1 Introduction

Effective perfect fluids in cosmology

Guillermo Ballesteros and Brando Bellazzini

Museo Storico della Fisica e Centro Studi e Ricerche Enrico Fermi,

Piazza del Viminale 1, I-00184 Rome, Italy

Via Marzolo 8, I-35131 Padova, Italy

SISSA, Via Bonomea 265, I-34136 Trieste, Italy

Université de Genève, Department of Theoretical Physics and

Center for Astroparticle Physics (CAP),

24 quai E. Ansermet, CH-1211 Geneva 4, Switzerland

We describe the cosmological dynamics of perfect fluids within the framework of effective field theories. The effective action is a derivative expansion whose terms are selected by the symmetry requirements on the relevant long-distance degrees of freedom, which are identified with comoving coordinates. The perfect fluid is defined by requiring invariance of the action under internal volume-preserving diffeomorphisms and general covariance. At lowest order in derivatives, the dynamics is encoded in a single function of the entropy density that characterizes the properties of the fluid, such as the equation of state and the speed of sound. This framework allows a neat simultaneous description of fluid and metric perturbations. Longitudinal fluid perturbations are closely related to the adiabatic modes, while the transverse modes mix with vector metric perturbations as a consequence of vorticity conservation. This formalism features a large flexibility which can be of practical use for higher order perturbation theory and cosmological parameter estimation.

## 1 Introduction

The success of effective field theories for studying physical systems comes from the fact that they allow to capture in a single picture the universal long distance (low energy) properties of models that are instead intrinsically different at much shorter scales. Since cosmological problems are often characterized by well separated scales, the language of effective field theories provides a powerful tool for the study of cosmological evolution, in particular inflationary and dark energy dynamics. Most of the works on cosmological evolution that are based on effective theories can be broadly classified in two different categories, depending on whether they aim to describe the full (effective) action [1, 2, 3, 4, 5, 6] or, conversely, focus on the dynamics of the perturbations around some background [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. The formalism we develop in this work pertains to the former class but, as we will see, gives a general and straightforward effective expansion for cosmological fluid perturbations on a Friedmann-Lemaître-Robertson-Walker (FLRW) metric (and may also be extended to other backgrounds).

There has recently been a renewed interest in the understanding of fluid dynamics from an effective field theory point of view [31, 32, 33, 34, 35, 36, 37, 38]. This general approach is based on the identification of the relevant long-distance degrees of freedom and their symmetries, possibly including spacetime symmetries such as Galileo, Poincaré, or diffeomorphism invariance. The dynamics is organized in a systematic derivative expansion that makes the theory predictive at low energies. This formalism (or variations of it) has already been applied to describe some aspects of perfect fluids and superfluids [34, 35, 36, 38]. However, although fluids are ubiquitous in cosmology, the cosmological applications of this framework have been, to the best of our knowledge, largely neglected111See however [39] for an application of this formalism to a Lorentz violating dark matter model.. In this work we describe the basic set-up for perfect fluids, adopting the general principles of effective field theory to describe long-distance relativistic dynamics in cosmology and their interplay with metric perturbations.

Neglecting chemical potentials, we consider perfect (dissipationless) fluids that carry no conserved charge. We thus focus on minimal fluids that involve only three degrees of freedom associated with the position in space of a fluid element. These degrees of freedom can be identified with comoving coordinates and, as we will see, the symmetries of the fluid determine the operators that appear in the action. In particular, perfect fluids are invariant under internal spatial diffeomorphisms that preserve the volume. As a result, the relativistic dynamics is fully described at lowest order in derivatives by a single operator. By construction, these fluids support both longitudinal (compressional) and transverse (vortices) excitations, each with its own dynamics. We discuss perturbation theory for a fluid with an arbitrary equation of state, and describe the coupling to scalar, vector and tensor metric perturbations. Vector metric perturbations mix with vortices, while adiabatic perturbations correspond to the compressional modes of the fluid. Our formalism can cover and extends other frameworks (e.g. theories, where the dynamics of fluid vortices is instead absent).

Following the formulation developed in [31, 32, 33, 34], which can be connected to earlier works [40, 41, 42, 43, 44, 45], we extend the effective theory of perfect fluids to generic metric backgrounds in Section 2. We then focus on the FLRW case in Section 3, and move on to discuss matter and metric perturbations in Sections 4 and 5. Adiabatic perturbations are introduced in Section 6; and the conclusions and future work directions are presented in 7. In the Appendix A we review perfect fluids in the context of relativistic hydrodynamics. In the Appendix B we comment on the relation between Eulerian and Lagrangian formulations for fluids, which we both use through the text. We discuss vorticity conservation using ADM variables [46] in Appendix C.

## 2 The effective theory of perfect fluids

In this section we will closely follow the formulation of perfect fluids of [31, 32, 33, 34] which recasts in the modern language of the effective field theories some earlier results of the pull-back approach [41, 42, 43, 44, 45] which is based on the action formalism pioneered in [47, 48].

A perfect fluid is described by a functional of three spacetime scalar functions with  , that define, at any time, an isomorphism between the three-dimensional coordinate space of an observer and a continuum of points (the fluid) [31, 41]. These functions label a generic fluid element222We are focusing on mechanical fluids that at each point in space are fully described by three degrees of freedom. See Appendix B for more details and the relation between these fluids and the continuum limit of relativistic uncharged point-like particles. in , whereas the map gives the position in real space of the fluid element at a given time . In other words, is the trajectory of the fluid element . This corresponds to a Lagrangian description of the dynamics, see Figure 1. Vice versa, the inverse map returns the fluid element that is sitting in at the time , providing an Eulerian description of the dynamics. In a stationary background state, the isomorphism can be chosen such that .

Since both and are isomorphic to , there is always a change of spacetime coordinates such that so that can be naturally interpreted as the comoving coordinates of the fluid. This means that their variation along the fluid four-velocity (being the proper time) is zero

 uμ∂μΦa(x,τ)=0,uμuμ=−1. (2.1)

As we will see next, these two conditions characterize completely the fluid four-velocity, once the symmetry properties of the system have been chosen. Indeed, the whole structure of the effective theory is fully determined by the symmetries of the fluid.

Since the fluid must be homogeneous and isotropic, the internal coordinates have to satisfy the symmetries

 Φa⟶ Φa+ca (2.2) Φa⟶ RabΦb,R∈SO(3) (2.3)

where and the matrix of elements are constant in space and time. In a homogeneous and isotropic model of the universe, even though the background solution , which represents the ground state of the system, spontaneously breaks spatial translations and rotations, the diagonal combination of internal (acting on ) and space (acting on ) symmetries is left unbroken [31]. These diagonal symmetries ensure that the perturbations (or in other words, the excitations of the fluid) propagate in a homogeneous and isotropic background.

In addition, we demand invariance under volume preserving spatial diffeomorphisms

 Φa⟶fa(Φ),det(∂fa∂Φb)=1 (2.4)

that distinguish a perfect fluid from a gel (or jelly), which is a homogeneous and isotropic solid [31]. It is clear that the spatial volume preserving diffeomorphisms (2.4) include the symmetry transformations (2.2) and (2.3), but we have highlighted the latter two because of their clear geometrical meaning.

The low energy effective theory for a perfect fluid is given by a Lagrangian organized in a derivative expansion. Since the dynamics is invariant under spacetime diffeomorphisms and the transformations (2.2), (2.3) and (2.4), the lowest order Lagrangian must be a function of the determinant of the matrix , whose elements are given by [40, 31]

 Bab≡gμν∂μΦa∂νΦb. (2.5)

The effective action that describes the low energy dynamics of the fluid is then

 Sm[Φ]=∫d4x√−gLm (2.6)

where the Lagrangian (density) is a function of the determinant of

 Lm= F(b) (2.7) b≡ √detB (2.8)

that encodes the long distance properties of the fluid. We are implicitly assuming that there are no extra symmetries that could forbid any possible Lagrangian . Otherwise, the low energy Lagrangian would start at the next order in derivatives: , where is an arbitrary function of [33]. However, this term can be recast into a higher derivative term by a field redefinition [33]. In the following we will consider only the lowest order Lagrangian (2.7).

The equations of motion that come from (2.7) are

 ∂μ[√−ggμνb Fb(B−1)cd∂νΦd]=0 (2.9)

where denotes the derivative of with respect to . The gravitational energy-momentum tensor of the system is

 Tμν=−2√−gδSmδgμν=gμνF−bFb(B−1)cd∂μΦc∂νΦd. (2.10)

This corresponds to the energy-momentum tensor of a perfect fluid

 Tμν=(ρ+p)uμuν+pgμν (2.11)

whose components can be easily identified. The conditions (2.1) determine the fluid four-velocity, which is [31]

 uμ=−16b√−gϵμαβγϵabc∂αΦa∂βΦb∂γΦc. (2.12)

Contracting with (and recalling that ) one gets, in agreement with early works on the action formalism for perfect fluids [47, 48], that

 ρ=−F (2.13)

and therefore the fluid is not only perfect but also isentropic [42]. For a physical interpretation of see Appendix B. Finally, from the trace we also obtain the pressure

 ρ+p=−bFb. (2.14)

The fluid is thus barotropic because the pressure depends only on the energy density given that and are both functions of only.

We also find that

 hμν≡(B−1)ab∂μΦa∂νΦb=gμν+uμuν (2.15)

is the standard projector on hypersurfaces orthogonal to the four-velocity of the fluid.

Notice that for dust (pressureless matter) and so , while for radiation and . In general, the relation between the background energy density and pressure is, as usual, given by the equation of state, which is defined as

 ¯p=w¯ρ (2.16)

and therefore using (2.13) and (2.14) this gives

 w=−1+¯b¯Fb¯F=−1+¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯dlogFdlogb. (2.17)

The current333We fix the normalization of such that the temperature in (2.21) matches the standard thermodynamics normalization, .

 Jμ=−buμ (2.18)

is covariantly and identically (i.e. off-shell) conserved

 Jμ;μ=0 (2.19)

and it is identified with the entropy current [37, 33]. The comoving entropy density becomes

 s=b=ρ+pT (2.20)

where is the fluid temperature, given by

 T=−Fb. (2.21)

Starting from the entropy current it is possible to construct infinitely many other currents [43] which are identically conserved, , because are comoving coordinates satisfying the equation (2.1). These currents do not define an independent flow since they are all aligned with the entropy current (and thus the fluid four-velocity). Therefore, the associated charge transfer occurs only along the direction of mechanical fluid displacement. To each of these currents one can associate a chemical potential generalizing the equation (2.20) to , including the contribution from the comoving charge density . In this work we focus on fluids that carry none of these comoving charges, except for the entropy; so all chemical potentials vanish whereas . While the approximation of vanishing chemical potentials is a good one for many cosmological applications, finite charge densities can be easily incorporated by allowing non-vanishing chemical potentials [43, 33].

The invariance under spatial diffeomorphisms (2.4) gives rise, via Noether’s theorem, to another set of infinite (on-shell) conserved currents [31]

 Jμ(ε)=−bFb(B−1)cd∂μΦdεc(Φ)Jμ(ε);μ=0 (2.22)

where is an arbitrary transverse function of

 ∂aεa(Φ)=∂εa∂Φa=0. (2.23)

This constraint, generically solved by , ensures that is an infinitesimal volume preserving diffeomorphism. In the Appendix C we explicitly construct the associated conserved charges and comment on their relation with the vorticity conservation.

## 3 Effective perfect fluids in FLRW

We have seen that the perfect fluid form of the energy-momentum tensor (2.10) comes from imposing the conditions of homogeneity, isotropy and invariance of the action under volume preserving diffeomorphisms of the internal coordinates. In cosmology, the first two assumptions directly lead to the FLRW metric. In fact, the FLRW metric is a purely geometric consequence of requiring that the universe appears homogeneous and isotropic to free falling observers [50, 51]. For simplicity we assume from now on that the background metric of the universe is of flat FLRW type and work with conformal time ,

 ds2=−a2(τ)(dτ2+δijdxidxj). (3.1)

The usual equations of motion that govern the background cosmology in the metric (3.1) simply read:

 H2=8πG3a2¯ρT˙H=−4πG3a2(¯ρT+3¯pT) (3.2)

where  . Newton’s gravitational constant is denoted by and the quantities and represent the total background energy density and pressure if several fluids are present. Let us point out that if there are several fluids that only interact among themselves through gravity, each of them satisfies the following Friedmann equation for the background

 ˙¯ρα+3(1+wα)H¯ρα=0. (3.3)

From the equations (3.2) and the expressions (2.13) and (2.14) one sees that all self-accelerating solutions (which require ) have to satisfy the condition

 3¯b¯FTb<2¯FT (3.4)

where is the function (2.7) that describes the total energy density of the fluid admixture. Notice also that (2.17) implies that a perfect fluid will have an equation of state that is close to that of a cosmological constant () if .

Given the FLRW metric (3.1), the matrix (2.5) reads

 B=1a2(∂Φ)T(I−v⊗v)∂Φ (3.5)

where is the identity matrix, is a matrix of elements , and we have defined

 (∂Φ)ai≡∂iΦa,v≡−(∂ΦT)−1˙Φ (3.6)

where the dots denote derivatives with respect to conformal time . Recalling that the total time derivative of vanishes, i.e.

one recognizes in (3.6) as the (Lagrangian) velocity

 vi=˙xi (3.8)

In the language of fluid dynamics, the equation (3.7) means that the material (or convective) derivative of is zero, which is nothing else than the statement that the label of a fluid element does not change along its trajectory, accordingly with the interpretation of as the comoving coordinates. This is equivalent to the first of the conditions (2.1). In fact, from the definition of the four-velocity, , we get, consistently with equation (3.7),

 u0=dτdη=1aγui=dxidη=dxidτdτdη=1aγvi (3.9)

where .

The (square root of the) determinant and the inverse of (3.5) are then given by

 b=1a3det(∂Φ)√1−v2 (3.10)
 B−1=a2∂Φ−1(I+11−v2v⊗v)(∂ΦT)−1. (3.11)

The equilibrium solution of the fluid in the FLRW background is given by and therefore

 ¯Bij=1a2δij¯b=1a3¯vi=0. (3.12)

Moreover, the conserved currents can be explicitly expressed in a simple form

 J0(ε)= (ρ+p)εa(Φ)∂xi∂Φavi1−v2 (3.13) Ji(ε)= (ρ+p)εa(Φ)∂xj∂Φa(δij+vivj1−v2), (3.14)

which is suitable for a Lagrangian formulation (see appendix B) of the conserved charges

 Q(ε)=−∫d3xa3Jμ(ε)nμ=∫d3Φb(ρ+p)εa(Φ)∂xi∂Φaui (3.15)

where is the unit normal vector to constant time hypersurfaces. As expected, the background carries, apart from the entropy, no charges: . The ground state configuration is thus neutral but supports charge excitations.

Notice that among the various solutions of the transversality constraint (2.23) there exists a class [31], with and arbitrary constants, such that the integral in (3.15) gives rise to a time independent exact 2-form (living in the internal three-dimensional -space):

The conserved charges associated to this class of solutions of (2.23) are nothing but the coordinates of the dual 1-form: . In turn, the circulation defined as the flux of over an arbitrary surface in the internal -space

is conserved on-shell

 ˙V=0. (3.18)

This result represents the generalization to FLRW metrics of the standard non relativistic Kelvin’s circulation theorem. Notice that the factor reduces to in the non relativistic limit for a system of particles of equal mass and number density , where is the entropy density (as discussed in the previous section).

## 4 Perturbations

In this section we present the basic set-up for the study of fluid perturbations within this formalism. We will now focus on fluid perturbations alone and include metric fluctuations in Section 5. For simplicity, we assume that there is a single fluid, since the generalization to an arbitrary number of non-interacting components is straightforward. The starting point is the background state444There actually exist infinitely many background states defined by the constant (in space and time) parameter which fixes the background value of . Instead of keeping track of all the various factors of by expanding around one of these vacua, e.g. by using , we simply work with and recover the general expressions valid also for by expressing the physical results in terms of the corresponding generic background values , , , etc. For example, two fluids described by the same function but different backgrounds ’s, depending on the values of the ’s, will support perturbations with different speeds of sound which, nevertheless, are still expressed by (4.6) that is just evaluated into different background values., , to which we add small independent fluctuations in each space direction

 Φa=xa+πa. (4.1)

The natural expansion parameters are and . Physically, this means that the actual fluid cannot be described only using the background comoving coordinates because there are small inhomogeneities at each point which effectively break the symmetry properties of the system. Recalling the introduction in Section 2, this assignment of coordinates corresponds to switching from the isomorphism between the system of coordinates and the unperturbed background fluid , to another one in which the target space is now the actual imperfect fluid.

We have

 ∂Φ=I+∂π˙Φ=˙πdet∂Φ=1+∂iπi−12∂iπj∂jπi+12(∂iπi)2+… (4.2)

and expanding the determinant of the matrix up to second order in we get

 b=¯b(1+∂iπi−12˙π2−12∂iπk∂kπi+12∂iπi∂jπj+…) (4.3)

where . Therefore, after eliminating total derivatives, the action for the perturbations is

 Sπ=∫d4x√−g(L(2)π+…) (4.4)

with

 Lπ(2)=−¯b¯Fb2(˙π2−c2s(∂iπi)2) (4.5)

being the quadratic part of the Lagrangian for perturbations, where we have defined the speed of sound:

 c2s=¯b¯Fbb¯Fb. (4.6)

In the case of dust , whereas for radiation the speed of sound reaches, as expected, the value . The no-ghost condition is simply  . Recalling (2.14), this inequality is equivalent to . Imposing also that gives in turn . Notice that at this level, neglecting metric fluctuations, the dynamics of the perturbations (as well as that of the background) is entirely given in terms of , and . Higher order derivatives enter only in the self-interaction terms.

Let us also point out that the contribution of to the action, the linear piece of the Lagrangian , vanishes because it gives rise to a total derivative that we can omit in (4.4), since the background solution depends just on time by construction.

It is straightforward to link the four-velocity of the fluid to the fields that we have introduced in (4.1). This can be done at any desired order in by solving iteratively the equations (2.1), which are sufficient to determine the four-velocity up to a sign. At second order in (and choosing to be positive) the result is

 u0=1a(1+12˙π2)ui=1a(−˙πi+˙πk∂kπi). (4.7)

Notice that , as defined in (3.6), can be expanded as

 v=−(I−∂π)˙π+… (4.8)

which is consistent with the expression  , by comparison with (4.7). From the last expression we recover (4.7) by neglecting , which is irrelevant up to third order in .

In the following, we shall discuss the transverse and longitudinal modes of the Lagrangian (4.5) by decomposing the coordinate perturbation:

 π=πL+π⊥∇⋅π⊥=0∇×πL=0 (4.9)

such that the second order Lagrangian for can be written as

 Lπ(2)=12(¯ρ+¯p)(˙π2⊥+˙π2L−c2s(∇⋅πL)2). (4.10)

The generalization to fluids with different density functions is simple. Each fluid will be described by internal coordinates which can be perturbed independently using different  . If we assume that the fluids do not interact directly among themselves, which means that the Lagrangian does not contain terms mixing different sets of coordinates, the generalization of (4.10) is just

 Lπ(2)=12¯ρTN∑α=1(1+wα)Ωα(˙π2α⊥+˙π2αL−c2αs(∇⋅παL)2) (4.11)

where we define the relative background energy density of each fluid as the following time function:  . As we will see later, when we introduce metric perturbations, the specific case in which the are the same for two or more fluids is of particular interest because it describes adiabatic perturbations between those species. We end this section writing the equations of motion for transverse555The equation (4.12) corresponds to the equation of vorticity conservation, see equation (3.16). In fact, at this order in , we have .

 ddτ(a4(¯ρα+¯pα)˙πiα⊥)=0 (4.12)

and longitudinal modes

 ¨πiαL+(1−3c2αA)H˙πiαL−c2αs∂i(∇⋅παL)=0i={1,2,3}α=1,…,N. (4.13)

In this equation represents the standard adiabatic speed of sound of the fluid with label , which for this type of fluids is simply the ratio between the time variation of the background pressure and energy density:

 ˙¯pα=c2αA˙¯ρα. (4.14)

In general, the adiabatic speed of sound is defined in hydrodynamics as the derivative of the pressure with respect to the density. In our case:

 c2αA≡dpαdρα∣∣∣b=¯b. (4.15)

This definition reduces to (4.14) for a fluid given by (2.7). Using the equations (2.13) and (2.14), it is straightforward to check that, in this case, the adiabatic speed of sound actually coincides with the speed of sound that we introduced in (4.6):

 c2αA=c2αs (4.16)

for any fluid of the form (2.7). This shows that the only possible speed of sound of a perfect fluid defined by the Lagrangian (2.7) is adiabatic. As we will see later, the longitudinal modes, which (in absence of metric perturbations) are given by the evolution equation (4.13), are intrinsically related to the adiabatic modes of the fluid.

If we use the expression (4.3) to expand at first order the energy density and the pressure of a fluid defined by (2.7), we find that the corresponding perturbations are

 δMρ=(¯ρ+¯p)∇⋅πLδMp=(¯ρ+¯p)¯b¯Fbb¯Fb∇⋅πL (4.17)

in agreement with (4.16). In these formulas we have introduced the subscript to indicate that these are the perturbations coming exclusively from the variation of the matter part of the action (2.6). In the next section we include metric fluctuations to achieve a complete description of the perturbations. The equations (4.17) will turn out to be modified by a correction coming from the metric inhomogeneities; see equation (5.16).

## 5 Including metric perturbations

Assuming that the theory of gravity is General Relativity666The formalism can also be applied to theories of modified gravity. (GR) the full action takes the form

 S=SEH+Sm (5.1)

where the first piece is the usual Einstein-Hilbert action

 SEH=116πG∫d4x√−gR (5.2)

and the second one is the matter part of , which for each perfect fluid of the type we have been studying is given by (2.6).

Let us now focus on  , with the aim of finding the terms that constitute the direct interaction between metric and matter perturbations. A straightforward way of doing this at any desired order is to expand the action using a functional series. We expand first with respect to the metric variations taking an arbitrary matter configuration

 Sm[φ,gμν+δgμν]= Sm[φ,gμν]+12∫d4x√−gTμνδgμν(x)+… (5.3)

where globally represents all matter fields (e.g. dark matter, dark energy, etc) and is the gravitational energy-momentum tensor. Expanding now the equation (5.3) around a matter background we end up with the following action for the matter-gravity coupling at linear order in metric perturbations (but all orders in the matter fields)

 Sm⊃12∫d4x√−gδMTμνδgμν(x) (5.4)

where is the variation of induced by the matter perturbations. The computation of the matter-metric mixing terms is then very simple provided that we know the form of the energy-momentum tensor. In particular, for a perfect fluid (2.7), this is just given by (2.11). We will now see in detail how this works in the conformal Newtonian gauge.

In the following part of this section we focus only on scalar perturbations that mix with longitudinal modes. Vector and tensor metric perturbations will be discussed in Section 5.1. In the conformal Newtonian gauge the perturbed FLRW metric is then diagonal:

 ds2=a2(−(1+2ψ)dτ2+(1−2ϕ)δijdxidxj) (5.5)

If the universe did not contain any imperfect fluids at all, we would have at linear order in the equations of motion. However, we want to include the more general possibility that some fluids with anisotropic stress could also be present and therefore we will treat these two potentials as distinct variables. This, for instance, happens at very early times when the anisotropic stress of neutrinos cannot be neglected.

Let us consider the energy momentum tensor of some species in their rest frame. Using that  , we find the following expression for the (lowest order in metric fluctuations) coupling between matter and metric perturbations (of scalar type)

 12√−gδMTμνδgμν=−a4(ψδMρ+3ϕδMp) (5.6)

where the fluctuations in the energy density and the pressure include all orders. Notice that for a perfect fluid defined by (2.7) this coupling gives the following contribution to the action for fluctuations at first order in the metric perturbations

 Sm⊃−∫d4xa4(ψ+3c2sϕ)¯ρδM (5.7)

where denotes the intrinsic (not metric) relative energy density perturbation at all orders and is given by (4.6). For linear matter perturbations of such a fluid we can use (4.17) and therefore the mixing between matter and metric perturbations is

 Sm⊃S(2)m−g=−∫d4xa4(¯ρ+¯p)(ψ+3c2sϕ)∂iπiL (5.8)

which shows that metric fluctuations and transverse modes do not couple at this order.

As a check of these results, we perform an explicit independent calculation by writing the square root and in the perturbed metric (5.5). The matrix has coefficients

 a2Bij=−(1+2ψ)−1˙Φi˙Φj+(1−2ϕ)−1∂kΦi∂kΦj (5.9)

and the square root of its determinant is

 b=1a31(1−2ϕ)3/2det(∂Φ)√1−1−2ϕ1+2ψv2 (5.10)

where we recall that the definition of the coordinate velocity is (3.6) and is still given by (4.2). In addition, the metric determinant is

 √−g=a4√(1+2ψ)(1−2ϕ)3 (5.11)

so that the mixing term at linear order in the metric perturbations is

 √−gLm⊃a4(ψ−3ϕ)δF+3a4ϕδ(bFb) (5.12)

which, using (2.13) and (2.14), is in agreement with the formulas above.

We can easily compute all (metric and matter) second order terms that come from the matter action (2.6) by simply doing a Taylor series expansion. The result is

 S(2)m= 12∫d4xa4(1+w)¯ρ[˙π2−c2s(∂iπiL)2−2(ψ+3c2sϕ)∂iπiL] +12∫d4xa4¯ρ[ψ2+3(w−3(1+w)c2s)ϕ2−6wϕψ]. (5.13)

The first line in this expression corresponds to the mixing term (5.8) that we have just computed plus the piece that comes from (4.10), which is the purely matter part of the action that we already obtained by neglecting metric perturbations. The second line of (5) is the contribution of the matter action (2.6) to the action of the metric fluctuations. In order to get the full action for perturbations at second order we have to complete the metric part by perturbing the Einstein-Hilbert action (5.2). This can be found, for any gauge, in [52].

The generalization of (5) to fluids of this kind, interacting only via gravity, is straightforward. We just have to sum the individual actions of the different components. Then, we can easily write down the equations of motion for the transverse and longitudinal modes of each component. The transverse modes are unaffected by the scalar metric perturbations at this order and their equations of motion are still given by (4.12):

 ddτ(a4(¯ρα+¯pα)˙πα⊥)=0α=1,…,N. (5.14)

On the other hand, the equation (4.13) for the longitudinal modes gains a contribution from the metric perturbation, becoming:

 ¨παL+H(1−3c2sα)˙παL−c2sα∇(∇⋅παL)−∇ψ−3c2sα∇ϕ=0α=1,…,N. (5.15)

The expressions in (4.17) for the density and pressure perturbations get modified when metric fluctuations are included. Concretely, they are replaced by777This expression for the energy density reminds of the Zel’dovich approximation, , in Newtonian Lagrangian perturbation theory (see for example [53]), with playing the role of the displacement field. The extra metric perturbation term in (5.16) comes from our relativistic formulation.

 δρα=(¯ρα+¯pα)(∇⋅πα+3ϕ)δpα=c2αsδρα (5.16)

with the speed of sound squared defined exactly as before, in (4.6). Notice that both the density and pressure perturbations gain a term that depends on the metric potential . In particular, for the total (matter plus metric) density perturbation of each fluid we write

 δρα=δMρα+3(¯ρα+¯pα)ϕ (5.17)

where the matter part is given by the first expression of (4.17).

The four-velocities of the fluids also change with respect to (4.7) due to the effect of the metric perturbations. The equations (2.1) are both still valid and we can again find the four-velocity solving them iteratively. Alternatively, proceeding in the same way that we have used to obtain (3.9) we get

 u0=1a~γui=1a~γvi~γ−2≡(1+2ψ)−(1−2ϕ)v2 (5.18)

where the coordinate velocity is still given by

 vi=dxidτ=−˙πi+˙πk∂kπi+… (5.19)

which is the same that we would derive from (4.7) (where metric perturbations are set to zero). This happens because is the solution of (3.7), which is the same with or without metric perturbations. Expanding at second order in perturbations we get

 u0=1a(1−ψ+32ψ2+12v2+…)ui=1a(1−ψ+…)vi. (5.20)

Let us point out that the expression (5.20) for differs (at second order) from the one found in the review [54] and other related papers in the literature.

In the next section, we will use the equation (5.15) to understand adiabatic modes. In order to do so, we will need to replace and by their expressions in terms of pure matter perturbations. The difference between the two metric potentials is given, at first order, by the total anisotropic stress of the fluid system. If all its components are perfect, the metric potentials are equal at first order. Any difference between the metric potentials will then be due to imperfect components. Another useful piece of information is the (general relativistic) Poisson equation, which allows to express the Laplacian squared of as a function of the density and velocity perturbations. This equation can be derived by combining the and Einstein equations (see for instance [14]) or from the variation of the second order perturbed action (5.1) with respect to a metric degree of freedom that is later set to zero in the conformal Newtonian gauge [52]. The Poisson equation reads:

 ∇4ϕ=32H2N∑α=1Ωα(∇2δα−3(1+wα)Hθα) (5.21)

where we sum over all fluid species and is the standard notation for the divergence of the velocity perturbation [55] , as given in (5.19):

 θα=∇⋅vα=−∇⋅˙πα+… (5.22)

Since for a perfect fluid, is given by (5.16), it turns out that the right-hand side of (5.21) contains a contribution that depends on the spatial metric perturbation . Hence the expression (5.21) becomes a partial differential equation for with second and fourth order derivatives. We will use such a form of the Poisson equation in Section 6 when discussing adiabatic perturbations, see equation (6.16).

Before moving to the description of adiabatic modes, it is interesting and useful to see how the equations of motion and (5.14) and (5.15) are related to the standard continuity and Euler equations for generic fluids that have the following general form in the conformal Newtonian gauge [55]:

 ˙δα =−(1+wα)(θα−3˙ϕ)−3(δPαδρα−wα)Hδα (5.23) ˙θα =−(1−3wα)Hθα−˙wα1+wαθα−11+wαδPαδρα∇2δα−∇2ψ+∇2σα. (5.24)

Notice that for these fluids, i.e. described by (2.7), the Euler equation (5.24) with is precisely the divergence of (5.15), while the continuity equation (5.23) is just an identity. This follows from the fact that the velocity perturbations in (5.22) are the time derivative of the longitudinal modes rather than independent variables.

### 5.1 Vector and tensor metric perturbations

We have derived the equations of motion (5.14) and (5.15) for an ensemble of non-interacting fluids, each with its own field, assuming only scalar metric perturbations. However, since can be written for each fluid as the curl of a vector potential, we can expect its dynamics to be affected by metric vector perturbations. For the longitudinal component this cannot occur because it has zero curl and hence it can be expressed as the gradient of a single scalar degree of freedom, which then couples only to and as we have already seen.

The perturbed FLRW metric in the Poisson gauge [56, 57] generalizes the conformal Newtonian gauge (5.5) to include vector and tensor degrees of freedom:

 ds2=a2(−(1+2ψ)dτ2+2νidτdxi+[(1−2ϕ)δij+χij]dxidxj) (5.25)

where and are respectively pure vector and tensor perturbations that satisfy

 ∂kνk=0∂kχkj=0χjj=0. (5.26)

Applying (5.4) to the full set of metric perturbations (transverse) vector and (transverse and traceless) tensor metric degrees of freedom we find

 12√−gδMTμνδgμν= a4[−(ψδMρ+3ϕδMp)+a(¯ρ+¯p)νiδMui] (5.27) = −a4(¯ρ+¯p)[(ψ+3c2sϕ)∂iπiL+˙πiTνi]+(total derivative) (5.28)

where tensor perturbations do not appear because of the tracelessness condition. This confirms the known result that tensor metric perturbations do not couple, at the lowest order in perturbation theory, to perfect fluids. On the other hand, the transverse modes (vortices) mix with the vector metric perturbations. The equations (5.14) get now replaced by a conservation equation of the linear combination

 ddτ(a4(¯ρα+¯pα)(νi−˙πiα⊥))=0α=1,…,N (5.29)

where, as before, the index lists the different perfect fluids from 1 to . This equation is nothing but the conservation of the charges associated with the vorticity, see (C.17). It corresponds also to the separate conservation of the three-momentum for each fluid [54, 58]:

 δ˙qjα+4Hδqjα=0α=1,…,N (5.30)

where

 δqiα≡(¯ρα+¯pα)(νi−˙πiα⊥)α=1,…,N (5.31)

are the three-momenta of the fluids. Notice that, if the fluids were not perfect, the right-hand side of (5.30) (or equivalently (5.29)) would have an anisotropic stress source term. The equation (5.30) tells us that in absence of such a term, the three-momentum decays due to the Hubble expansion.

The equation for the evolution of the vector metric perturbations is [58]

 ∇2νi=6H2∑α(1+wα)(νi−˙πiα⊥) (5.32)

consistent with the redshift of the three-momenta at large scales.

For completeness we include the equation for tensor modes, which in absence of any source of anisotropic stress reads:

 ∇2χij=¨χij+2H˙χij. (5.33)

As it is well known, these modes do not couple to any fluid degree of freedom if there is no anisotropic stress.

Adiabatic perturbations correspond to modes associated with equal time shifts. They are therefore constructed by perturbing any homogeneous (intensive) fluid variable in the following way

 ¯Ij(τ)→Ij(τ,x)=¯Ij(τ+^π(τ,x))=¯Ij(τ)+^π(τ,x)˙¯Ij+… (6.1)

using the same for all  , where the subscript lists any intensive variable pertaining to the fluid. Therefore, we say that two intensive fluid variables and are adiabatically related if their perturbations can be constructed from the same time shift . If we focus on the energy density and the pressure of any fluid, at first order in this implies

 c2A=c2s. (6.2)

Moreover, for several fluids, we have that adiabatic density modes satisfy

 δρα˙¯ρα=δρβ˙¯ρβ (6.3)

for any pair of fluids and interacting only through gravity. Describing the perturbations as in (4.1), we get that the adiabatic mode is nothing but a common longitudinal degree of freedom

 ^π=δρα˙¯ρα=δpα˙¯pα=−13H(∇⋅π+3ϕ)∀α. (6.4)

In other words, the condition of adiabatic modes translates into “flavour” (or species) independence of the longitudinal modes

 παL=πβL. (6.5)

In the next subsection we show how this result extends to imperfect fluids that have vanishing non-adiabatic pressure perturbations.

For convenience, we use the curvature perturbation on uniform density hypersurfaces, which is defined to be [59]

 ζα=−ϕ−Hδρα˙¯ρα. (6.6)

Remarkably, in our case this is just one third of the divergence of

 ζα=13∇⋅πα (6.7)

and taking its derivative with respect to conformal time, we get the following relation with the divergence of the velocity (at first order in ):

 θα+3˙ζα=0. (6.8)

Defining the entropy perturbation between two species in the usual way [60]

 Sαβ=3(ζα−ζβ) (6.9)

we see that for the kind of fluids that we consider, the entropy perturbation among two species is zero if the coordinate perturbations ) of the two fluids differ by at most a divergenceless three-vector. This is exactly the condition (6.5) for adiabatic modes that we have found before.

### 6.1 Adiabatic modes for imperfect fluids

Let us recall that for more general (imperfect) fluids888For a study of cosmological perturbations of imperfect fluids, in particular in connection to scalar fields, see [61]. entropy modes may come as well from an intrinsic non-adiabatic pressure perturbation. The total pressure perturbation of any species can be decomposed as

where the second term is the product of the density perturbation and the usual adiabatic speed of sound. For perfect fluids like the ones we have studied in this work, (the non-adiabatic part of the pressure perturbation) is zero. However, when a fluid has other internal degrees of freedom different from the coordinates999See Appendix A for the general form of the energy-momentum tensor in this case., an intrinsic non-adiabatic pressure will typically arise. Nevertheless, there exists an interesting class of imperfect fluids that have vanishing non-adiabatic pressure perturbations. They fail to be perfect only because they have anisotropic stress

 ∑α(¯ρα+¯pα)∇2σα=∑i,j∂i∂jTij−13∑k∇2Tkk, (6.11)

that enters in the Einstein equations for the scalar potentials

 ∇4(ψ−ϕ)=92H2∑αΩα(1+wα)∇2σα. (6.12)

At early times, neutrinos fall into this class of fluids: while .

As we are going to see next, in an adiabatic mode, the expressions that occur for the density and velocity perturbations of perfect fluids are also valid at first order in for imperfect fluids of that kind. From the equation (3.3) we know that is a species independent ratio for all fluids involved in an adiabatic mode. Since the definition (6.6) can be written as we have that

 ζα=ζβ (6.13)

for any two fluids in an adiabatic mode. Moreover, if at least one perfect fluid is present, we have, thanks to equation (5.16), . For imperfect fluids with vanishing non-adiabatic pressure:

 ddτ