which we repeat here: (2) ˆ = (2) + C (1) , (28a) (2) ˆ ξ N • = (2) ξ N • − (1) ξ N • ∂ N ( (1) ξ N • ) . (28b)The details of this unified formulation lie in the coefficients C in (28a) and inthe remaining source terms rem, • in (27). The expressions for these quantities areobtained by comparing the transformation rules obtained from (17b) for each choiceof in (25) with the unified form (27).First, the coefficients C in (28a) are given by: C ψ = 2 , C B = C V = 1 + q, C ρ = ∂ N ρ ∂ N ρ , C ϕ = ∂ N ϕ ∂ N ϕ . (29)For a non-interacting fluid or a non-interacting scalar field, so that energy conser-vation holds in the background: ∂ N ρ = − ρ + p ) , (30)it follows that C ρ = − c s ) , (31a) C ϕ = (1 + q ) − (1 + c s ) = ( w − c s ) . (31b)Second, the remaining source terms rem, • have the following form for the differentchoices of in (25): ψ rem, • = D ( B ) − D ( B • ) , (32a) H B rem, • = ( ∂ N + 2 q ) ( D ( H B • ) − D ( H B ))+ 2 S i [( φ p + φ • ) D i ( H B • ) − ( φ p + φ ) D i ( H B )] , (32b) H V rem, • = 2 S i [ φ v D i ( H V • − H V )] , (32c) ρ rem, • = 0 , (32d) ϕ rem, • = 0 , (32e) The stress-energy tensor of a minimally coupled scalar field is equivalent to that of a perfectfluid, thereby defining an energy density and pressure for the scalar field. This equivalence leadsto the following relation between ϕ and ρ + p : ( ∂ N ϕ ) = a H − ( ρ + p ). For the first equation differentiate (30) and with respect to N and use the definition of c s expressed in terms of N . For the second equation differentiate ( ∂ N ϕ ) = a H − ( ρ + p ) usingthe result of the first equation. One also requires ∂ N ( a H − ) = (1 + q )( a H − ) , which followsfrom (9) and (20). The second equality in (31b) depends on (11). UNIFIED FORM FOR GAUGE TRANSFORMATIONS TO SECOND ORDER p and v stands for the Poisson and total matter gauge, re-spectively, which are defined in section 3. The scalar mode extraction operator S i and the spatial differential operators D and D that appear in equations (32) aredefined in equations (78a) and (79) in appendix B. We note that the temporal gaugeon the right side of equations (32) is unspecified and can be chosen to be one of thestandard gauges.At this stage we introduce the normalized density perturbation ( r ) δ according to ( r ) δ = ( r ) ρ ( − ∂ N ρ ) , r = 1 , , (33)which means that the kernel that is associated with the density perturbationin (25) is given by ( r ) =
13 ( r ) δ . (34)The factor of is included so that if conservation of energy holds in the background( ∂ N ρ = − ρ + p )) then (33) becomes ( r ) δ = ( r ) ρρ + p . (35)In this case the usual fractional perturbed energy density ( r ) δ is easily obtained via ( r ) δ = ( r ) ρρ = (1 + w ) ( r ) δ . (36)For convenience we now explicitly list the normalized source-compensated secondorder variables given by (28a), where the kernels are given by (25) and (34): (2) ˆ ψ = (2) ψ + 2 (1) ψ , (37a) H (2) ˆ B = H (2) B + (1 + q )( H (1) B ) , (37b) H (2) ˆ V = H (2) V + (1 + q )( H (1) V ) , (37c)
13 (2) ˆ δ =
13 (2) δ − c s ) (cid:0)
13 (1) δ (cid:1) , (37d) λ (2) ˆ ϕ = λ (2) ϕ + ( w − c s )( λ (1) ϕ ) . (37e)Here we have introduced the notation λ = − ( ∂ N ϕ ) − , (38)for the scale factor associated with the scalar field in equation (25). We note thatequations (37d) and (37e) depend on the conservation of energy in the background.One feature of equation (27) requires comment. In this equation one can replace (1) • on the right side by (1) using (24), and then modify the definition of (2) ˆ ξ N • by changing the sign in (28b). Although this form of the equation may look more We decided to introduce this shorthand notation because these expressions occur frequentlyin second order cosmological perturbation theory. See appendix B for some notational motivationand historical background concerning these expressions.
PERFORMING A CHANGE OF GAUGE
Equation (27) with (37) and (32) forms the first main result of this paper andprovides the basis for the change of gauge formulas in section 3. For ease of referencewe now write out the unified formula (27) with having the values in (25) and (34): (2) ˆ ψ • = (2) ˆ ψ − (2) ˆ ξ N • + 2 (1) ξ N • ∂ N (1) ψ • + ψ rem, • . (39a) H (2) ˆ B • = H (2) ˆ B − (2) ˆ ξ N • + 2 (1) ξ N • ∂ N ( H (1) B • ) + H B rem, • , (39b) H (2) ˆ V • = H (2) ˆ V − (2) ˆ ξ N • + 2 (1) ξ N • ∂ N ( H (1) V • ) + H V rem, • , (39c)
13 (2) ˆ δ • =
13 (2) ˆ δ − (2) ˆ ξ N • + 2 (1) ξ N • ∂ N (cid:0)
13 (1) δ • (cid:1) , (39d) λ (2) ˆ ϕ • = λ (2) ˆ ϕ − (2) ˆ ξ N • + 2 (1) ξ N • ∂ N ( λ (1) ϕ • ) , (39e)where λ is given by (38) and the remainder terms by (32). However, we need toaugment this set of equations with transformation equations for the metric variable φ , which has to be treated separately since its transformation law involves the timederivative of the gauge field. At first order we have: (1) φ • = (1) φ + ( ∂ N + 1 + q )( (1) ξ N • ) , (40a)and at second order, (2) ˆ φ • = (2) ˆ φ + ( ∂ N + 1 + q )( (2) ˆ ξ N • ) + 2 (1) ξ N • ∂ N (1) φ • + φ rem, • , (40b)where (2) ˆ φ = (2) φ − (1) φ , (40c) φ rem, • = (cid:0) ∂ N (1) ξ N • (cid:1) − ( ∂ N q )( (1) ξ N • ) . (40d)We end this section by noting that there are other ways defining the curvatureperturbation ψ . In this paper we write the scalar part of the perturbed spatial metricas (1 − ψ ) δ ij (which we refer to as the Malik-Wands form, see for example Malikand Wands (2009) [16]), while another choice is an exponential form e − ψ SB δ ij firstintroduced by Salopek and Bond (1990) [22] (see for example, Lyth and Rodriguez(2005) [14], section IIB). Equating the two forms, Taylor expanding the exponentialand performing a perturbation expansion for ψ SB yields (1) ψ SB = (1) ψ and (2) ψ SB = (2) ψ + 2( (1) ψ ) , showing that (2) ψ SB = (2) ˆ ψ . We refer to section 2.1 in Carrilho andMalik (2015) [5] for two other possibilities. Having fixed the spatial gauge (see Eq. (18)), we can now choose a temporal gaugeto second order by setting to zero the first and second perturbations of one thevariables B , ψ , V , δ , ϕ , thereby specifying the gauge uniquely. We use the followingterminology and subscripts to label the gauges: PERFORMING A CHANGE OF GAUGE p , defined by B p = 0,ii) uniform curvature gauge, subscript c , defined by ψ c = 0,iii) total matter gauge, subscript v , defined by V v = 0,iv) uniform density gauge, subscript ρ , defined by δ ρ = 0,v) uniform scalar field gauge, subscript sc , defined by ϕ sc = 0.In order to introduce a specific gauge labelled by • we must determine the transitionfunction (2) ˆ ξ N • using equations (39). (We will not list the expressions for (1) ξ N • belowsince they can easily be read off from the second order equations: replace (2) by (1) , omit the hats and drop the rem terms.) Referring to the above definition of thegauges we choose • = p in (39b), • = v in (39c), • = c in (39a), • = ρ in (39d) and • = s c in (39e), to obtain the following results: ( r ) B p = 0 = ⇒ (2) ˆ ξ N p = H (2) ˆ B + H B rem, p , (41a) ( r ) V v = 0 = ⇒ (2) ˆ ξ N v = H (2) ˆ V + H V rem, v , (41b) ( r ) ψ c = 0 = ⇒ (2) ˆ ξ N c = (2) ˆ ψ + ψ rem, c , (41c) ( r ) δ ρ = 0 = ⇒ (2) ˆ ξ Nρ =
13 (2) ˆ δ , (41d) ( r ) ϕ sc = 0 = ⇒ (2) ˆ ξ N sc = λ (2) ˆ ϕ. (41e) These expressions for the gauge fields at second order represent the second mainresult of this paper.
Their concise form is a consequence of using the hatted variables.The final step is to successively substitute the expressions (41) into (27). Thisimmediately gives the following change of gauge formulas at second order: (2) ˆ p = (2) ˆ − H (2) ˆ B + 2 H (1) B ∂ N (1) p + rem, p − H B rem, p , (42a) (2) ˆ v = (2) ˆ − H (2) ˆ V + 2 H (1) V ∂ N (1) v + rem, v − H V rem, v , (42b) (2) ˆ c = (2) ˆ − (2) ˆ ψ + 2 (1) ψ ∂ N (1) c + rem, c − ψ rem, c , (42c) (2) ˆ ρ = (2) ˆ −
13 (2) ˆ δ + 2(
13 (1) δ ) ∂ N (1) ρ + rem,ρ , (42d) (2) ˆ s c = (2) ˆ − λ (2) ˆ ϕ + 2( λ (1) ϕ ) ∂ N (1) s c + rem, s c , (42e)where the rem, • terms are given by (32), and represents any of the symbols inequation (25) and (34). The gauge on the right side is unspecified and can be chosento be one of the standard gauges. For example if one wishes to transform from thetotal matter gauge to the uniform curvature gauge, one would use the third equationwith subscripts v added on the right side: (2) ˆ c = (2) ˆ v − (2) ˆ ψ v + 2 (1) ψ v ∂ N (1) c + rem, c , v − ψ rem, c , v , (43)with (1) c = (1) v − (1) ψ v , (44) This gauge is naturally only available in a perturbed universe with a scalar field. In thiscontext it is in fact equivalent to the total matter gauge, but it is helpful to give it a separatename. This equivalence is established in a subsequent paper [27].
EXAMPLES The remainder terms are obtained from equations (32) by choosingthe total matter gauge on the right side. In the present example we obtain ψ rem, c , v = D ( B v ) − D ( B c ) . (45)In summary equation (42) , in conjunction with the definition (37) of the hattedvariables, represent the main goal of this paper. They provide an efficient algorithmfor calculating any of the gauge invariants in any of the five gauges, as illustratedin the next section. Although our primary motivation was to simplify and unifythe change of gauge formulas at second order, we note that equations (42) also givea useful overview of the situation at linear order. By replacing (2) with (1) and byomitting the hats and dropping the rem terms one can read off familiar relationssuch as ψ v = ψ p − H V p , ψ p = −H B c , δ v = δ p − H V p , ψ ρ = − δ c , ψ sc = − λϕ c . (46)Finally, we recall that the change of gauge formulas for the metric perturbation φ have to be treated separately and are given by equations (40), with the specific gaugefield to be obtained from equations (41) once the two gauges have been chosen. Anexample at linear order is (1) φ v = (1) φ c + ( ∂ N + 1 + q )( (1) ξ N v , c ) , with (1) ξ N v , c = H V c . (47) In this section we give examples of using the general equations (42) to calculatesecond order gauge invariants of interest in current research in cosmology. Theexpressions we obtain are more concise than those in the literature because of ouruse of the hatted variables and the differential operators D and D . The latterfeature, in particular, simplifies the representation of the terms involving spatialderivatives. In order to make comparisons with the literature it is necessary toexpand our expressions by using the definition (37) of the hatted variables and thedefinition (79) of D and D . The latter definitions lead to the following identitiesthat will be useful when making comparisons: D ( A ) − D ( B ) = S ij [ D i ( A + B ) D j ( A − B )] , (48a) D ( A ) − D ( B ) = ( D S ij − δ ij )[ D i ( A + B ) D j ( A − B )] , (48b)where the scalar mode extraction operator S ij is defined by (78a). We will frequentlychange from e -fold time N to conformal time η using equation (20) in order to makecomparisons with the literature. We note the process of expanding our expressions tomake comparisons with the literature as illustrated in section 4 and in appendix A,can be tedious. We regard this as a measure of how concise our expressions are, andwe emphasize that this is not something one has to do when using our formalism in As with equations (41), the first order formulas can be read off from the second order formulasby inspection, since they correspond to the leading order terms.
EXAMPLES (2) ψ ρ , the second order curvature perturbation inthe uniform density gauge, which is important as a conserved quantity on super-horizon scales. Choosing = ψ in (42d) and using (32a) for the remainder termimmediately gives (2) ˆ ψ ρ = (2) ˆ ψ −
13 (2) ˆ δ +
23 (1) δ ∂ N (1) ψ ρ − D ( (1) B ρ ) + D ( (1) B ) . (49)Equation (49) is a concise version of equation (7.71) in Malik and Wands (2009) [16]. If we choose the arbitrary temporal gauge to be the uniform curvature gauge ( ψ c = 0)equation (49) becomes (2) ˆ ψ ρ = −
13 (2) ˆ δ c +
23 (1) δ c ∂ N (1) ψ ρ − D ( (1) B ρ ) + D ( (1) B c ) , (50)which is a concise version of equation (3.3) in Christofferson et al (2015) [7], whichrelates (2) ψ ρ to (2) δ c , the second order density perturbation in the uniform curvaturegauge. To compare with earlier literature we change from N to η and make thereplacement ( r ) δ = − H /ρ ′ ) ( r ) ρ , which leads to (2) ψ ρ = H ρ ′ ρ c − H ( ρ ′ ) ρ c (cid:0) (1) ρ ′ c + (5 + 3 c s ) H (1) ρ c (cid:1) − D ( (1) B ρ ) + D ( (1) B c ) , (51a)where D ( (1) B ρ ) − D ( (1) B c ) = H − ( D S ij − δ ij )[ D i ( (1) δ c − H (1) B c ) D j (1) δ c ] , (51b)the latter relation following from (48b) and H (1) B ρ = H (1) B c −
13 (1) δ c . We findthat equation (3.3) in [7] agrees with (51). Equation (51) has also been given byCarrilho and Malik (2015) [5] (see equation (3.3)).Our second example concerns the density perturbation at second order in thetotal matter gauge δ v which is used when deriving the generalized Poisson equationin second order perturbation theory in relativistic cosmology (see Hidalgo et al (2013) [10]). Choosing = δ in (42b) and using (32d) for the remainder term weexpress δ v in terms of an arbitrary temporal gauge: (2) ˆ δ v = (2) ˆ δ − H (2) ˆ V + 2 H (1) V ∂ N (1) δ v + 6 S i [ (1) φ v D i H (1) V ] , (52)where the mode extraction operator S i is defined in equation (78a).Expanding our equation, changing from N to η and using (33) leads to (2) ρ v = (2) ρ + 3 ρ ′ V + (1) V [2 (1) ρ ′ v + 3 ρ ′ (1 + c s + (1 + w )) H (1) V ] − ρ ′ S i [ (1) φ v D i (1) V ] . (53)Equation (3.10) in [10] can be simplified to have the form (53) (subject to a fewdiffering coefficients) when we choose the arbitrary temporal gauge on the right side See, for example, Bartolo et al (2010) [3], equation (36). Set E = 0 in their equation to fix the spatial gauge. There are a number of typos. In rearranging the O ( D ) terms one has to use (33), and the definition of S ij . EXAMPLES et al form. As our third example, by choosing = δ in (42c) and using (32d) for theremainder term, we express δ c in terms of an arbitrary temporal gauge: (2) ˆ δ c = (2) ˆ δ − (2) ˆ ψ + 2 (1) ψ∂ N (1) δ c + 3( D ( (1) B c ) − D ( (1) B )) . (54)Expanding our equation, changing from N to η and using (33) leads to (2) ρ c = (2) ρ + ρ ′ H (2) ψ + (1) ψ H (cid:20) (1) ρ ′ + ρ ′ H (cid:0) (1) ψ ′ − H (1 + 3 c s ) (1) ψ (cid:1)(cid:21) − ρ ′ H ( D ( (1) B c ) − D ( (1) B )) , (55a)where D ( (1) B c ) − D ( (1) B ) = H − ( D S ij − δ ij )[ D i ( (1) ψ − H (1) B ) D j (1) ψ ] , (55b)the latter relation following from (48b) and H (1) B c = H (1) B − (1) ψ . Equation (55)agrees with equation (7.35) in Malik and Wands (2009) [16] with the gauge fixed sothat E = 0. Our final example in this section concerns the curvature perturbation in theuniform scalar field gauge (2) ˆ ψ s c which is a conserved quantity on super-horizonscales in a scalar field dominated universe (Vernizzi (2005) [30]). Choosing = ψ in (42e) and using (32a) for the remainder term immediately gives (2) ˆ ψ s c = (2) ˆ ψ − λ (2) ˆ ϕ + 2 λ (1) ϕ∂ N (1) ψ s c − D ( (1) B s c ) + D ( (1) B ) . (56)Expanding our equation leads to ψ s c = (2) ψ − λ (2) ϕ + λ (1) ϕ (cid:2) − λ∂ N (1) ϕ − ( λ∂ N ϕ + 2) λ (1) ϕ + 2( ∂ N + 2) (1) ψ ) (cid:3) − D ( (1) B s c ) + D ( (1) B ) , (57)with λ = − ( ∂ N ϕ ) − . Converting to η as time variable yields equation (30) in [30],when specialized to the long wavelength limit. We note in passing that this trans-formation formula plays a central role in finding a conserved quantity and explicitsolutions at second order, as discussed in the next section and in detail in the followup papers [28] and [29], called UW3 and UW4, respectively, below. Use (1) φ v = (1) φ + (1) V ′ + H (1) V , introduce ( r ) δ = ( r ) ρ/ρ , use 2 S i [ V D i V ] = V , replace 1 + c s using w ′ = 3 H (1 + w )((1 + w ) − (1 + c s )), and assume conservation of energy ρ ′ = − H (1 + w ) ρ . To make the comparison note that H ρ ′′ − ρ ′ H ′ = − ρ ′ H (1 + c s ), and write the spatialderivative terms using our notation D i , D and S ij . Here we have used equation (28a) for λ (2) ˆ ϕ and equation (29) for C ϕ , as well as the first orderrelation (1) ψ s c = (1) ψ − λ (1) ϕ. Note that H ( λ∂ N ϕ + 2) = λϕ ′′ + H ′ + 2 H , where λ = − ( ∂ N ϕ ) − = −H ( ϕ ′ ) − in ournotation. However, Vernizzi uses the convention ˙ f = ∂ η . In addition, since Vernizzi restrictsconsideration to long wavelength perturbations the terms D ( B s c ) − D ( B ), which are O ( D ), donot appear. DISCUSSION The present paper is the first of four closely connected papers dealing with scalarperturbations up to second order. In the present paper, which we will refer to asUW1, we have introduced new second order variables and a new second order gaugevector field, which simplifies the change of gauge formulas at second order, andprovides an efficient unifying algorithm for calculating any gauge invariant in thecommonly used gauges. This is important since it is often desirable to use severalgauges when addressing a given problem. In the second paper, called UW2 [27],we present five ready-to-use systems of governing equations for second order per-turbations. These two papers constitute the foundation for subsequent physicalapplications, illustrated by UW3 [28] and UW4 [29].In UW3 we use the new variables and gauge transformation formulas, and applythem to the equations given in UW2 to produce new dimensionless gauge-invariantconserved quantities and explicit general solutions, containing both the so-calledgrowing and decaying modes, for second order perturbations in the super-horizonregime. This is made possible due to that the dimensionless source-compensated“hatted” second order perturbation variables simplify the analysis of perturbationsin the super-horizon regime in a number of ways. For example, in this regime theperturbed energy conservation equations can be written in the following form interms of hatted variables: ∂ N ( (1) δ − (1) ψ ) ≈ − (1) Γ , (58a) ∂ N ( (2) ˆ δ − (2) ˆ ψ ) ≈ − (2) Γ − (1) Γ ) + 2 ∂ N ( (1) Γ (1) δ ) , (58b)with a simple quadratic source term in the second order equation (see UW3 [28]).Here ( r ) Γ , r = 1 ,
2, are the non-adiabatic pressure perturbations (see UW2 [27]). Itfollows from (58), specialized to the uniform curvature gauge ( ( r ) ψ = 0 , r = 1 , (1) δ c and the hatted second order perturbation (2) ˆ δ c are conserved for adiabaticperturbations ( ( r ) Γ = 0, r = 1 ,
2) in the super-horizon regime. Note, however, thatthe unhatted second order density perturbation (2) δ c is not conserved unless c s isconstant, as follows from (37d). Another example is that when using the uniformcurvature gauge in the super-horizon regime, the perturbed Einstein equations as-sume a particularly simple form when expressed in terms of hatted variables, whichleads to further conserved quantities. In particular if the source is a minimally cou-pled scalar field with an arbitrary scalar field potential we obtain a new second orderconserved quantity for the scalar field, which is used in UW4 [29] to obtain new phys-ical results for scalar fields for second order perturbations in the long wavelengthlimit, without imposing the slow-roll approximation. A Relation with Malik and Wands (2009)
We begin by listing the main differences between Malik and Wands (2009) [16]and the present paper. First, they give a more general framework for the gaugetransformation formulas than we do in that they do not require the perturbationsat first order to be purely scalar. Second, they do not use the gauge freedom to set
RELATION WITH MALIK AND WANDS (2009) C = 0, C i = 0 as we have done ( C and C i are labeled E and F i in their paper).Their generating vector field ξ at first order, given by their equation (6.17), ξ µ = ( α , ∂ i β + γ i ) , (59)is thus more general than ours, which is given by equation (19) (also note thatthey use subscripts to denote the order of the perturbation). As a result of thesedifferences when comparing equations it is necessary to set E = 0, F = 0, S i = 0, h ij = 0 at first order in the metric perturbations (their equations (2.8)-(2.12)),and to set β = 0, γ i = 0 in their generating gauge vector field. Then identify α ≡ (1) ξ η = H − ξ N . Their generating gauge vector field at second order isgeneral, ξ µ = ( α , ∂ i β + γ i ). In accordance with (19), we identify α = (2) ξ η = H − ξ N , β ≡ (2) ξ, γ i = (2) ˜ ξ i . (60)The third difference is in the treatment of the density perturbation. Malik andWands begin with the standard unscaled perturbation expansion (their equation(6.16)): ρ = ρ + ρ + ρ , (61)and do not assume local energy conservation. We, on the other hand, in severalequations assume local energy conservation and then define the scaled perturbations ( r ) δ = ρ r ρ + p , r = 1 , . (62)However, normalizing with ρ + p is equivalent to normalizing with − ρ ′ / (3 H ), whichsuggests a natural generalization in the case local energy conservation does not hold.The fourth difference is in the treatment of the velocity perturbation. Malik andWands begin with the perturbation expansion (their equation (4.4)): u i = a − ( v i + v i ) , v ir = ∂ i v r + ˜ v jr , r = 1 , , (63)of the contravariant spatial components of the 4-velocity, whereas we expand thecovariant components as in (12), using V instead of v as the scalar perturbation ofthe velocity. It follows that v and V are related as follows: V = v + B , (64a) V = v + B − S i [ φ D i B + 2 ψ D i v ] . (64b)We are using the same letters for the scalar metric perturbations, namely ( φ, B, ψ )as Malik and Wands. Their transformation laws for these variables are given byequations (6.37)-(6.39) at first order and (6.47), (6.51) and (6.58) at second order.In order to facilitate a comparison we write our formulas using the Malik and Wandsvariables and notation. For (2) φ , given by equation (40b), we use (60) to obtain˜ φ = φ + α ′ + H α + α ( α ′′ +5 H α ′ +( H ′ +2 H ) α +2 φ ′ +4 H φ )+2 α ′ ( α ′ +2 φ ) . (65) Expand the relation u a = g ab u b . Here we are using the Malik and Wands subscript conventionto indicate the order of the perturbation. RELATION WITH MALIK AND WANDS (2009) (2) ψ , given byequation (39a), we use (60) to obtain˜ ψ = ψ − H α − α (cid:0) H α ′ + ( H ′ + 2 H ) α − ψ ′ − H ψ (cid:1) + ( D S ij − δ ij )[ D i (2 B − α ) D j α ] . (66)To obtain agreement with Malik and Wands we write their (6.58) in the form˜ ψ = ψ − H α + ( D S ij − δ ij ) χ ij , (67)where χ ij is given by (6.54) specialized to scalar perturbations at first order bysetting C ij = − ψδ ij , B i = D i B and ξ k = 0, which yields χ ij = 2 α (cid:0) H α ′ + ( H ′ + 2 H ) α − ψ ′ − H ψ (cid:1) δ ij + 2( D i B D j α + D i α D j B − D i α D j α ) . (68)Substituting (68) in (67) yields (66).For (2) B , given by equation (39b), we use (60) to obtain˜ B = B − α + β ′ − α ( α ′ + H α ) − S i [ − φ D i α + α D i ( B ′ + H B ) + ( α ′ + H α ) D i ( B − α )] , (69)where we have introduced the Malik-Wands notation β = (2) ξ , which is given by β = −S ij [ D i (2 B − α ) D j α ] . (70)To obtain agreement with Malik and Wands we write their (6.50) in the form˜ B = B − α + β ′ + S i χ Bi , (71)where β is given by (6.59) with the spatial gauge fixed so that ˜ E = 0 = E , β = − S ij χ ij , (72)and where χ Bi is given by (6.49) specialized to scalar perturbations at first order bysetting B i = D i B , and ξ k = 0, which yields χ Bi = − D i [ α ( α ′ + H α )]+2[ − φ D i α + α D i ( B ′ + H B )+( α ′ + H α ) D i ( B − α )] . (73)Note that (72) with (68) yields (70). Substituting (73) in (71) yields (69).We now consider the matter perturbations. For (2) V the “Malik and Wandsform” of our equation (39c) is as follows:˜ V = V − α − α ( α ′ + H α ) + 2 S i [ − φ D i α + α D i ( V ′ + H V )] . (74)To compare with Malik and Wands we need to use equations (64) and (69) to obtainthe transformation law for v :˜ v = v − β ′ + 2 S i [ α D i ( v ′ − H v )] , (75) SPATIAL DIFFERENTIAL OPERATORS β is given by (70). This agrees with Malik and Wands equations (6.27) and(6.28) apart from a differing sign in (6.27). Finally, for (2) δ the un-scaled form of our equation (39d) in Malik and Wandsnotation is as follows:˜ ρ = ρ + ρ ′ α + α ( ρ ′′ α + ρ ′ α ′ + 2 ρ ′ ) , (76)which agrees with Malik and Wands equation (6.20) after setting ξ i = 0. B Spatial differential operators
In this appendix we introduce the spatial differential operators that are used in thispaper. First, we require the spatial Laplacian and the trace-free symmetric secondorder derivative: D := γ ij D i D j , D ij := D ( i D j ) − γ ij D . (77)We use these to define the scalar mode extraction operators (see [25], equations(85)): S i = D − D i , S ij = ( D − ) D ij , (78a)where D − is the inverse spatial Laplacian. Note that S i is the inverse operator of D i and S ij is the inverse operator of D ij : S i D i C = C, S ij D ij C = C. (78b)We can now define the following spatial differential operators:( D C ) := γ ij ( D i C )( D j C ) , (79a) D ( C ) := S ij ( D i C )( D j C ) , (79b) D ( C ) := (cid:0) D D ( C ) − ( D C ) (cid:1) , (79c)which act on an arbitrary function C .The operator ( D C ) is familiar, being the square of the magnitude of the gradient D i C , whereas the other two are less so. The expression D ( C ) can be viewed as thescalar mode of the rank two tensor ( D i C )( D j C ) while D ( C ) is defined by takingthe Laplacian of D ( C ). We mention one property of these operators that suggeststheir physical role. When taking limits such as the long wavelength limit and theNewtonian limit it is essential to count how quantities change under a scaling of thespatial coordinates. More precisely, if some expression L ( D i ) involving D i scales as L ( λ D i ) = λ p L ( D i ) under a rescaling of spatial coordinates x i → λ − x i , we say that L ( D i ) has weight p in D i . It follows from equations (78a) and (79) that D ( C )has weight 0 while D ( C ) has weight 2 in D i . We thus expect that D ( C ) will be We find that after setting ξ i = 0 (6.27) should read χ v i = 2 α ( v ′ i − H v i ). If the background is not flat, S ij = D − ( D + K ) − D ij . Note that D and D ij both have weight 2 while in contrast S i and S ij have weights − −
2, respectively.
SPATIAL DIFFERENTIAL OPERATORS D ( C ) will be negligible in the long wavelength limit, while the reversewill be true in the Newtonian limit.In this paper the operators D and D serve to simplify the quadratic sourceterms in the gauge change formulas at second order. They play a similar role inother source terms, for example the source terms of the perturbed Einstein tensor,and hence in the solutions of the perturbed Einstein equations at second order. Thefact that these operators occur frequently motivates our choice of notation: thesymbol D suggests a spatial differential operator acting on an arbitrary function C ,while the subscript or indicates the weight in D i . The use of D i in general, ratherthan ,i also helps to clarify the structure of the spatial derivative terms.It turns out that the operators D and D , as defined in (79), are related totwo quantities, denoted Ψ and Θ , that have been used in the literature on secondorder perturbations since 1997. These quantities were defined in the case of a flatbackground by Mollerach and Matarrese (1997) [19] as follows: Ψ := D − (cid:0) D i D j C D i D j C − ( D C ) (cid:1) , (80a)Θ := D − (cid:0) Ψ − ( D C ) (cid:1) , (80b)where C is a function that determines the spatial dependence of the perturbations.The explicit relations are simple, namelyΘ = − D ( C ) , Ψ = − D ( C ) , (81)but their derivation requires the use of some complicated identities satisfied by D i ,as follows. Expand the second derivatives on the left sides to get D i D j ( D i C D j C ) = 2( D i C ) D i ( D C ) + ( D i D j C )( D i D j C ) + ( D C ) , (82a) D ( D C ) = 2( D i C ) D i ( D C ) + 2( D i D j C )( D i D j C ) . (82b)Next take the difference and replace D i D j by D ij on the left side to obtain: D ij ( D i C D j C ) − D ( D C ) = − ( D i D j C )( D i D j C ) + ( D C ) , (83)which is the key identity. The desired relations (81) follow immediately from (83)on using the definitions (79) and (80).Although Θ and Ψ were first introduced to describe second order perturbationsof the Einstein-de Sitter universe, since 2005 they have also been used to describeperturbed ΛCDM universes. See, for example, Bartolo et al (2005) [2] followingequation (9), Tomita (2005) [23], equations (2.11), Villa and Rampf (2016) [31],equations (5.21) and (5.38), and Tram et al (2016) [24], equations (D.11) and (D.12).In these references one sees that Θ ( i.e. D ( C )) contributes to perturbations onsuper-horizon scales, while D Ψ ( i.e. D D ( C )) contributes to the Newtonian partof the second order density perturbation. As mentioned earlier, this physical inter-pretation is a consequence of the fact that D ( C ) is of weight zero in D i while See equation(3.7) for Ψ and equation (3.11) for Θ in [19]. See also Materrese et al (1998) [18],equations (4.36) and (6.6). In a suggestion of physical interpretation Villa and Rampf (2016) [31] (see page 15 followingequation (5.39)) refer to Θ as the ”GR kernel” and to Ψ as the “Newtonian kernel.” EFERENCES D ( C ) is of weight two and D D ( C ) is of weight four. The important point, how-ever, is that D ( C ) and D ( C ) are not restricted in use to the ΛCDM universes:they arise in the general change of gauge formulas in this paper, in the source termsof the second order perturbations of the Einstein tensor, and D ( C ) contributes toperturbations on super-horizon scales in general. For example, D ( ψ p ) contributesto the CMB anisotropy at second order on large scales. References [1] N. Bartolo, E. Komatsu, S. Matarrese, and A. Riotto. Non-Gaussianity frominflation: theory and observations.
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