Cosmic microwave background with Brans-Dicke Gravity: I. Covariant Formulation

# Cosmic microwave background with Brans-Dicke Gravity: I. Covariant Formulation

Feng-Quan Wu    Li-e Qiang    Xin Wang    Xuelei Chen National Astronomical Observatories, Chinese Academy of Sciences,
20A Datun Road, Chaoyang District, Beijing 100012, China
July 13, 2019
###### Abstract

In the covariant cosmological perturbation theory, a 1+3 decomposition ensures that all variables in the frame-independent equations are covariant, gauge-invariant and have clear physical interpretations. We develop this formalism in the case of Brans-Dicke gravity, and apply this method to the calculation of cosmic microwave background (CMB) anisotropy and large scale structures (LSS). We modify the publicly available Boltzmann code CAMB to calculate numerically the evolution of the background and adiabatic perturbations, and obtain the temperature and polarization spectra of the Brans-Dicke theory for both scalar and tensor mode, the tensor mode result for the Brans-Dicke gravity are obtained numerically for the first time. We first present our theoretical formalism in detail, then explicitly describe the techniques used in modifying the CAMB code. These techniques are also very useful to other gravity models. Next we compare the CMB and LSS spectra in Brans-Dicke theory with those in the standard general relativity theory. At last, we investigate the ISW effect and the CMB lensing effect in the Brans-Dicke theory. Constraints on Brans-Dicke model with current observational data is presented in a companion paper Wu and Chen (2009)(paper II).

Brans-Dicke theory, alternative gravity, cosmological perturbation theory, covariant, cosmic microwave background, large scale structure

## I Introduction

The Jordan-Fierz-Brans-Dicke theory Jordan (1949, 1959); Fierz (1956); Brans and Dicke (1961); Dicke (1962) (hereafter the Brans-Dicke theory for simplicity) is a natural alternative and a simple generalization of Einstein’s general relativity theory, it is also the simplest example of a scalar-tensor theory of gravity Bergmann (1968); Nordtvedt (1970); Wagoner (1970); Bekenstein (1977); Bekenstein and Meisels (1978). In the Brans-Dicke theory, the purely metric coupling of matter with gravity is preserved, thus ensuring the universality of free fall (equivalence principle) and the constancy of all non-gravitational constants. From early on, testing the Brans-Dicke theory with CMB anisotropy has been considered Peebles and Yu (1970). However, the usual approach is to use a metric-based and gauge-dependent method, i.e. making the calculation with a particular gauge, see. e.g., Ref. Nariai (1969); Baptista et al. (1996); Chen and Kamionkowski (1999); Nagata et al. (2002); Hwang (1997).

The covariant approach to general relativity is an elegant solution to the “gauge problem”, which has plagued the study of linear perturbation in gauge-dependent methods since the pioneering work of LifshitzLifshitz (1946). Before this problem was recognized, contradictory predictions of the behavior of perturbation of Friedmann-Lemaître-Robertson-Walker (FLRW) cosmologies were made. In 1980, Bardeen reformulated the metric approach using gauge-invariant variables Bardeen (1980) (see also Ref. Mukhanov et al. (1992) for a review on the variables which has been widely used in recent perturbation calculations). However, as pointed out by Ellis Ellis and Bruni (1989), although the Bardeen variables are related to the density perturbations, they are not those perturbations themselves, since they include metric tensor Fourier components and other quantities in cunning combinations. The physical meaning of Bardeen’s gauge-invariant variables are not always transparent. As emphasized by Hawking Hawking (1966), the metric tensor can not be measured directly, so it is not surprising that the variables used in the metric-based method do not always have a clear physical interpretation.

The covariant approach to general relativity and cosmology has its origins in the work of Heckmann, Schücking, Raychaudhuri and Hawking Heckmann and Schücking (1955); Raychaudhuri (1957); Hawking (1966). In 1989, Ellis and Bruni proposed using the spatial gradient of matter density () as the basic variable to describe density perturbations Ellis and Bruni (1989). Subsequently, the cosmological applications have been developed extensively by Ellis and others in recent years Ellis et al. (1990); Ellis and Tsagas (2002); Ellis and van Elst (1999); Bruni et al. (1992); Dunsby et al. (1992); Maartens (1997); Maartens and Bassett (1998); Maartens (2000); Tsagas and Barrow (1997); Tsagas and Maartens (2000); Tsagas (2005); Tsagas et al. (2007); Li and Chu (2006); Li et al. (2007a, b); van Elst and Ellis (1999); Leong et al. (2002). The method also has been applied to problems in CMB physics Challinor (2000a, b, c); Maartens et al. (1999). Instead of using the components of metric as basic variables, the covariant formalism performs a 1+3 split of the Bianchi and Ricci identities, using the kinematic quantities, energy-momentum tensors of the fluid(s) and the gravito-electromagnetic parts of the Weyl tensor to study how perturbations evolve. The most notable advantage of this method is that the covariant variables have transparent physical definitions, which ensures that predictions are always straightforward to interpret physically. Other advantages include the unified treatment of scalar, vector and tensor modes, a systematic linearization procedure which can be extended to consider higher-order effects (this means the covariant variables are exactly gauge-invariant, independent of any perturbative expansion), and the ability to linearize about a variety of background models, e.g. either the FLRW or the Bianchi models Challinor and Lasenby (1998, 1999).

A pioneering work in applying the covariant approach to Brans-Dicke theory is Ref. Hirai and Maeda (1994), in which a conformal transformation was performed, and calculation was done in the Einstein frame. More recently, Ref. Carloni et al. (2006); Carloni and Dunsby (2007) chose the effective fluid frame, implying and , i.e. their foliation selects vorticity-free spacelike hypersurfaces in which const, hence greatly simplifies the calculations.

The aim of this paper is to construct a full set of covariant and gauge-invariant linearized equations to calculate angular power spectra of CMB temperature and polarization anisotropies in the cold dark matter frame, and to show that the covariant method will lead to a clear, mathematically well-defined description of the evolution of density perturbations. In a companion paper Wu and Chen (2009) (heretofore denoted paper II), we shall apply the formalism developed in this paper to the latest CMB and large scale structure data to obtain constraint on the Brans-Dicke parameter.

In §2, we briefly review the Brans-Dicke theory and its background cosmological evolution. The formalism of covariant perturbation theory is presented in §3, and the numerical implementation in §4. We discuss the results on primary anisotropy in §5. The Integrated Sachs-Wolfe effect and gravitational lensing is discussed in §6. Finally, we summarize and conclude in §7.

Throughout this paper we adopt the metric signature . Our conventions for the Riemann and Ricci tensor are fixed by , where denotes the usual covariant derivative, and . We use to represent ordinary derivative. The spatially projected part of the covariant derivative is denoted by . The index notation denotes the index string . Round brackets around indices denote symmetrization on the indices enclosed, square brackets denote anti-symmetrization, and angled brackets denote the projected symmetric and tracefree (PSTF) part. We adopt and use units with throughout. In the numerical work we use  Mpc as unit for distance.

## Ii The Brans-Dicke Theory and Background Cosmology

The Brans-Dicke theory is a prototype of the scalar-tensor theory of gravity. One of its original motivations is to realize Mach’s principle of inertia Brans and Dicke (1961); Dicke (1962). It introduced a new degree of freedom of the gravitational interaction in the form of a scalar field non-minimally coupled to the geometry. The action for the Brans-Dicke theory in the usual (Jordan) frame is

 S=116π∫d4x√−g[−ϕR+ωϕgμν∇μϕ∇νϕ]+S(m), (1)

where is the Brans-Dicke field, is a dimensionless parameter, and is the action for the ordinary matter fields Matter is not directly coupled to , in the sense that the Lagrangian density does not depend on . For convenience, we also define a dimensionless field

 φ=Gϕ, (2)

where is the Newtonian gravitational constant measured today. The Einstein field equations are then generalized to

 Gμν = 8πGφT(m)μν+ωφ2(∇μφ∇νφ−12gμν∇λφ∇λφ) (3) +1φ(∇μ∇νφ−gμν∇λ∇λφ),

where is the stress tensor for all other matter except for the Brans-Dicke field, and it satisfies the energy-momentum conservation equation, The equation of motion for is

 ∇a∇aφ=κ2ω+3T(m), (4)

here is the trace of the energy-momentum tensor. The action (1) and the field equation (3) suggests that the Brans-Dicke field plays the role of the inverse of the gravitational coupling, , which becomes a function of the spacetime point.

For background cosmology, we treat the ordinary matter as the perfect fluid with the energy density and pressure ,

 T(m)μν=(ρ+P)uμuν−Pgμν. (5)

The equations describing the background evolution are

 ρ′+3H(ρ+P)=0, (6)
 H2=κS23φρ+ω6(φ′φ)2−Hφ′φ, (7)
 φ′′+2Hφ′=κS22ω+3(ρ−3P), (8)

where the prime denotes derivative with respect to conformal time , S is the scale factor, and . General relativity is recovered in the limits

 ω→∞,φ′→0,φ′′→0. (9)

To recover the value Newton’s gravitational constant today which is determined by Cavendish type experiments, we also require that the present day value of is given by

 φ0=2ω+42ω+3. (10)

## Iii Perturbation Theory

### iii.1 The 1+3 covariant decomposition

The main idea of the decomposition is to make space-time splits of physical quantities with respect to the 4-velocity of an observer. There are many possible choices of the frame, for example, the CMB frame in which the dipole of CMB anisotropy vanishes, or the local rest-frame of the matter. These frames are generally assumed to coincide when averaged on sufficiently large scale. Here it will be convenient to choose to coincide with the velocity of the CDM component, since is then geodesic, and acceleration vanishes. From the 4-velocity , we could construct a projection tensor into the space perpendicular to (the instantaneous rest space of observers whose 4-velocity is ):

 hab≡gab−uaub, (11)

where is the metric of the spacetime. Since is a projection tensor, it can be used to obtain covariant tensor perpendicular to , and it satisfies

 hab=h(ab),  uahab=0,  hcahcb=hab,  haa=3. (12)

With the timelike 4-velocity and its tensor counterpart , one can decompose a spacetime quantity into irreducible timelike and spacelike parts. For example, we can use to define the covariant time derivatives of a tensor :

 ˙Tb…cd…e≡ua∇aTb…cd…e, (13)

furthermore, we can exploit the projection tensor to define a spatial covariant derivative which returns a tensor which is orthogonal to on every index:

 DaTb…cd…e≡haphbr…hcshtd…hue∇pTr…st…u, (14)

If the velocity field has vanishing vorticity, reduces to the covariant derivative in the hypersurfaces orthogonal to . The projected symmetric tracefree (PSTF) parts of vectors and rank-2 tensors are

 V⟨a⟩=habVb, (15)
 T⟨ab⟩=h⟨achb⟩dTcd=h(achb)dTcd−13hcdTcdhab. (16)

One can also define a volume element for the observer’s instantaneous rest space:

 εabc=ηabcdud=ε[abc] , (17)

where is the 4-dimensional volume element (, ). Note that . The skew part of a projected rank-2 tensor is spatially dual to the projected vector , and any projected second-rank tensor has the irreducible covariant decomposition

 Tab=13Thab+εabcTc+T⟨ab⟩, (18)

where is the spatial trace. In the 1+3 covariant formalism, all quantities are either scalars, projected vectors or PSTF tensors. The covariant decomposition of velocity gradient are

 ∇aub=Daub+uaAb, (19)
 Daub=ωab+σab+13θhab, (20)

where is the shear tensor which satisfies , and ; is the vorticity tensor, which satisfies and . One can also define the vorticity vector (with ). The scalar is the volume expansion rate, is the local Hubble parameter; and is the acceleration, which satisfies . We note that the tensor describes the relative motion of neighbouring observers. The volume scalar determines the average separation between two neighbouring observers. The effect of the vorticity is to change the orientation of a given fluid element without modifying its volume or shape, therefore it describes the rotation of matter flow. Finally, the shear describes the distortion of matter flow, it changes the shape while leaving the volume unaffected Tsagas et al. (2007).

Gauge-invariant quantities can be constructed from scalar variables by taking their projected gradients. The comoving fractional projected gradient of the density field of a species is the key quantity of covariant method Ellis and Bruni (1989),

 X(i)a≡Sρ(i)Daρ(i), (21)

which describes the density variation between two neighbouring fundamental observers. The comoving spatial gradient of the expansion rate orthogonal to the fluid flow is

 Za≡SDaθ, (22)

which describes perturbations in the expansion. These quantities are in principle observable, characterizing inhomogeneity in a covariant way, and vanishes in the FLRW limit.

The matter stress-energy tensor can be decomposed irreducibly with respect to as follows:

 T(m)ab≡ρuaub+2u(aqb)−Phab+πab, (23)

where is the density of matter measured by an observer moving with 4-velocity , is the relativistic momentum density or heat (i.e. energy) flux and is orthogonal to , is the isotropic pressure, and the projected symmetric traceless tensor is the anisotropic stress, which is also orthogonal to . The quantities , , , are referred to as dynamical quantities and , , , as kinematical quantities. In the FLRW limit, isotropy restricts to the perfect-fluid form, so the heat flux and anisotropic stress must vanish.

The remaining first-order gauge-invariant variables which we need are derived from the Weyl tensor , which is associated to the long-range gravitational field and vanishes in an exact FLRW universe due to isotropy. In analogy to the electromagnetic field, the Weyl tensor can be split into electric and magnetic parts, denoted by and respectively. They are both symmetric traceless tensors and orthogonal to ,

 Eab ≡ Cacbducud=E, (24) Bab ≡ −∗Cacbducud=−12εaefCefbdud (25) = B.

Here denotes the dual, .

For the radiation field, we can make a 1+3 covariant decomposition of the photon 4-momentum as

 pa=E(ua+ea), (26)

where is the energy of the photon. describes the propagation direction of photon in the instantaneous rest space of the observer. The observer can introduce a pair of orthogonal polarization vectors and , which are perpendicular to and , to form a right-handed orthonormal tetrad at the observation point. The (screen) projection tensor is defined as

 Hab=gab−uaub+eaeb, (27)

which is perpendicular to both and , and satisfies .

Using the polarization basis vectors, the observer can decompose an arbitrary radiation field into Stokes parameters , , and Challinor (2000a). Therefore one can introduce a second-rank transverse polarization tensor

 Pab(ei)a(ej)b=12(I+QU+VU−VI−Q), (28)

for and , and we have omitted the arguments and . , where is photon distribution function. Decomposing into its irreducible components, one obtains

 Pab(E,ed) = −12I(E,ed)Hab+Pab(E,ed) (29) +12V(E,ed)ϵabcec,

where the linear polarization tensor satisfies

 Pab(ei)a(ej)b=12(QUU−Q). (30)

It is convenient to define energy-integrated multipole for total intensity brightness and the electric part of the linear polarization:

 IAl=∫∞0dE∫dΩ   I(E,ec)e , (31)
 EAl = Ml2∫∞0dE∫dΩ  e⟨Al−2Pal−1al⟩(E,ec) , (32)

where and .

### iii.2 The linearized perturbation equations

In the 1+3 covariant approach, the fundamental quantities are not the metric, which is gauge-dependent, but the kinematic quantities of the fluid, namely the shear , the vorticity , the volume expansion rate and the acceleration , the energy-momentum of matter and the gravito-electromagnetic parts of the Weyl tensor. The fundamental equations governing these quantities are the Bianchi identities and the Ricci identities. The Riemann tensor in these equations is expressed in terms of , and the Ricci tensor . The modified Einstein equation connects the Ricci tensor to the matter energy-momentum tensor. In the following, we have linearized all the perturbation equations. We should also note that the definitions of covariant variables do not assume any linearization, and exact equations can be found for their evolution.

The first set of equations are derived from the Ricci identities for the vector field , i.e.

 2∇[a∇b]uc=Rabcdud. (33)

Substituting the 4-velocity gradient (19) and the decomposition of the Riemann tensor, and separating out the time-like projected part into the trace, the symmetric trace-free and the skew symmetric parts, we obtain three propagation equations. The first propagation equation is the Raychaudhuri equation,

which is the key equation of gravitational collapse, accounting for the time evolution of . The second is the vorticity propagation equation,

 ˙ωab−D[a˙ub]+23θωab=0. (35)

The last one is the shear propagation equation,

 ˙σ+23θσab−D+Eab+κ2πabφ +12φDφ+12˙φφσab=0, (36)

which describes the evolution of kinematical anisotropies. It shows that the tidal gravitational field and the anisotropic stress would induce shear directly, and the shear will change the spatial inhomogeneity of the expansion through the constraint equations (37).

The propagation equations are complemented by three constraint equations, which are spacelike components of Eq.(33). The first is the shear constraint,

 Dbωab+Dbσab−23Daθ−κφqa−ω˙φφ2Daφ −1φ(Daφ).−˙φφ˙ua=0, (37)

which shows the relation between the momentum flux , the shear and the spatial inhomogeneity of the expansion. The second constraint equation is the vorticity divergence identity,

 Dc(εabcωab)=0. (38)

The last one is the equation,

 Bab+(Dcωd(a+Dcσd(a)η   db)ceue=0, (39)

which shows that the magnetic Weyl tensor can be constructed from the vorticity tensor and the shear tensor. With this last equation may be eliminated from some equations in favor of the vorticity and the shear.

So far we have only discussed propagation and constraint equations for the kinematic quantities. The second set of equations arises from the Bianchi identities of the Riemann tensor,

 ∇[eRcd]ab=0 , (40)

which gives constraint on the curvature tensor and leads to the Bianchi identities for Weyl tensor after contracting once,

 ∇dCabcd=∇[bRa]c+16gc[b∇a]R. (41)

The splitting of the once contracted Bianchi identities leads to two propagation and two constraint equations which are similar in form to the Maxwell field equations in an expanding universe, governing the evolution of the long range gravitational field. The first propagation equation is the -equation,

 ˙Eab+θEab+DcBd(aη   db)ceue+κ6φ[3(ρ+P)σab+ 3D−3˙πab−θπab]+12σab(ω+32)˙φ2φ2− 16σabφDμDμφ+12(ω+32)˙φφ2Dφ +12˙φφEab+34κ˙φφ2πab=0 , (42)

and the second propagation is the -equation

 ˙Bab+(θ+˙φ2φ)Bab−[DcEd(a+κ2φDcπd(a+ 12φDcDdD(aφ−16φDcDμDμφhd(a]η   db)ceue =0 . (43)

This pair of equations for electric and magnetic parts of the Weyl tensor would give rise wavelike behavior for its propagation: if we take the time derivative of the -equation, commuting the time and spatial derivatives of term and substituting from the -equation to eliminate , we would obtain a term and a double spatial derivatives term, which together give the wave operator acting on ; similarly we can obtain a wave equation for by taking time derivative of the -equation. These waves are also subjected to two constraint equations, which emerge from the spacelike component of the decomposed Eq.(41). The first constraint is

 DbEab−κ6φ(2Daρ+2θqa+3Dbπab)+2κ3ρDaφφ2− κ2˙φφ2qa−(ω3+12)˙φφ2[43θDaφ+(Daφ).+˙u˙φ]=0 . (44)

This is the div E equation, with the source term given by the spatial gradient of energy density. It can be regarded as a vector analogue of the Newtonian Poisson equation, and shows that the scalar modes will result in a non-zero divergence of , and hence a non-zero gravitational E-field. The second constraint equation is

 DbBab−κ2φ[(ρ+P)η  cdabubωcd+ηabcdubDcqd]− 12[(ω˙φ2φ2−13φDμDμφ−θ3˙φφ+¨φφ)η  cdabubωcd+ ηabcdub(ω˙φφ2DcDdφ+1φ(DcDdφ). +θ3φDcDdφ+˙φφDc˙ud)]=0 . (45)

This is the div B equation, with the fluid vorticity serving as source term. It shows that the vector modes will result in non-zero divergence of , and hence a non-zero gravitational B-field. The above equations are remarkably similar to the Maxwell equations of the electromagnetism, so we have chosen to use and as the symbols.

The last set of equations arises from the twice-contracted Bianchi identities. Projecting parallel and orthogonal to , we obtain two propagation equations,

 ˙ρ+θ(ρ+P)+Daqa=0 , (46)
 ˙qa+43θqa+(ρ+P)˙ua+Dbπab−DaP=0 , (47)

respectively. For perfect fluids, these reduce to

 ˙ρ+θ(ρ+P)=0 , (48)
 (ρ+P)˙ua−DaP=0 , (49)

which are the energy conservation equation and momentum conservation equation respectively.

The background field equation for Brans-Dicke field is given in Eq.(8). The first order covariant and gauge-invariant perturbation variable of the Brans-Dicke field is defined as the spatial derivative of the Brans-Dicke field,

 Va≡SDaφ . (50)

Taking the covariant spatial derivative of Eq.(8), commuting the spatial and time derivatives of term, we could obtain the first order perturbation equation for Brans-Dicke field after linearization,

where the upper index labels the particle species.

In the absence of rotation, , one can define a global 3-dimensional spacelike hypersurfaces that are everywhere orthogonal to . This 3-surfaces is meshed by the instantaneous rest space of comoving observers. The geometry of the hypersurfaces is determined by the 3-Riemann tensor defined by

 [Da,Db]uc=(3)Rabdcud , (52)

which is similar to the definition of Riemann tensor but with a conventional opposite sign. The relationship between and is

 (3)Rabcd = −haqhbshcfhdpRqsfp−vacvbd+vadvbc (53) = (3)R[ab][cd],

where is the relative flow tensor between two neighbouring observers. In analogy to 4-dimension, the projected Ricci tensor and Ricci scalar are defined by

 (3)Rab=(3)Racbdhcd=(3)Rcacb , (54)

and

 (3)R=(3)Rabhab. (55)

The is determined by the Gauss-Codacci formula

 (3)Rab=13(3)Rhab−13θσab−κ2πab+Eab , (56)

where

 (3)R=2(κρ−13θ2) . (57)

The Eq.(57) is also the generalized Friedmann equation, showing how the matter tensor determines the 3-space average curvature.

The last first-order covariant and gauge-invariant variables can be obtain from the spatial derivative of the projected Ricci scalar,

 ηa≡12SDa(3)R , (58)

after a tedious calculation, we obtain

 ηa = κρXaφ−κρVaφ2−1S(2H+φ′φ)Za (59) +1S2(ωφ′φ2−3Hφ)(V′a−HVa) +(ω+3)1S2φ′φ2HVa+1S(ωφ′2φ2−3Hφ′φ)Wa −ωS2φ′2φ3Va−1φDaDνVν−3SH2Vaφ .

### iii.3 Mode expansion in spherical harmonics

In the linear perturbation theory it is convenient to expand the variables in harmonic modes, since it splits the perturbations into scalar, vector or tensor modes and decouples the temporal and spatial dependencies of the 1+3 equations. This converts the constraint equations into algebraic relations and the propagation equations into ordinary differential equations along the flow lines. In this paper we focus on the scalar and tensor perturbation modes, since the vector modes would decay in an expanding universe in the absence of sources such as topological defects.

#### iii.3.1 Scalar mode

For scalar perturbations we expand in the scalar eigenfunctions of the generalized Helmholtz equation

at zero order. They are defined so as to be constant along flow lines, i.e. independent of proper time , and orthogonal to the fluid 4-velocity .

For the th multipoles of the radiation anisotropy and polarization, we expand in the rank- PSTF tensor, , derived from the scalar harmonics with

 Q(k)Al=(Sk)lDQ(k) , (61)

where the index notation denotes the index string . The recursion relation for the ,

 Q(k)Al=SkD (62)

follows directly from Eq.(61). The factor of in the definition of the ensures that at zero-order. The also satisfies some other zero-order properties,

 uaiQ(k)a1..ai..al=0,haiajQ(k)a1..ai..aj..al=0. (63)

We also have the following differential relations which can be derived from Eqs.(60) and (62):

 Da1Q(k)a1a2…al = l2l−1kS[1−(l2−1)Kk2]Q(k)a2…al , (64) D2Q(k)a1…al = k2S2[1−l(l+1)Kk2]Q(k)a1…al . (65)

Now we can expand the gauge-invariant variable in the following dimensionless harmonic coefficients:

 X(i)a = ∑kkX(i)kQ(k)a , (66) Za = ∑kk2SZkQ(k)a , (67) q(i)a = ρ(i)∑kq(i)kQ(k)a , (68) v(i)a = ∑kv(i)kQ(k)a , (69) π(i)ab = ρ(i)∑kπ(i)kQ(k)ab , (70) Eab = ∑kk2S2ΦkQ(k)ab , (71) σab = ∑kkSσkQ(k)ab , (72) Aa = ∑kkSWkQ(k)a , (73) Va = ∑kkVkQ(k)a , (74) ηa = ∑kk3S2ηkQ(k)a , (75) IAl = ργ∑kI(l)kQ(k)Al , (76) EAl = ργ∑kE(l)kQ(k)Al , (77)

where the upper index labels the particle species. The scalar expansion coefficients, such as , are first-order gauge-invariant variables, and their spatial gradients are second-order, for example . In the covariant and gauge-invariant approach, we characterize scalar perturbations by requiring that the vorticity and the magnetic part of the Weyl tensor be at most second-order. Demanding ensures that density gradients are not from kinematic effects due to vorticity, and setting ensures that gravitational waves are excluded to the first order.

To obtain the scalar equations for the scalar expansion coefficients, one could substitute the harmonic expansions of the covariant variables into the propagation and constraint equations given in the section above. Here we will consider only the adiabatic modes. For the (i) fluid,

 DaP(i)=c(i)2sDaρ(i), (78)

where is the adiabatic sound speed of the (i) fluid. For the spatial gradients of the total density , we find

 X′k+3Hρ∑iρ(i)X(i)k(c(i)2s−Pρ)+k[(1+Pρ)Zk +∑iq(i)k]−3H(1+Pρ)Wk=0 . (79)

For the individual fluid of the species, the propagation equation satisfies

 X′(i)k+3H(c(i)2s−P(i)ρ(i))X(i)k+k[(1+P(i)ρ(i))Zk+q(i)k] −3H(1+P(i)ρ(i))Wk=0 . (80)

For the heat fluxes, we have

 q′(i)k+H(1−3P(i)ρ(i))q(i)k+(1+P(i)ρ(i))kWk+ 23k(1−3Kk2)π(i)k−kc(i)2sX(i)k=0. (81)

The heat flux for each fluid component is often given by , so we can derive the propagation equations for from Eq.(81).

We also can obtain the time evolution of the spatial gradient of the expansion

 Z′k+HZk+Wkk(3H′−3H2−k2)+1kκS22φ∑i(1+ 3c(i)2s)ρ(i)X(i)k+12k{Vk[−4ωφ′2φ3−3φ′′φ2−S2κφ2(ρ+ 3P)+k