Second-order perturbations of cosmological fluids:
Relativistic effects of pressure, multi-component, curvature, and rotation
Jai-chan Hwang∗
Department of Astronomy and Atmospheric Sciences, Kyungpook National University, Taegu, Korea
Hyerim Noh†
Korea Astronomy and Space Science Institute, Daejon, Korea
(Dated: October 31, 2018)
We present general relativistic correction terms appearing in Newton’s gravity to the second-order
perturbations of cosmological fluids. In our previous work we have shown that to the second-order
perturbations, the density and velocity perturbation equations of general relativistic zero-pressure,
irrotational, single-component fluid in a flat background coincide exactly with the ones known in
Newton’s theory. We also have shown the effect of gravitational waves to the second-order, and pure
general relativistic correction terms appearing in the third-order perturbations. Here, we present
results of second-order perturbations relaxing all the assumptions made in our previous works. We
derive the general relativistic correction terms arising due to (i) pressure, (ii) multi-component,
(iii) background curvature, and (iv) rotation. In case of multi-component zero-pressure, irrotational
fluids under the flat background, we effectively do not have relativistic correction terms, thus the
relativistic result again coincides with the Newtonian ones. In the other three cases we generally
have pure general relativistic correction terms. In case of pressure, the relativistic corrections appear
even in the level of background and linear perturbation equations. In the presence of background
curvature, or rotation, pure relativistic correction terms directly appear in the Newtonian equations
of motion of density and velocity perturbations to the second order. In the small-scale limit (far
inside the horizon), relativistic equations including the rotation coincide with the ones in Newton’s
gravity. All equations in this work include the cosmological constant in the background world
model. We also present the case of multiple minimally coupled scalar fields, and properly derive
the large-scale conservation properties of curvature perturbation variable in various temporal gauge
conditions to the second order.
PACS numbers: PACS numbers: 04.50.+h, 04.62.+v, 98.80.-k
Contents
I. Introduction 2
II. Newtonian nonlinear perturbations 3
III. Summary of previous work 5
IV. Relativistic fully nonlinear equations 6
A. Covariant equations 6
B. ADM equations 8
V. Second-order perturbations 8
A. Basic equations in the energy-frame 10
B. Decomposition 14
C. Comoving gauge and irrotational condition 16
VI. Effects of pressure 17
A. Irrotational case 17
∗Electronic address: jchan@knu.ac.kr
†Electronic address: hr@kasi.re.kr
http://arxiv.org/abs/0704.1927v1
mailto:jchan@knu.ac.kr
mailto:hr@kasi.re.kr
B. Comoving gauge 17
C. Newtonian correspondence 19
D. Linear-order relativistic pressure corrections 20
VII. Effects of multi-component 20
A. Irrotational case 21
B. Linear perturbations 22
1. Exact solutions 24
C. Comoving gauge 25
D. Newtonian correspondence 26
VIII. Effects of curvature 28
A. Newtonian correspondence 28
IX. Effects of vector-type perturbation 29
A. Linear perturbations 29
B. Second-order perturbations 30
C. Zero-pressure case 31
D. Newtonian correspondence 31
E. Pure vector-type perturbations 33
X. Equations with fields 33
A. A minimally coupled scalar field 33
B. Minimally coupled scalar fields 35
C. Generalized gravity case 35
XI. Curvature perturbations and large-scale conservations 36
A. Linear-order equations 36
B. Second-order equations 38
C. Large-scale solutions 39
XII. Discussion 39
Acknowledgments 40
References 40
I. INTRODUCTION
Large amount of cosmological data on the large-scale structures and motions of galaxies [1, 2], and the temperature
and polarization anisotropies of cosmic microwave background radiation [3, 4] have been accumulating recently. In
current standard cosmological scenario such structures are explained as small (linear) or large (nonlinear) deviations
from spatially homogeneous and isotropic Friedmann background world model. In order to explain these data theo-
retically, researchers rely on the linear perturbation theory based on relativistic gravity, and quasi-linear perturbation
theories and nonlinear simulations based on Newton’s gravity. To the linear order in perturbation the general rel-
ativistic result was first derived by Lifshitz in 1946 [5], and later shown to coincide with the Newtonian result in a
zero-pressure medium [6]. The same is also known to be true for the background world model. That is, the general
relativistic result was first derived by Friedmann in 1922 [7], and later shown to coincide with Newtonian result in a
zero-pressure medium [8].
The observed large-scale distribution of galaxies shows that in the largest observed scale (say, larger than several
hundred mega-parsec scale) the distribution may not be inconsistent with the linear assumption around the Friedmann
background. However, as the scale becomes smaller the distribution apparently shows quasi-linear to fully nonlinear
structures. The fully nonlinear processes occur in small scale where the relativistic effects characterized by GM/rc2 ∼
v2/c2 are quite small. If we could ignore such relativistic effects, Newton’s gravity would be sufficient to handle
the relevant nonlinear processes. If we need to consider the weakly relativistic correction terms in fully nonlinear
stage, instead of the relativistic perturbation approach which can handle the fully relativistic processes under weakly
nonlinear assumption, we can use the post-Newtonian approximation developed in the context of cosmology in [9].
For structures in the quasi-linear evolution phase, previous researches were based on Newton’s gravity especially
assuming the single component zero-pressure fluid without rotational perturbation [10]. In our previous works in
[11, 12] we have shown that, in the single component zero-pressure fluid without rotational perturbation, cosmological
scalar-type perturbation equations in a spatially flat background coincide exactly with the Newtonian ones up to the
second order in perturbation. In [11, 12] we also have shown the contribution of gravitational wave perturbations to
the hydrodynamic parts in the second-order perturbations. In Newton’s gravity the hydrodynamic equations of zero-
pressure fluid contain only the quadratic order nonlinearity. In [13] we presented pure general relativistic correction
terms appearing in the third-order perturbation, and showed that all third-order correction terms are 10−5 times
smaller than the second-order relativistic/Newtonian terms, and independent of the horizon scale.
In this work we will take into account of the pure general relativistic effects appearing in the second-order perturba-
tions which were ignored in our previous work in [12]. We will consider general relativistic effects of (i) pressure, (ii)
multi-component, (iii) background curvature, (iv) rotation in cosmological fluids to the second-order perturbations.
As results we will show that, although in [12] we have shown the exact relativistic/Newtonian correspondence to the
second-order perturbations by ignoring the above four conditions and the gravitational waves, as we take these four
effects into account we often encounter pure general relativistic effects appearing in the corresponding Newtonian equa-
tions even to the second order in perturbations. Our results will show that the relativistic/Newtonian correspondence
continues even in the multi-component situation assuming zero-pressure irrotational fluid in a flat background, but
in the presence of the cosmological constant. This is a practically useful result because the matter content of present
universe is dominated by collisionless dark matter and baryon both of which practically have zero-pressure. In the
other three cases, relaxing any of the assumptions about pressure, rotation, and background curvature generally leads
to pure general relativistic correction terms to the second order. We will present such correction terms in the context
of Newtonian hydrodynamics. One additional relativistic/Newtonian correspondence occurs in the case of rotation in
small-scale (sub-horizon-scale) limit which is another practically important result. This correspondence allows us to
use the Newtonian equations safely in such a small-scale limit even in the presence of rotational perturbation to the
second order.
In Sec. II we summarize Newtonian hydrodynamic perturbation equations valid to fully nonlinear order. In Sec.
III we briefly summarize our previous result of relativistic/Newtonian correspondence to the second order, and pure
general relativistic correction terms appearing in the third order. In Sec. IV we present parts of the covariant and
the ADM (Arnowitt-Deser-Misner) equations which are valid in multi-component situation. In Sec. V we present
the basic perturbation equations valid to second order. In [11] the basic set of equations was presented using fluid
quantities based on the normal-frame four-vector. The fluid quantities in the present work are based on the energy-
frame four-vector, and in this section we present the basic equations using such fluid quantities. In Secs. VI-IX we
analyse the effects of the pressure, the multi-components, the background curvature, and the rotational perturbation,
respectively. In Sec. X we present equations in the scalar fields and generalized gravity theories using the energy-frame
fluid quantities. In Sec. XI we properly derive conservation properties of curvature perturbation in various temporal
gauge (hypersurface) conditions to the second order in perturbations. Section XII is a discussion. In this work we
follow notations used in [11, 12]. We set c ≡ 1, but when we compare with Newtonian case we often recover the speed
of light c.
II. NEWTONIAN NONLINEAR PERTURBATIONS
In order to compare properly the relativistic results with the Newtonian ones, in this section we summarize the
Newtonian cosmological perturbation theory in fully nonlinear context. We consider multi-component fluids in the
presence of isotropic pressure. In case of n-fluids with the mass densities ̺i, the pressures pi, the velocities vi
(i = 1, 2, . . . n), and the gravitational potential Φ, we have
˙̺i +∇ · (̺ivi) = 0, (1)
v̇i + vi · ∇vi = −
∇pi −∇Φ, (2)
∇2Φ = 4πG
̺j . (3)
Assuming the presence of spatially homogeneous and isotropic but temporally dynamic background, we introduce
fully nonlinear perturbations as
̺i = ¯̺i + δ̺i, pi = p̄i + δpi, vi = Hr+ ui, Φ = Φ̄ + δΦ, (4)
where H ≡ ȧ/a, and a(t) is a cosmic scale factor. We move to the comoving coordinate x where
r ≡ a(t)x, (5)
∇ = ∇
−Hx · ∇
. (6)
In the following we neglect the subindex x. To the background order we have
˙̺i + 3H̺i = 0,
̺j , H
, (7)
where E is an integration constant which can be interpreted as the specific total energy in Newton’s gravity; in
Einstein’s gravity we have 2E = −Kc2 where K can be normalized to be the sign of spatial curvature. To the
perturbed order we have [14]
δ̇i +
∇ · ui = −
∇ · (δiui) , (8)
u̇i +Hui +
ui · ∇ui = −
a ¯̺i
1 + δi
∇δΦ, (9)
∇2δΦ = 4πG
¯̺jδj . (10)
By introducing the expansion θi and the rotation
−→ω i of each component as
∇ · ui,
−→ω i ≡
∇× ui, (11)
Eq. (9) gives
θ̇i + 2Hθi + 4πG
¯̺jδj = −
∇ · (ui · ∇ui)−
a2 ¯̺i
1 + δi
, (12)
−̇→ω i + 2H
−→ω i = −
∇× (ui · ∇ui) +
a2 ¯̺i
(∇δi)×∇δpi
(1 + δi)2
. (13)
By introducing decomposition of perturbed velocity into the potential- and transverse parts as
ui ≡ ∇ui + u
i , ∇ · u
i ≡ 0; θi =
−→ω i =
i , (14)
instead of Eq. (13) we have
i +Hu
i = −
ui · ∇ui +
1 + δi
−∇∆−1∇ ·
ui · ∇ui +
1 + δi
. (15)
We note that the pure ui contributions in the right-hand-side of Eq. (13) or Eq. (15) vanish. Thus, under van-
ishing pressure, pure irrotational perturbation cannot generate the rotational perturbation. Equation (13) shows
that presence of pressure perturbation oblique (i.e., non-parallel) to the density perturbation can generate rotational
perturbation. Combining Eqs. (8),(12) we can derive
δ̈i + 2Hδ̇i − 4πG
¯̺jδj = −
[a∇ · (δiui)]
∇ · (ui · ∇ui) +
a2 ¯̺i
1 + δi
. (16)
Equations (8)-(16) are valid to fully nonlinear order. Notice that for vanishing pressure these equations have only
quadratic order nonlinearity in perturbations.
III. SUMMARY OF PREVIOUS WORK
In [11, 12] we have derived second-order perturbation equations valid for the single component, irrotational, and
zero-pressure medium in zero-curvature background. These are
∇ · u = −
∇ · (δu) , (17)
∇ · (u̇+Hu) + 4πG̺δ = −
∇ · (u · ∇u)− Ċ(t)αβ
uα|β + Ċ
, (18)
δ̈ + 2Hδ̇ − 4πG̺δ =
∇ · (u · ∇u)−
[a∇ · (δu)]
+ Ċ(t)αβ
uα|β + Ċ
. (19)
Except for the presence of tensor-type perturbation, Eqs. (17)-(19) are exactly the same as the ones known in the
Newtonian theory. We note that these equations are valid in the presence of Λ. To the linear order, these are valid in
the presence of general K, see Sec. VIII. We have correctly identified the relativistic density and velocity perturbation
variables which correspond to the Newtonian counterparts to the second order. In the relativistic context, our δ and
u are the density perturbation and (related to) the perturbed expansion scalar, respectively, in the comoving gauge;
the variables are equivalently gauge-invariant. However, we were not able to identify relativistic variable which
corresponds to the Newtonian gravitational potential to the second order; this is understandable if we consider the
factor of two difference between Einstein’s (post-Newtonian) and Newton’s gravity theories in predicting the light
bending under the gravitational field. To the linear order the spatial curvature perturbation in the zero-shear gauge
can be identified as the perturbed Newtonian potential [15, 16]. Equations (17)-(19) include effects of gravitational
waves to the density and velocity perturbations. Equations of the gravitational waves can be found in [12].
To the third order, we have [13]
∇ · u = −
∇ · (δu) +
2ϕu−∇
· ∇δ, (20)
+ 4πGµδ = −
∇ · (u · ∇u)
ϕu · ∇ (∇ · u) +
u · ∇u−
u∇ · u
u · ∇
u · ∇X +
X∇ · u,(21)
δ̈ + 2
δ̇ − 4πGµδ = −
[a∇ · (δu)] +
∇ · (u · ∇u) +
2ϕu−∇
ϕu · ∇ (∇ · u)−
u · ∇u−
u∇ · u
u · ∇
u · ∇X −
X∇ · u,(22)
where
X ≡ 2ϕ∇ · u− u · ∇ϕ+
∆−1∇ · [u · ∇ (∇ϕ) + u∆ϕ] . (23)
In these equations we ignored the role of tensor-type perturbation; for a complete set of equations, see [13]. The
variable ϕ is a perturbed-order metric (spatial curvature) variable in the comoving gauge condition, see later.
All the third-order correction terms in Eqs. (20)-(22) are simply of ϕ-order higher than the second-order rela-
tivistic/Newtonian terms. Thus, the pure general relativistic effects are at least ϕ-order higher than the relativis-
tic/Newtonian ones in the second order equations. Thus, we only need the behavior of ϕ to the linear order which is
related to the other hydrodynamic variables as
ϕ = −δΦ+ ȧ∆−1∇ · u. (24)
It also satisfies [17]
ϕ̇ = 0, (25)
thus ϕ = C(x) with no decaying mode; this is true considering the presence of the cosmological constant, see [17].
IV. RELATIVISTIC FULLY NONLINEAR EQUATIONS
In this section, for convenience, we present some additional covariant or ADM equations not available in [11]. In
the multi-component situation, we have
T̃ab ≡
T̃(j)ab. (26)
The energy-momentum conservation gives
T̃ b(i)a;b = Ĩ(i)a,
Ĩ(j)a = 0. (27)
Tildes indicate the covariant quantities.
A. Covariant equations
We introduce the fluid quantities as
T̃(i)ab ≡ µ̃(i)ũ(i)aũ(i)b + p̃(i)
g̃ab + ũ(i)aũ(i)b
+ q̃(i)aũ(i)b + q̃(i)bũ(i)a + π̃(i)ab, (28)
where
ũa(i)ũ(i)a ≡ −1, ũ
(i)q̃(i)a ≡ 0 ≡ ũ
(i)π̃(i)ab, π̃
(i)a ≡ 0. (29)
The fluid quantities of each component are based on the fluid four-vector ũ(i)a as
µ̃(i) ≡ T̃(i)abũ
(i)ũ
(i), p̃(i) ≡
T̃(i)abh̃
(i), q̃(i)a ≡ −T̃(i)cdũ
(i)h̃
(i)a, π̃(i)ab ≡ T̃(i)cdh̃
(i)ah̃
(i)b − p̃(i)h̃(i)ab, (30)
where h̃(i)ab ≡ g̃ab + ũ(i)aũ(i)b. Equation (27) gives
µ(i) +
µ̃(i) + p̃(i)
θ̃(i) + q̃
(i);a + q̃
(i)ã(i)a + π̃
(i)σ̃(i)ab = −ũ
(i)Ĩ(i)a, (31)
µ̃(i) + p̃(i)
ã(i)a + h̃
p̃(i),b +
q(i)b + π̃
(i)b;c
+ q̃b(i)
ω̃(i)ab + σ̃(i)ab +
θ̃(i)h̃(i)ab
= h̃ b(i)aĨ(i)b, (32)
where the kinematic quantities are also based on the fluid four-vector ũ(i)a as
h̃ c(i)ah̃
(i)b ũ(i)c;d = h̃
(i)[ah̃
(i)b]ũ(i)c;d + h̃
(i)(ah̃
(i)b)ũ(i)c;d ≡ ω̃(i)ab + θ̃(i)ab = ũ(i)a;b + ã(i)aũ(i)b,
θ̃(i) ≡ u
(i);a, σ̃(i)ab ≡ θ̃(i)ab −
θ̃(i)h̃(i)ab, ã(i)a ≡ ũ(i)a;bũ
(i) ≡
u(i)a,
µ(i) ≡ µ̃(i),aũ
(i). (33)
In the multi-component situation, we can derive the corresponding equation of Raychaudhury equation for the indi-
vidual component. From ũ(i)a;bc − ũ(i)a;cb ≡ ũ(i)dR̃
abc we can derive
θ(i) +
θ̃2(i) − ã
(i);a + σ̃
(i)σ̃(i)ab − ω̃
(i)ω̃(i)ab = 4πG
µ̃− 3p̃− 2T̃abũ
(i)ũ
+ Λ. (34)
In the energy-frame we take q̃(i)a ≡ 0 for each component of the fluids without losing any physical degree of freedom.
In a single component situation, taking the energy-frame, the energy conservation equation, the momentum con-
servation equation, and the Raychaudhury equation are [18]
µ+ (µ̃+ p̃) θ̃ + π̃abσ̃ab = 0, (35)
(µ̃+ p̃) ãa + h̃
p̃,b + π̃
= 0, (36)
θ̃2 − ãa;a + σ̃
abσ̃ab − ω̃
abω̃ab + 4πG (µ̃+ 3p̃)− Λ = 0. (37)
By combining Eqs. (35)-(37) we can derive
µ+ π̃abσ̃ab
µ̃+ p̃
µ+ π̃abσ̃ab
µ̃+ p̃
= 4πG (µ̃+ 3p̃)− Λ + σ̃abσ̃ab − ω̃
abω̃ab +
h̃ab(p̃,b + π̃
µ̃+ p̃
. (38)
This equation was derived in Eq. (88) of [19], see also [20]. In the multi-component case, in the energy-frame,
combining Eqs. (31)-(34) we can derive
µ(i) + π̃
σ̃(i)ab + ũ
Ĩ(i)a
µ̃(i) + p̃(i)
µ(i) + π̃
σ̃(i)ab + ũ
Ĩ(i)a
µ̃(i) + p̃(i)
= −4πG
µ̃− 3p̃− 2T̃abũ
(i)ũ
− Λ + σ̃ab(i)σ̃(i)ab − ω̃
(i)ω̃(i)ab +
(p̃(i),b + π̃
(i)b;c
− Ĩ(i)b)
µ̃(i) + p̃(i)
. (39)
In [21] Langlois and Vernizzi derived a simple covariant relation which leads to one of the conserved variable in the
large-scale limit. These authors introduced
ζ̃a ≡ h̃
α̃,b +
3(µ̃+ p̃)
α ≡ α̃,aũ
θ̃. (40)
Using only the energy conservation in Eq. (35) we can derive
£ũζ̃a = −
3(µ̃+ p̃)
p̃,a +
(µ̃+ p̃)θ̃
ũaπ̃
bcσ̃bc
3(µ̃+ p̃)
π̃bcσ̃bc
3(µ̃+ p̃)
µ̃,aπ̃
bcσ̃bc
3(µ̃+ p̃)2
, (41)
where £ũ is a Lie derivative along ũa with £ũζ̃a ≡ ζ̃a;bũ
b + ζ̃bũb;a. Thus, for vanishing anisotropic pressure, we have
the Langlois-Vernizzi relation [21]
£ũζ̃a = −
3(µ̃+ p̃)
p̃,b −
. (42)
These equations are valid in a single component fluid, or in multiple component fluids for the collective fluid variables.
We can easily extend the relation to the individual fluid component as follows. We introduce
ζ̃(i)a ≡ h̃
α̃(i),b +
µ̃(i),b
3(µ̃(i) + p̃(i))
α(i) ≡ α̃(i),aũ
(i) ≡
θ̃(i). (43)
Using only Eq. (31) we can derive
£ũ(i) ζ̃(i)a = −
θ̃(i)
3(µ̃(i) + p̃(i))
p̃(i),a +
(µ̃(i) + p̃(i))θ̃(i)
µ̃(i),a
ũ(i)a
σ̃(i)bc + ũ
Ĩ(i)b
3(µ̃(i) + p̃(i))
σ̃(i)bc + ũ
Ĩ(i)b
3(µ̃(i) + p̃(i))
µ̃(i),a
σ̃(i)bc + ũ
Ĩ(i)b
3(µ̃(i) + p̃(i))
. (44)
Thus, for vanishing anisotropic pressure and direct interactions among fluids, i.e., π̃(i)ab = 0 = Ĩ(i)a, we have
£ũ(i) ζ̃(i)a = −
θ̃(i)
3(µ̃+ p̃)
h̃ b(i)a
p̃(i),b −
µ̃(i),b
. (45)
Application of these compact relations to large-scale conservation properties to the second order will be studied in
Sec. XI.
B. ADM equations
The ADM formulation [22] is presented in Eqs. (2)-(13),(47),(48) of [11]. Interpretation of the ADM fluid quantities
in Eqs. (45),(46) of [11] was based on the normal-frame fluid quantities; for relations to the energy-frame fluid
quantities, see Eq. (57) below. The ADM fluid quantities of individual component are introduced as
E(i) ≡ ñañbT̃
(i) = N
2T̃ 00(i) , J(i)α ≡ −ñbT̃
(i)α = NT̃
(i)α,
S(i)αβ ≡ T̃(i)αβ , S(i) ≡ h
αβS(i)αβ , S̄(i)αβ ≡ S(i)αβ −
hαβS(i). (46)
Equation (27) gives Eqs. (12),(13),(47),(48) in [11]. Equations (10),(12),(47) in [11] can be arranged as
(∂0 −N
α∂α)K
(Kαα )
= 4πG (E + S)− Λ + K̄αβK̄αβ −
N :αα, (47)
Kαα =
)−1 [
(∂0 −N
α∂α)E +
− S̄αβK̄αβ
, (48)
Kαα =
E(i) +
)−1 [
(∂0 −N
α∂α)E(i) +
N2Jα(i)
K̄αβ +
Ĩ(i)0 − Ĩ(i)αN
. (49)
Momentum conservation equations for the collective and individual components can be found in Eqs. (13),(48) of
[11]. By combining Eqs. (47)-(49) we can derive the ADM counterpart of the density perturbation equations in Eqs.
(38),(39).
V. SECOND-ORDER PERTURBATIONS
We use a metric convention in Eq. (49) of [11]
g̃00 ≡ −a
2 (1 + 2A) , g̃0α ≡ −a
2Bα, g̃αβ ≡ a
+ 2Cαβ
. (50)
The subindex 0 indicates the conformal time η with adη ≡ cdt. To the second-order in perturbation we introduce the
fluid four-vector of individual component as
ũα(i) ≡
V α(i), ũ
(i) =
1− A+
V α(i)V(i)α −B
αV(i)α
ũ(i)α = a
V(i)α −Bα +ABα + 2V
, ũ(i)0 = −a
1 +A−
V α(i)V(i)α
. (51)
In this definition of fluid four-vector we follow the notation in Eq. (53) of [11]. If we introduce ũα
≡ V̄ α
, we have
V α(i) = (1−A)V̄
(i). The fluid quantities of individual component are introduced as
µ̃(i) ≡ µ(i) + δµ(i), p̃(i) ≡ p(i) + δp(i), π̃(i)αβ ≡ a
2Π(i)αβ , π̃(i)0α = −a
2Π(i)αβV
, π̃(i)00 = 0, (52)
where from π̃ a
≡ 0 we have
Π α(i)α − 2C
αβΠ(i)αβ = 0. (53)
The energy-momentum tensor of individual component in the energy-frame follows from Eq. (28) as
T̃ 0(i)0 = −µ(i) − δµ(i) −
µ(i) + p(i)
V α(i) −B
V(i)α,
T̃ 0(i)α =
µ(i) + p(i)
V(i)α −Bα −AV(i)α + 2ABα + 2V
δµ(i) + δp(i)
V(i)α −Bα
+Π(i)αβ
T̃ α(i)β =
p(i) + δp(i)
δαβ +
µ(i) + p(i)
V α(i)
V(i)β −Bβ
+Π α(i)β − 2C
αγΠ(i)βγ . (54)
Using T̃ ab =
, and the total fluid quantities in Eq. (82) of [11] we have
µ(j), p =
p(j), (55)
for the background order fluid quantities, and
δµ(j) +
µ(j) + p(j)
V α(j) −B
V(j)α − Vα
δp(j) +
µ(j) + p(j)
V α(j) −B
V(j)α − Vα
(µ+ p)Vα =
µ(j) + p(j)
V(j)α +
δµ(j) + δp(j)
V(j)α − Vα
V(j)β − Vβ
Παβ =
Π α(j)β +
µ(j) + p(j)
vα(j) −B
V(j)β − Vβ
V(j)γ − Vγ
, (56)
for perturbed order fluid quantities to the second-order. From Eq. (46) we have the ADM fluid quantities based on
the energy-frame fluid quantities
E(i) = µ(i) + δµ(i) +
µ(i) + p(i)
V α(i) −B
V(i)α −Bα
J(i)α = a
µ(i) + p(i)
V(i)α −Bα +ABα + 2V
δµ(i) + δp(i)
V(i)α −Bα
+ Π(i)αβ
S(i) = 3
p(i) + δp(i)
µ(i) + p(i)
V α(i) −B
V(i)α −Bα
S̄(i)αβ = a
Π(i)αβ +
µ(i) + p(i)
V(i)α −Bα
V(i)β −Bβ
V(i)γ −Bγ
. (57)
We can compare Eq. (57) with the ADM fluid quantities based on the normal-frame fluid quantities in Eq. (76) of
[11]; in a single component case we simply delete (i) subindices.
In [11] the fluid quantities are based on the normal-frame vector. By taking ũ(i)α ≡ 0, ũ(i)a becomes the normal-
frame vector ña, see Eq. (54) in [11]. Based on the normal-frame vector, to the second order, the fluid quantities have
contributions due to the frame choice: for example, even in the zero-pressure fluid, the perturbed pressure based on
the normal-frame does not necessarily vanish to the second-order, see Eq. (58) below. By comparing Eq. (57) with
Eq. (76) of [11] we have
δµN(i) = δµ(i) +
µ(i) + p(i)
V α(i) −B
V(i)α −Bα
δpN(i) = δp(i) +
µ(i) + p(i)
V α(i) −B
V(i)α −Bα
Q(i)α =
µ(i) + p(i)
V(i)α −Bα +ABα + 2V
δµ(i) + δp(i)
V(i)α −Bα
+Π(i)αβ
ΠN(i)αβ = Π(i)αβ +
µ(i) + p(i)
V(i)α −Bα
V(i)β −Bβ
V(i)γ −Bγ
. (58)
For the total fluid quantities the relations between the two frames are presented in Eq. (87) of [11]. Thus, by replacing
all fluid quantities in Eqs. (99)-(107) of [11] using Eq. (58) and Eq. (87) of [11] we have the equations in the energy
frame. Using Q(i)α in Eq. (58), Eq. (54) gives
T̃ 0(i)α = (1−A)Q(i)α. (59)
As the fluid four-velocity of i-th component we can use either Q(i)α or V(i)α −Bα related by Eq. (58).
The kinematic quantities in the energy-frame are presented in Eqs. (63)-(66) of [11]. In Eq. (33) we introduced
kinematic quantities for the individual component. To the second order we can show
θ̃(i) = 3
Cα′α +
V α(i)|α +
A2 − 3
BαV(i)α −
ACα′α +
V α(i)
A,α + C
V α(i)V(i)α +
V α(i) −B
V(i)α −Bα
CαβC′αβ ,
σ̃(i)αβ = a
V(i)(α|β) + C
αβ −AC
V(i)(α −B(α
V(i)β) −Bβ)
+ V(i)(αA,β) + V
Cαβ|γ + 2V
(i)|(α
C(β)γ
Cγ′γ + V
(i)|γ
(i)|γ
+ Cγ′γ −AC
V(i)γ −Bγ
A,γ + V
Cδδ|γ − 2C
γδC′γδ
ω̃(i)αβ = a
V(i)[α −B[α +AB[α + 2V
V(i)[α −B[α
V(i)β] −Bβ]
+A,β]
ã(i)α = A,α +
V(i)α −Bα +ABα + 2V
− 2AA,α −A
V(i)α − Bα
V(i)α −Bα
+Bβ|α
σ̃(i)α0 = −V
σ̃(i)αβ , σ̃(i)00 = 0; ω̃(i)α0 = −V
ω̃(i)αβ , ω̃(i)00 = 0; ã(i)0 = −V
(i)ã(i)α, (60)
where a prime indicates the time derivative based on η.
The gauge transformation properties of the fluid quantities are presented in Eqs. (232)-(235) of [11] for the normal-
frame, and Eqs. (238) of [11] for the energy-frame. A prescription to get the gauge transformation properties for
individual fluid quantities is also presented below Eq. (235) of [11]. Under the gauge transformation we have
δµ̂(i) = δµ(i) −
µ′(i) + δµ
ξ0 − δµ(i),αξ
µ′′(i)ξ
0ξ0 + µ′(i)
ξ0ξ0′ + ξαξ0,α
δp̂(i) = δp(i) −
p′(i) + δp
ξ0 − δp(i),αξ
p′′(i)ξ
0ξ0 + p′(i)
ξ0ξ0′ + ξαξ0,α
V̂(i)α − B̂α + ÂB̂α + 2V̂
Ĉαβ = V(i)α −Bα +ABα + 2V
Cαβ + ξ
V(i)α −Bα
V(i)α −Bα
V(i)β −Bβ
ξβ,α −
V(i)α −Bα
A− ξ0′ −
ξ0,α − ξ
,β − ξ
0ξ0′,α − ξ
βξ0,αβ ,
Π̂(i)αβ = Π(i)αβ −
Π′(i)αβ + 2
Π(i)αβ
ξ0 −Π(i)αβ,γξ
γ − 2Π(i)γ(αξ
. (61)
A. Basic equations in the energy-frame
The basic set of equations with fluid quantities based on the normal-frame is presented in Eqs. (99)-(107) of [11].
By using Eq. (58) we can recover the equations with fluid quantities based on the energy frame. For convenience, in
the following we present the complete set of equations with fluid quantities in the energy frame. These equations are
written without taking any gauge conditions yet, thus in a sort of gauge-ready form. To the linear order this method
was suggested by Bardeen [23, 24].
Definition of δK:
K̄ + 3H + δK − 3HA+ Ċαα +
HA− Ċαα −
HBαBα +
+ 2Cαβ
Ċαβ +
≡ n0. (62)
Energy constraint equation:
16πGµ+ 2Λ− 6H2 −
R(3) + 16πGδµ+ 4HδK −
R(3)Cαα
δK2 − 16πG (µ+ p) (V α −Bα) (Vα −Bα)−
Ċαβ +
B(α|β)
Ċαβ +
Ċαα +
R(3)Cαγ C
− Cα|βα
+ Cαβ|γ
3Cαβ|γ − 2Cαγ|β
≡ n1. (63)
Momentum constraint equation:
Ċβα +
+B |βα
Ċγγ +
δK,α + 8πGa (µ+ p)
Vα −Bα +ABα + 2V
AδK,α − 8πGa
(δµ+ δp) (Vα −Bα) + (µ+ p)A (Vα −Bα) + Παβ
V β −Bβ
Ċβα +
+ B |βα
Ċγα +
B |γα +B
+ 2Cβγ
Ċαγ +
B(α|γ)
+ Cγ|βα − C
Ċβγ +
Ċγγ +
+ 2Cγδ
Ċγδ +
≡ n2α. (64)
Trace of the ADM propagation equation:
3Ḣ + 3H2 + 4πG (µ+ 3p)− Λ
+ δK̇ + 2HδK − 4πG (δµ+ 3δp) +
3Ḣ +
= AδK̇ −
δK,αB
δK2 + 8πG (µ+ p) (V α −Bα) (Vα −Bα)
3A2 −BαBα
2A∆A+A,αA,α −
∆ (BαBα) +A
+ 2CαβA,α|β
Ċαβ +
B(α|β)
Ċαβ +
Ċαα +
≡ n3. (65)
Tracefree ADM propagation equation:
Ċαβ +
Bα|β +B
Ċαβ +
Bα|β +B
Ċγγ +
Ċγγ +
A|γ γ
R(3)Cαβ −
R(3)Cγγ
− 8πGΠαβ
Ċαβ +
Bα|β +B
A+ 2Cαγ
Ċβγ +
B(β|γ)
Ċαβ +
Bα|β +B
A+ 2Cαγ
Ċβγ +
B(β|γ)
Ċαβ +
Bα|β +B
Ċαβ +
Bα |β +B
Bγ + δK
Ċαβ +
Bα|β +B
−A2 +BγBγ
− 2CαγA,β|γ −
Ċγγ +
A+ 2Cγδ
Ċγδ +
Ċγγ +
A+ 2Cγδ
Ċγδ +
Ċγγ +
Ċγγ +
Bδ + δK
Ċγγ +
−AA|γ γ +
−A2 +BδBδ
− 2CγδA,γ|δ −
− Cγ|δγ
Ċαγ +
Bα|γ +B
Cαδ|βγ + C
− Cαβ|δγ − C
+ 2Cαγ
Cδγ|βδ + C
β|γδ − C
− Cδδ|γβ
R(3)Cαγ C
Cαδ |β + C
− Cγδ|βC
γδ|α + 2Cαγ|δ
Cβδ|γ − Cβγ|δ
Cǫγ|δǫ + C
γ|ǫδ − C
− Cǫǫ|γδ
R(3)CδγC
2Cǫδ|ǫ − C
− Cγ|δγ
+ Cγδ|ǫ
2Cγǫ|δ − 3Cγδ|ǫ
−16πGCαγΠβγ + 8πG (µ+ p)
(V α −Bα) (Vβ −Bβ)−
δαβ (V
γ −Bγ) (Vγ −Bγ)
≡ n α4β . (66)
Energy conservation equation:
[µ̇+ 3H (µ+ p)] + δµ̇+ 3H (δµ+ δp)− (µ+ p) (δK − 3HA) +
(µ+ p)
V α −Bα +ABα + 2V βCαβ
δµ,αB
α + (δµ+ δp) (δK − 3HA) + (µ+ p)AδK +
H (µ+ p)
A2 −BαBα
a4 (µ+ p) (V α −Bα) (Vα −Bα)
(δµ+ δp) (V α −Bα) + Παβ (Vβ −Bβ)
(µ+ p)
−A (V α −Bα)|α − 2A,α (V
α −Bα) + 2
Cαβ (Vβ −Bβ)
− Cαα|β
V β −Bβ
Ċαβ +
≡ n5. (67)
Momentum conservation equation:
a4 (µ+ p)
Vα −Bα +ABα + 2V
(µ+ p)A,α +
δp,α +Π
= (µ+ p) (δK − 3HA) (Vα −Bα)−
(δµ+ δp) (Vα −Bα) + Π
α (Vβ −Bβ)
δp,α +Π
A− (δµ+ δp)A,α
− (µ+ p)
−AA,α +Bβ|αV
β + (Vα −Bα)|β V
β + (Vα −Bα)
V β −Bβ
CβγΠαγ
Πγα + C
Πβγ −A,βΠ
≡ n6α. (68)
In the multi-component situation we additionally have the energy and the momentum conservation of individual
component. Using the energy-frame fluid quantities, Eqs. (106),(107) in [11] become
µ̇(i) + 3H
µ(i) + p(i)
I(i)0
+ δµ̇(i) + 3H
δµ(i) + δp(i)
µ(i) + p(i)
(δK − 3HA)
µ(i) + p(i)
V α(i) −B
α +ABα + 2V
δI(i)0
δµ(i),αB
δµ(i) + δp(i)
(δK − 3HA) +
µ(i) + p(i)
AδK +
µ(i) + p(i)
A2 −BαBα
µ(i) + p(i)
V α(i) −B
V(i)α −Bα
δµ(i) + δp(i)
V α(i) −B
V(i)β −Bβ
µ(i) + p(i)
V α(i) −B
− 2A,α
V α(i) −B
V(i)β −Bβ
− Cα|βα
V(i)β −Bβ
Ċαβ +
δI(i)αB
α ≡ n(i)5, (69)
µ(i) + p(i)
V(i)α −Bα +ABα + 2V
µ(i) + p(i)
A,α +
δp(i),α +Π
(i)α|β
− δI(i)α
µ(i) + p(i)
(δK − 3HA)
V(i)α −Bα
δµ(i) + δp(i)
V(i)α −Bα
V(i)β −Bβ
δp(i),α +Π
(i)α|β
− δI(i)α
δµ(i) + δp(i)
µ(i) + p(i)
−AA,α +Bβ|αV
V(i)α −Bα
V(i)α −Bα
CβγΠ(i)αγ
−A,βΠ
≡ n(i)6α. (70)
By removing indices indicating the components in Eqs. (69),(70) we recover equations for the collective component
which coincide with the equations in a single component situation in Eqs. (67),(68).
To the background order, from Eqs. (69),(65) we have
µ̇(i) + 3H
µ(i) + p(i)
I(i)0, (71)
µ̇+ 3H (µ+ p) = 0, (72)
3Ḣ + 3H2 = −
µ(j) + 3p(j)
+ Λc2, (73)
µ(j) −
, (74)
where we recovered the speed of light c. Dimensions are
[G̺] = T−2, [c] = LT−1, [η] = 1, [p] = [µ] = [̺c2], [a] = L, [K] = 1, [Λ] = L−2, [I(i)0] = [µ]. (75)
Equation (72) follows from the sum of Eq. (71) over components. Equation (74) follows from integrating Eq. (73)
where K-term can be regarded as an integration constant; in Einstein’s gravity K-term can be normalized as the
sign of spatial curvature. Compared with the Newtonian background equations in Eq. (7), ignoring the direct inter-
action terms in Eq. (71), the presence of pressure terms in Eqs. (71)-(74) is the pure general relativistic effect. The
cosmological constant Λ can be introduced by hand even in the Newtonian case.
B. Decomposition
We decompose the metric to three perturbation types
A ≡ α, Bα ≡ β,α +B
α , Cαβ ≡ ϕg
+ γ,α|β + C
(α|β)
, (76)
where superscripts (v) and (t) indicate the transverse vector-type, and transverse-tracefree tensor-type perturbations,
respectively. We introduce
χ ≡ a
β + c−1aγ̇
, Ψ(v)α ≡ B
α + c
−1aĊ(v)α , (77)
which are spatially gauge-invariant to the linear order. We set
Kαα ≡ −3H + κ. (78)
We will identify κ with Newtonian velocity variable which will be an important step in our analysis, see Eqs.
(120),(194),(215).
For the fluid quantities we decompose
ũ(i)α = a
V(i)α −Bα +ABα + 2V
≡ av(i)α ≡ a
−v(i),α + v
, (79)
Π(i)αβ ≡
Π(i),α|β −
∆Π(i)
(i)(α|β)
(i)αβ
, δI(i)α ≡ δI(i),α + δI
. (80)
The perturbed fluid velocity variables v(i) and v
subtly differ from the ones introduced in [11]; see Sec. VC. For
the collective fluid component or for a single component case, we simply delete (i) subindices. For isotropic pressure
we introduce
δp ≡ c2sδµ+ e, c
. (81)
The perturbation variable e is called an entropic perturbation. Defined in this way e is gauge-invariant only to the
linear order. To the second order, from Eq. (61) we can derive the following gauge-invariant combination
δpδµ ≡ e−
. (82)
In our notation, δpδµ is a gauge-invariant combination which is the same as δp in the δµ = 0 slicing (temporal gauge)
condition to the second order; as the spatial gauge we take γ ≡ 0 to the second order; for the derivation, see the
prescription below Eq. (266) of [11]. In the multi-component case, we similarly have
δp(i)δµ(i) ≡ e(i) −
δµ(i)
µ̇(i)
ė(i) +
c2(i)
δµ(i)
, (83)
where
δp(i) ≡ c
(i)δµ(i) + e(i), c
(i) ≡
ṗ(i)
µ̇(i)
. (84)
We have [25]
e = erel + eint, erel ≡
c2(j) − c
δµ(j), eint ≡
e(j). (85)
Equation (56) gives
δµ(j) +
µ(j) + p(j)
vα(j)
v(j)α − vα
δp(j) +
µ(j) + p(j)
vα(j)
v(j)α − vα
(µ+ p) vα =
µ(j) + p(j)
v(j)α +
δµ(j) + δp(j)
v(j)α − vα
v(j)β − vβ
Παβ =
Π α(j)β +
µ(j) + p(j)
vα(j)
v(j)β − vβ
δαβ v
v(j)γ − vγ
, (86)
By recovering c, dimensions of the variables are
[g̃ab] = [g̃
ab] = [ũa] = 1, [T̃ab] = [µ], [g
] = 1, [∇] = [∆] = 1, [w] = [c2s] = 1,
[A] = [Bα] = [Cαβ ] = 1, [α] = [ϕ] = [β] = [γ] = [B
α ] = [C
α ] = [Ψ
α ] = [C
] = 1, [χ] = L, [κ] = T−1,
[δµ] = [δp] = [e] = [Παβ ] = [Π
] = [µ], [δ] = [Vα] = 1, [v] = [v
α ] = 1, [Π] = L
2[µ], [Π(v)α ] = L[µ]. (87)
Scalar-type perturbation equations can be derived from Eqs. (62)-(70)
κ− 3Hα+ 3ϕ̇+
χ = n0, (88)
4πGδµ+Hκ+
∆+ 3K
n1, (89)
∆+ 3K
χ− 12πG (µ+ p) av = n2 ≡
∆−1∇αn2α, (90)
κ̇+ 2Hκ− 4πG (δµ+ 3δp) +
3Ḣ +
α = n3, (91)
χ̇+Hχ− ϕ− α− 8πGΠ = n4 ≡
a2 (∆ + 3K)
∆−1∇α∇βn
4α, (92)
δµ̇+ 3H (δµ+ δp)− (µ+ p)
κ− 3Hα+
= n5, (93)
a4 (µ+ p) v
a4 (µ+ p)
a (µ+ p)
∆+ 3K
= n6 ≡ −
∆−1∇αn6α, (94)
δµ̇(i) + 3H
δµ(i) + δp(i)
µ(i) + p(i)
κ− 3Hα+
δI(i)0 = n5(i), (95)
µ(i) + p(i)
µ(i) + p(i)
µ(i) + p(i)
δp(i) +
∆ + 3K
Π(i) − δI(i)
= n6(i) ≡ −
µ(i) + p(i)
∆−1∇αn6α(i). (96)
Equations for the vector-type perturbation follow from Eqs. (64),(66),(68),(70)
∆ + 2K
Ψ(v)α + 8πG(µ+ p)v
n2α −∇α∆
−1∇βn2β
2α , (97)
Ψ̇(v)α + 2HΨ
α − 8πGΠ
α = 2a (∆ + 2K)
4α −∇α∆
−1∇γ∇βn
4α , (98)
[a4(µ+ p)v
a4(µ+ p)
∆+ 2K
n6α −∇α∆
−1∇βn6β
6α , (99)
[a4(µ(i) + p(i))v
a4(µ(i) + p(i))
∆ + 2K
µ(i) + p(i)
µ(i) + p(i)
µ(i) + p(i)
n6(i)α −∇α∆
−1∇βn6(i)β
6(i)α
.(100)
Equations for the tensor-type perturbation follow from Eq. (66)
+ 3HĊ
∆− 2K
− 8πGΠ
= n4αβ −
∇α∇β −
(∆ + 3K)
∆−1∇γ∇δn
−2∇(α (∆ + 2K)
∇γn4β)γ −∇β)∆
−1∇γ∇δn
4αβ . (101)
In order to derive eqs. (92,98,101) it is convenient to show
∇α∇β −
(χ̇+Hχ− ϕ− α− 8πGΠ) +
(α|β)
− 8πG
(α|β)
+ 3HĊ
∆− 2K
− 8πGΠ
= n4αβ , (102)
which follows from Eq. (66). Quadratic combinations of linear-order perturbation variables of all three types of
perturbations contribute to all three types of perturbation to the second order.
C. Comoving gauge and irrotational condition
In Eq. (180) of [11] we introduced
Q(i)α ≡
µ(i) + p(i)
−v̄(i),α + v̄
, (103)
where we put overbars to v̄(i) and v̄
in order to distinguish these from our new notations to be used in this paper;
in [11] we didn’t have overbars. From Eq. (58) we have
−v̄(i),α + v̄
≡ V(i)α −Bα +ABα + 2V
Cαβ +
δµ(i) + δp(i)
µ(i) + p(i)
V(i)α −Bα
Π(i)αβ
µ(i) + p(i)
. (104)
It is more convenient to introduce the decomposition in Eq. (79). Thus, we have
−v̄(i),α + v̄
≡ −v(i),α + v
δµ(i) + δp(i)
µ(i) + p(i)
−v(i),α + v
Π(i)αβ
µ(i) + p(i)
. (105)
The variable v̄
introduced in [11] cannot be regarded as a proper vector-type perturbation. However, if we also
take the temporal comoving gauge in [11] which sets v̄ ≡ 0 together with v̄
= 0, we have v = 0 = v
; these
are the same as taking the proper irrotational condition (v
= 0) and the temporal comoving gauge (v = 0). Our
analyses in [12, 13] are, in fact, based on taking these two conditions together. In the irrotational fluids, the temporal
comoving gauge v ≡ 0 leads to ũα = 0, thus ũa coincides with the normal frame four-vector ña.
From Eq. (60) we have
ω̃(i)αβ = av
(i)[α|β]
µ(i) + p(i)
−v(i),[α + v
(i)[α
δp(i) +
∆ + 3K
Π(i) − δI(i)
∆+ 3K
(i)β]
(i)β]
(106)
where we used Eq. (70) to the linear order. For vanishing vector-type perturbation we set v
≡ 0, etc. In this case,
we have
ω̃(i)αβ = −
µ(i) + p(i)
v(i),[α
δp(i) +
∆+ 3K
Π(i) − δI(i)
, (107)
which vanishes for δp(i) = 0 = Π(i) and δI(i) = 0.
VI. EFFECTS OF PRESSURE
We consider a single component situation without rotational perturbations. Equations (65),(67),(68), and Eq. (64)
to the linear order provide a complete set of equations we need in the following.
A. Irrotational case
As an irrotational fluid we ignore all vector-type perturbations, thus v
α = B
α = C
α = Π
α = 0. Quadratic
combinations of linear-order perturbations of all three-types of perturbations contribute to each type of second-order
perturbations. Thus, concerning the second-order scalar-type and vector-type perturbations, ignoring the pure vector-
type perturbation corresponds to ignoring the vector-type perturbation only to the linear order; to the linear order the
vector-type perturbation only has a decaying solution in the expanding phase. Assuming the background equations
are valid Eqs. (67),(68),(65),(64) give
δ̇ + 3H
− (1 + w)
v − 3Hα
(κ− 3Hα) + (1 + w)
H (1 + w)
α2 − β,αβ,α
χ,α|β + Ċ
(1 + w) v,α|β
+ γ,α|β + C
(1 + w) v,α
v,α −
δp,α +Π
+ ϕ,α − [(∆ + 4K)γ],α
(δµ+ δp) v,α +Παβv,β
, (108)
a4 (µ+ p) v,α
(µ+ p)α,α + δp,α +Π
(µ+ p) (v − β)
(v − β)
+(µ+ p) v,α
v − 3Hα
(δµ+ δp) v,α +Π
− (δµ+ δp)α,α + α
(µ+ p)α,α − δp,α −Π
− α,βΠ
α + 2
CγβΠαγ
−ΠγαC
+ΠβγC
, (109)
κ̇+ 2Hκ− 4πG (δµ+ 3δp) +
3Ḣ +
α = −
,α + ακ̇+
3α2 − β,αβ,α
+ 8πG (µ+ p) v,αv,α
2∆α (α+ ϕ) + α,α [α− ϕ+ (∆ + 4K)γ]
+ 2α,α|β
γ,α|β + C
∆ (β,αβ,α)
χ,α|β + Ċ(t)αβ
χ,α|β + Ċ
, (110)
κ = −
∆+ 3K
χ+ 12πGa (µ+ p) v. (111)
Equation (111) is valid to the linear order.
B. Comoving gauge
As the temporal and spatial gauge conditions we set
v ≡ 0 ≡ γ, (112)
thus, β = χ/a. Under these gauge conditions Eq. (109) gives
α = −
χ,αχ,α −
1 + w
1 + 3c2s
2(1 + w)
+ αΠ, (113)
where the imperfect fluid contributions (stresses) are denoted by αΠ with
∆ + 3K
1 + w
∆ + 3K
∆−1∇α
1 + c2s
1 + w
e,α +Π
1 + w
δ,β +
e,β +Π
Πβα + 2
CβγΠαγ
−ΠγαC
+ΠβγC
(114)
Using this, Eqs. (108)-(111) give
δ̇ − 3Hwδ − (1 + w) κ = −
,α + κδ +
1 + w
δ2 + δΠ, (115)
κ̇+ 2Hκ− 4πGµδ −
1 + w
∆+ 3K
δ = −
2(1 + w)
4πGµ−
1 + 2c2s
1 + w
∆+ 3K
1 + w
2Hδκ+
ϕ,αδ,α − 2ϕ∆δ − 2δ
,α|βC
χ,α|β + Ċ(t)αβ
χ,α|β + Ċ
+κΠ, (116)
κ = −
∆+ 3K
χ, (117)
where Eq. (117) is valid only to the linear order, and
δΠ ≡ 2H
∆+ 3K
κ+ 3H
1 + w
χ,α|β + Ċ
∆+ 3K
3Hδ − (1 + w) κ+
∆ + 3K
∆−1∇α
1 + c2s
1 + w
e,α +Π
1 + w
δ,β +
e,β +Π
Πβα + 2
CβγΠαγ
−ΠγαC
+ΠβγC
κΠ ≡ 12πGe
1 + w
3Ḣ +
∆ + 3K
1 + w
∆ + 3K
∆−1∇α
1 + c2s
1 + w
e,α +Π
1 + w
δ,β +
e,β +Π
CβγΠαγ
−ΠγαC
+ΠβγC
−2Hκ+ 4πG
1 + 6c2s
µδ + 12πGe−
1 + w
∆ + 3K
∆+ 3K
∆ + 3K
∆+ 3K
1 + w
∆ + 3K
ϕ,α − 2
∆+ 3K
),α|β
. (118)
Combining Eqs. (115),(116) we can derive
1 + w
(µ+ p)a
− c2s
δ = (1 + w)
χ,α|βχ,α|β −
1 + w
a2κδ − δ,αχ
1 + w
4πGµ−
1 + 2c2s
1 + w
∆+ 3K
2a2Hδκ+ ϕ,αδ,α − 2ϕ∆δ − 2δ
,α|βC
+ (1 + w) Ċ(t)αβ
χ,α|β + Ċ
+(1 + w) κΠ +
1 + w
1 + w
δΠ,αχ
,α. (119)
Notice that the equations above are valid in the presence of general K and Λ.
C. Newtonian correspondence
Guided by our success in the zero-pressure case, we continue to identify
κ ≡ −
∇ · u, (120)
to the second order. Using Eq. (117), assuming K = 0, we can identify
κ = −c
χ = −
∇ · u, u ≡
∇χ = −c∇vχ, (121)
to the linear order. We have recovered the speed of light c. We have [u] = LT−1. Equation (113) becomes
α = −
1 + w
1 + 3c2s
2(1 + w)
+ αΠ. (122)
Equations (115), (116) give
δ̇ − 3wHδ + (1 + w)
∇ · u = −
∇ · (δu) +
1 + w
Hδ2 + δΠ, (123)
∇ · (u̇+Hu) +
1 + w
δ = −
∇ · (u · ∇u)− Ċ(t)αβ
uα|β + Ċ
1 + w
1 + 2c2s
1 + w
δ2 + 2Hδ
∇ · u+
2ϕ∆δ − (∇ϕ) · ∇δ + 2δ,α|βC
− κΠ.(124)
We have [αΠ] = 1, [δΠ] = T
−1, and [κΠ] = T
−2. Combining these equations or Eq. (119) gives
1 + w
(µ+ p)a
− c2sc
1 + w
∇ · (u · ∇u)−
1 + w
1 + w
∇ · (δu)−
1 + w
+ (1 + w) Ċ(t)αβ
uα|β + Ċ
1 + 2c2s
1 + w
δ2 − c2s
2aHδ∇ · u+ 2ϕc2∆δ − c2 (∇ϕ) · ∇δ + 2c2δ,α|βC
+(1 + w) κΠ +
1 + w
1 + w
u · ∇δΠ. (125)
We note that, to the linear order, Eq. (125) is valid in the presence of general K and Λ, see Eq. (119).
D. Linear-order relativistic pressure corrections
To the linear order, ignoring the entropic perturbation e and the anisotropic stress Π, Eqs. (123)-(125) become
δ̇ − 3wHδ + (1 + w)
∇ · u = 0, (126)
∇ · (u̇+Hu) + 4πG̺δ = −
1 + w
, (127)
1 + w
a(1 + w)̺
, (128)
where ignoring the specific internal energy density ǫ we used µ = ̺c2, thus δ = δ̺/̺; in general we have µ = ̺
c2 + ǫ
[18]. Equation (128) can be expanded to give
1 + w
a(1 + w)̺
= δ̈ +
2− 6w + 3c2s
Hδ̇ −
1 + 8w − 6c2s − 3w
4πG̺− 12
w − c2s
) Kc2
5w − 3c2s
δ, (129)
which is valid in the presence of K and Λ. Equation of density perturbation in the comoving gauge was first derived
by Nariai in [26].
In a single component case the Newtonian equations in Eqs. (8),(12),(16) give
∇ · u = 0, (130)
∇ · (u̇+Hu) + 4πG̺δ = −
, (131)
δ̈ + 2Hδ̇ − 4πG̺δ =
. (132)
Comparing Eqs. (126),(127) with Eqs. (130),(131), we notice the presence of w ≡ p/(̺c2) term in three places in
the relativistic equations. Even to the linear order the presence of these terms should be regarded as pure general
relativistic effect of the isotropic pressure. The effects of pressures to the second order compared with the Newtonian
equations can be found in Eqs. (123)-(125) which should be compared with Newtonian equations in Eqs. (8),(12),(16).
Pressure has the genuine relativistic role in cosmology even in the background level.
VII. EFFECTS OF MULTI-COMPONENT
We assume zero-pressure medium, thus set
p(i) ≡ 0, δp(i) ≡ 0 ≡ Π(i)αβ . (133)
From Eq. (86) we notice that to the second order the collective fluid quantities differ from simple sum of individual
fluid quantities. Even in the zero-pressure mediums, we have
δµ(j) + µ(j)v
v(j)α − vα
µ(j)v
v(j)α − vα
µvα =
µ(j)v(j)α + δµ(j)
v(j)α − vα
Παβ =
vα(j)
v(j)β − vβ
δαβ v
v(j)γ − vγ
, (134)
thus δp 6= 0 6= Παβ to the second order.
Equations (69),(70),(65),(67),(68),(64) give
δ̇(i) − δK + 3HA+
1 + δ(i)
vα(i)
δ(i),αB
α + δ(i) (δK − 3HA) +AδK +
A2 −BαBα
−v(i)α
2v̇α(i) +Hv
Avα(i)|α −
(i) +
Cαβv(i)β
Cα|βα v(i)β , (135)
1 + δ(i)
v(i)α
A,α = (δK − 3HA) v(i)α
AA,α − δ(i)A,α −B
βBβ|α −
v(i)αv
− v(i)α|βB
β − v(i)βB
, (136)
δK̇ + 2HδK − 4πG (δµ+ 3δp) +
3Ḣ +
A = AδK̇ −
δK,αB
δK2 +
3A2 −BαBα
2A∆A+A,αA,α −
∆ (BαBα) +A
+ 2CαβA,α|β
Ċαβ +
B(α|β)
Ċαβ +
Ċαα +
+ 8πGµvαvα, (137)
δ̇ − δK + 3HA+
[(1 + δ) vα]|α = −
α + δ (δK − 3HA) +AδK +
A2 −BαBα
−vα (2v̇
α +Hvα)−
Avα|α −
Cαβvβ
Cα|βα vβ − 3H
, (138)
[a (1 + δ) vα]
A,α = (δK − 3HA) vα
AA,α − δ(i)A,α −B
βBβ|α −
− vα|βB
β − vβB
δp,α +Π
, (139)
Ċβα +
+B |βα
Ċγγ +
δK,α = −8πGa
µ(j)v(j)α. (140)
where Eq. (140) is valid to the linear order.
A. Irrotational case
Assuming an irrotational condition we ignore all vector-type perturbations. As the spatial gauge condition we take
γ ≡ 0, (141)
thus, β ≡ χ/a. Equations (135)-(140) become
δ̇(i) − κ+ 3Hα−
1 + δ(i)
δ(i),αχ
,α + δ(i) (κ− 3Hα) + ακ+
χ,αχ,α
+Hv(i),αv
α∆v(i) +
ϕ,αv(i),α −
ϕ∆v(i) −
C(t)αβv(i),α|β , (142)
1 + δ(i)
= (κ− 3Hα) v(i),α +
−αα,α +
χ,βχ,β|α + δ(i)α,α +
v(i),αv
v(i),βχ
, (143)
κ̇+ 2Hκ− 4πG (δµ+ 3δp) +
3Ḣ +
α = ακ̇−
3α2 −
χ,αχ,α
2α∆α+ α,αα,α −
(χ,αχ,α)− α
,αϕ,α + 2ϕ∆α+ 2C
(t)αβα,α|β
Ċ(t)αβ +
χ,α|β
χ,α|β
+ 8πGµv,αv,α, (144)
δ̇ − κ+ 3Hα−
[(1 + δ) v,α]|α = −
,α + δ (κ− 3Hα) + ακ+
χ,αχ,α
+Hv,αv
α∆v +
ϕ,αv,α −
ϕ∆v −
C(t)αβv,α|β − 3H
, (145)
[a (1 + δ) v]
α,α = (κ− 3Hα) v,α
−αα,α +
χ,βχ,β|α + δα,α +
δp,α +Π
, (146)
∆ + 3K
χ+ κ = 12πGa
µ(j)v(j). (147)
Equation (147) is valid to the linear order.
B. Linear perturbations
To the linear order Eqs. (142)-(146) give
δ̇(i) − κ+ 3Hα− c
v(i) = 0, (148)
v̇(i) +Hv(i) −
α = 0, (149)
κ̇+ 2Hκ− 4πG
̺(j)δ(j) +
3Ḣ + c2
α = 0, (150)
∆+ 3K
χ+ κ =
̺(j)v(j), (151)
δ̇ − κ+ 3Hα− c
v = 0, (152)
v̇ +Hv −
α = 0, (153)
where we have recovered the speed of light c. The following additional equations can be found in Eqs. (195),(196),(199)
of [11]
κ− 3Hα+ 3ϕ̇+ c
χ = 0, (154)
4πG̺δ +Hκ+ c2
∆+ 3K
ϕ = 0, (155)
(χ̇+Hχ)− ϕ− α = 0. (156)
Equation (156) gives
αχ = −ϕχ. (157)
Equation (153) gives
αv = 0. (158)
Equations (151), (154) give
ϕ̇v =
χv. (159)
Thus, for K = 0 we have
ϕ̇v = 0, (160)
which is valid even in the presence of multi-components and the cosmological constant. Equations (149), (153) give
v̇χ +Hvχ +
ϕχ = 0, (161)
v̇(i)χ +Hv(i)χ +
ϕχ = 0. (162)
Equations (151), (155) give
∆+ 3K
ϕχ = −4πG̺δv = −4πG
̺(j)δ(j)v = −4πG
̺(j)δ(j)v(j) . (163)
We can derive density perturbation equation in many different temporal gauge (hypersurface) conditions all of which
naturally correspond to gauge-invariant variables. In a single component case, density perturbation in the comoving
gauge (v = 0) is known to give Newtonian result. In the multi-component situation we have many different comoving
gauge conditions. Here, we consider two such gauges for δ(i) variable: one based on v = 0 gauge, and the other based
on v(ℓ) = 0 gauge for a specific ℓ.
(I) Equation (148) evaluated in the v = 0 gauge, and using Eq. (151) we can derive
δ̇(i)v − c
v(i)χ − c
vχ = 0, (164)
where we used χv ≡ χ − av ≡ −avχ, vv(i) ≡ v − v(i), and αv ≡ α − c
−1 (av)
= 0. Using Eqs. (161),(162),(163), we
δ̈(i)v + 2Hδ̇(i)v − 4πG
̺(j)δ(j)v = 0. (165)
This coincides exactly with the Newtonian result in Eq. (16) to the linear order, even in the presence of K. Thus, we
may identify δ(i)v as the Newtonian density perturbation δi to the linear order even in the presence of K. For K = 0
we may also identify −c∇v(i)χ as the Newtonian velocity perturbation ui. However, in the presence of K, we cannot
identify the relativistic variables which correspond to the Newtonian velocity perturbation of individual component.
Therefore, to the linear order we have the following Newtonian correspondences
δi = δ(i)v, ui = −c∇v(i)χ, (166)
where the latter one is valid only for K = 0.
(II) Evaluating Eq. (148) in the v(ℓ) = 0 gauge for a specific ℓ, and using Eq. (151) we can derive
δ̇(i)v(ℓ) − c
∆+ 3K
v(ℓ)χ =
v(j) − v(ℓ)
v(i) − v(ℓ)
, (167)
where we used χv(ℓ) ≡ χ − av(ℓ) ≡ −av(ℓ)χ, v(i)v(ℓ) ≡ v(i) − v(ℓ), and αv(ℓ) ≡ α − c
av(ℓ)
= 0. Using Eqs.
(161),(162),(163), we have
δ̈(i)v(ℓ) + 2Hδ̇(i)v(ℓ) − 4πG
̺(j)δ(j)v(j) =
v(ℓ) − v(j)
. (168)
The terms in right-hand-sides of Eqs. (167),(168) look like relativistic correction terms present even to the linear
order based on the variable δ(i)v(ℓ) . Since no such correction terms appear in Eq. (165) based on the variable δ(i)v,
the relativistic correction terms in Eq. (168) can be regarded as being caused by a complicated hypersurface (gauge)
choice.
1. Exact solutions
Assuming K = 0, we can identify Newtonian perturbation variables as
δ ≡ δv, κv ≡ −
∇ · u, u ≡ −c∇vχ, δΦ ≡ −c
2ϕχ, δi ≡ δ(i)v, ui ≡ −c∇v(i)χ. (169)
Equations (152),(150),(148),(149),(163) become
δ̇ = −
∇ · u, (170)
u̇+Hu = −
∇δΦ, (171)
δ̇i = −
∇ · ui, (172)
u̇i +Hui = −
∇δΦ, (173)
δΦ = 4πG̺δ = 4πG
̺jδj . (174)
Under the identification in Eq. (169) these equations are valid in both Newton’s and Einstein’s gravity theories.
Equations (170), (171), (174), and Eqs. (172),(173),(174), respectively, give
δ̈ + 2Hδ̇ − 4πG̺δ =
= 0, (175)
δ̈i + 2Hδ̇i − 4πG
̺jδj = 0. (176)
Equation (175) has an exact solution
δ(x, t) = H
cg(x)
∫ t dt
+ cd(x)
, (177)
where cg and cd are integration constants which indicate the relatively growing and decaying solutions in expanding
phase; we do not consider the lower bound of integration which is absorbed to the cd mode. Equations (175), (176),
and the solution in Eq. (177) are valid considering general K and Λ in the background world model. Equation (174)
can be solved to give
δΦ = −G̺a2
δ(x′, t)
|x′ − x|
d3x′. (178)
From Eqs. (170),(171),(174) we can show [14]
u = −a
4πG̺a2
D(x), ∇ ·D ≡ 0, (179)
where the D term is the solution of the homogeneous part of Eq. (171); it decouples from the density inhomogeneity
and corresponds to the peculiar velocity in the background world model. Since the D term is not connected to the
density inhomogeneity and simply decays, we may ignore it to the linear order.
Now, for the individual component, from Eqs. (172),(173) we have
ui = −a
4πG̺a2
∇di(x) +
Di(x), ∇ ·Di ≡ 0, (180)
δi = δ + ci(x)−∆di(x)
∫ t dt′
a2(t′)
, (181)
̺jdj ≡ 0 ≡
̺jcj . (182)
The ci and di are the two isocurvature-type (δ = 0, thus δΦ = 0) solutions. It happens that the relatively decaying
isocurvature-type solution, i.e., di-mode, temporally behaves the same as the peculiar velocity in the background, i.e.,
Di-mode. The relatively growing isocurvature-type solution ci does not contribute to the ui, see Eq. (172). From
Eqs. (179),(180) we have
ui − u =
[∇di(x) +Di(x)−D(x)] , (183)
which simply decays; the Di and D solutions are divergence-free and decoupled from the density perturbation, and
are the peculiar velocity perturbation present in the background world model.
C. Comoving gauge
To the linear order, only the hypersurface condition v = 0 (the comoving temporal gauge) allows the density
perturbation equation presented in the Newtonian form. Thus, we take
v ≡ 0, (184)
even to the second order. We take γ ≡ 0 in Eq. (141) as the spatial gauge condition. Equation (146) gives
α = −
χ,αχ,α −
vα(j)v(j)α +∆
vα(j)v
(j)|β
. (185)
Using this, Eqs. (142)-(145) give
δ̇ − κ = −
,α + δκ+
µ(j)v
(j)v(j)α + 3H
∆−1∇α
vα(j)v
(j)|β
, (186)
κ̇+ 2Hκ− 4πG̺δ = −
Ċ(t)αβ +
χ,α|β
χ,α|β
3Ḣ + 8πG̺+ c2
µ(j)v
(j)v(j)α +
3Ḣ + c2
∆−1∇α
vα(j)v
(j)|β
, (187)
δ̇(i) − κ+ c
1 + δ(i)
vα(i)
δ(i),αχ
,α + δ(i)κ+Hv(i)αv
ϕ,αv(i)α − 2ϕv
(i)|α − 2v
µ(j)v
(j)v(j)α + 3H
∆−1∇α
vα(j)v
(j)|β
, (188)
1 + δ(i)
v(i)α
v(i)βχ
+ κv(i)α −
v(i)αv
µ(j)∇α
v(j)β
(j)|β
. (189)
From Eqs. ((186),187), and Eqs. (187)-(189) we can derive, respectively,
− 4πG̺δ (1 + δ) = −
Ċ(t)αβ +
χ,α|β
χ,α|β
2Ḣ + 4πG̺+
µ(j)v
(j)v(j)α +
6Ḣ + c2
∆−1∇α
vα(j)v
(j)|β
, (190)
δ̇(i) +
δ(i),αχ
− 4πG̺δ
1 + δ(i)
Ċ(t)αβ +
χ,α|β
χ,α|β
(κ+ ϕ̇)
v(i)α + 2 (κ− ϕ̇) v
(i)|α
v(i)αχ
vα(i)v
+ Ḣv(i)αv
(i) +
3Ḣ + 4πG̺
µ(j)v
(j)v(j)α + 6Ḣ
∆−1∇α
vα(j)v
(j)|β
. (191)
Notice the O(vα
v(j)α) correction terms are present in Eqs. (186),(187),(190) even in the single component situation.
Except for these O(vα
v(j)α) terms, the remaining parts of these equations coincide with the ones in the single
component situation.
D. Newtonian correspondence
For K = 0, to the linear order, we have
χv ≡ χ− av ≡ −avχ, κv = −c
χv = c
vχ, v(i)v ≡ v(i) − v = v(i)χ − vχ, ϕ̇v = 0. (192)
To the linear order we identify
u ≡ −c∇vχ, ui ≡ −c∇v(i)χ. (193)
Now, to the second order, we attempt identifying the Newtonian perturbation variables δ, δi, u, and ui as
κv ≡ −
∇ · u, χv ≡
u, u ≡ ∇u, v(i)v ≡
(ui − u) , δ ≡ δv, δi ≡ δ(i)v. (194)
Using these identifications Eqs. (186)-(191) can be written as
∇ · u = −
∇ · (δu) +H
|uj − u|
2 + 3∆−1∇ · [(uj − u)∇ · (uj − u)]
, (195)
∇ · (u̇+Hu) + 4πG̺δ = −
∇ · (u · ∇u)− Ċ(t)αβ
uα|β + Ċ
4πG̺− c2
|uj − u|
12πG̺− c2
∆−1∇ · [(uj − u)∇ · (uj − u)]
, (196)
δ̇i +
∇ · ui = −
∇ · (δiui) +
2ϕ∇ · (ui − u)− (ui − u) · ∇ϕ+ 2 (u
i − u
|ui − u|
2 + 3H
|uj − u|
2 +∆−1∇ · [(uj − u)∇ · (uj − u)]
, (197)
∇ · (u̇i +Hui) + 4πG̺δ = −
∇ · (ui · ∇ui)− Ċ
(t)αβ
uα|β + Ċ
|uj − u|
2 + 3∆−1∇ · [(uj − u)∇ · (uj − u)]
, (198)
− 4πG̺δ = −
[a∇ · (δu)]
∇ · (u · ∇u) + Ċ(t)αβ
uα|β + Ċ
4πG̺−
|uj − u|
24πG̺− c2
∆−1∇ · [(uj − u)∇ · (uj − u)]
, (199)
a2δ̇i
− 4πG̺δ = −
[a∇ · (δiui)]
∇ · (ui · ∇ui) + Ċ
i + Ċ
(t)αβ
{∆ [u · (ui − u)]−∇ · [(ui − u) · ∇u+ u · ∇ (ui − u)]}
|ui − u|
2 − 8πG
|uj − u|
2 + 3∆−1∇ · [(uj − u)∇ · (uj − u)]
. (200)
In Sec. VII B 1 we have shown that, to the linear order, (ui − u) simply decays (∝ a
−1) in an expanding phase.
From Eq. (183) we have
ui − u =
[∇di(x) +Di(x)−D(x)] ,
̺jdj ≡ 0, ∇ ·D ≡ 0 ≡ ∇ ·Di. (201)
Thus, ui−u simply decays in an expanding background. If we ignore these contributions from the velocity differences,
except for the presence of the tensor-type perturbation, Eq. (200) coincides exactly with the zero-pressure limit of
Newtonian result in Eq. (16). Terms in the last two lines of Eq. (200) are relativistic correction terms which vanish
for a single component case leading to Eq. (199); the second line in Eq. (199) also vanishes in the single component
case.
Ignoring quadratic combination of (ui − u) terms, we have
∇ · u = −
∇ · (δu) , (202)
∇ · (u̇+Hu) + 4πG̺δ = −
∇ · (u · ∇u)− Ċ(t)αβ
uα|β + Ċ
, (203)
− 4πG̺δ = −
[a∇ · (δu)]
∇ · (u · ∇u) + Ċ(t)αβ
uα|β + Ċ
, (204)
δ̇i +
∇ · ui = −
∇ · (δiui) +
2ϕ∇ · (ui − u)− (ui − u) · ∇ϕ+ 2 (u
i − u
, (205)
∇ · (u̇i +Hui) + 4πG̺δ = −
∇ · (ui · ∇ui)− Ċ
(t)αβ
uα|β + Ċ
, (206)
a2δ̇i
− 4πG̺δ = −
[a∇ · (δiui)]
∇ · (ui · ∇ui) + Ċ
i + Ċ
(t)αβ
{∆ [u · (ui − u)]−∇ · [(ui − u) · ∇u+ u · ∇ (ui − u)]} . (207)
Equations (202)-(204) coincide with the density and velocity perturbation equations of a single component medium
[12]; thus, except for the contribution from gravitational waves, these equations coincide with ones in the Newtonian
context.
If we further ignore (ui − u) terms appearing in the pure second-order combinations, Eqs. (205)-(207) become
δ̇i +
∇ · ui = −
∇ · (δiui) , (208)
∇ · (u̇i +Hui) + 4πG̺δ = −
∇ · (ui · ∇ui)− Ċ
(t)αβ
uα|β + Ċ
, (209)
a2δ̇i
− 4πG̺δ = −
[a∇ · (δiui)]
∇ · (ui · ∇ui) + Ċ
i + Ċ
(t)αβ
. (210)
Notice that, by ignoring i-indices, Eqs. (208)-(210) coincide with Eqs. (202)-(204). In this context, except for the
contribution from gravitational waves, the above equations coincide exactly with ones in the Newtonian context even
in the multi-component case; compare with Eqs. (8),(9),(16) without pressure. In the single component situation such
a relativistic/Newtonian correspondence to the second order was shown in [11, 12]. In the present case, the same
equation valid in the single component is now valid in the multi-component case for the collective fluid variables. This
justifies our identifications of Newtonian perturbation variables in Eq. (194).
VIII. EFFECTS OF CURVATURE
We consider a single zero-pressure, irrotational fluid. We take the temporal comoving gauge (v ≡ 0) and the spatial
γ ≡ 0 gauge. In the presence of background curvature the basic equations are presented in Eqs. (115)-(117) for
nonvanishing pressure, or Eqs. (188)-(191) for zero-pressure multiple component fluids. By setting pressures equal to
zero in Eqs. (115)-(117), or from Eqs. (187)-(190), we have
δ̇ − κ = −
,α + δκ, (211)
κ̇+ 2Hκ− 4πG̺δ = −
Ċ(t)αβ +
χ,α|β
χ,α|β
, (212)
δ̈ + 2Hδ̇ − 4πG̺δ = 4πG̺δ2 −
(δ,αχ
Ċ(t)αβ +
χ,α|β
χ,α|β
, (213)
∆+ 3K
χ+ κ = 0, (214)
where Eq. (214) is valid to the linear order. Compared with the situation with vanishing curvature, the effects of
curvature in the above perturbed set of equations appear only in the linear-order relation between κ and χ in Eq.
(214).
A. Newtonian correspondence
Considering the successful Newtonian correspondence to the linear order even in the presence of the background
curvature, we assume the identification in Eq. (120) is valid to the second order. Then, to the linear order, from Eq.
(214) we have
κ ≡ −
∇ · u ≡ −
u = −c
∆+ 3K
χ, (215)
where u ≡ ∇u and χ = χv = −avχ. Thus, to the linear order, formally we have
∆+ 3K
u. (216)
In the presence of curvature, the scalar-type perturbation can be handled by solving Eqs. (211),(212),(214) together
with the identifications made above. We can formally separate the effects of pure curvature contribution. Using Eqs.
(215),(216), Eqs. (211),(212),(214) become
∇ · u = −
∇ · (δu) +
(∇δ) · ∇
∆+ 3K
, (217)
∇ · (u̇+Hu) + 4πG̺δ = −
∇ · (u · ∇u)− Ċ(t)αβ
u,α|β
∆+ 3K
∆+ 3K
∇ · u
2∆+ 3K
∆+ 3K
∇ · u
∆+ 3K
),α|β
∆+ 3K
. (218)
From Eq. (74), the K term can be written as
(Ωt − 1) , Ωt ≡ Ω+ ΩΛ, Ω ≡
, ΩΛ ≡
. (219)
IX. EFFECTS OF VECTOR-TYPE PERTURBATION
The spatial C-gauge sets
γ ≡ 0 ≡ C(v)α . (220)
The remaining variables under this gauge condition are completely free of the spatial gauge modes and have unique
spatially gauge-invariant counterparts. If we simultaneously take any temporal gauge which also removes the temporal
gauge mode completely, all the remaining variables have corresponding unique gauge-invariant counterparts. The
above statements are true to all orders in perturbations, see Sec. VI of [11]. From Eq. (77) we have
χ, B(v)α ≡ Ψ
α . (221)
Thus, we have
χ,α +Ψ
α , Cαβ ≡ ϕg
. (222)
As the temporal comoving gauge we set
v ≡ 0. (223)
As we mentioned, the remaining variables under these gauge conditions are completely free of the gauge modes and
have unique gauge-invariant counterparts to all orders in perturbation, see [11].
A. Linear perturbations
To the linear order, the three types of perturbations decouple, and evolve independently. The rotational perturbation
is described by Eqs. (206)-(209) in [11]:
∆ + 2K
Ψ(v)α +
(µ+ p) v(v)α = 0, (224)
a4 (µ+ p) v
a4 (µ+ p)
∆+ 2K
, (225)
Ψ̇(v)α + 2HΨ
Π(v)α , (226)
where the last equation follows from the first two. Compared with Bardeen’s notation in [16], we have
B(v)α = B
(1)Q(1)α , C
α = −
α , Ψ
α = ΨQ
B(1) −
Q(1)α ,
v(v)α = vcQ
v(1) −B(1)
Q(1)α , v
α = v
v(1) −
Q(1)α , (227)
where Bardeen’s v
c and v
s , thus our v
α and v
α + Ψ
α , are related to the vorticity and the shear, respectively.
From Eq. (60), to the linear order, we have
ω̃αβ = av
[α|β]
σ̃αβ = a
∇α∇β −
. (228)
Bardeen called Ψ
α a ‘frame-dragging potential’. The difference between v
c and v
s is crucially important to show
Mach’s principle including the linear order rotational perturbation in [27].
In the absence of anisotropic stress which can act as a sink or source of the angular momentum, we have
Angular momentum ∝ a4 (µ+ p) v(v)α ∝ a
2Ψ(v)α ∝ constant in time. (229)
Thus, for vanishing anisotropic stress, we have
v(v)α ∝
a4(µ+ p)
, Ψ(v)α ∝
. (230)
In the zero-pressure limit we have
v(v)α ∝
, Ψ(v)α ∝
. (231)
B. Second-order perturbations
We consider a single component situation with general pressure. We set K ≡ 0. To the linear-order we use
χ ≡ χv ≡ −avχ, κ ≡ κv = c
vχ. (232)
The scalar-type perturbation is described by Eqs. (67),(65) which give
δ̇ + 3H
c2s − w
δ + 3H
− (1 + w) κ
= −3H (1 + w)
v ,αχ −Ψ
vχ,α −Ψ
v ,αχ − Ψ
vχ − 3Hα
+ (1 + w)αc
vχ + (1 + w) v
1− 3c2s
Hv(v)α +
Π(v)α
(1 + w)
− (2α+ ϕ)
v(v)α + 2C(t)αβv
Παα −
vχ,α|β +
(δµ+ δp)
v(v)α +
, (233)
κ̇+ 2Hκ−
(δµ+ 3δp) = −
3Ḣ + c2
v ,αχ −Ψ
vχ,α −Ψ
− 2Hαc
(δµ+ 3δp)α+
Ḣα2 + (α+ 2ϕ) c2
α,α (α− ϕ)
C(t)αβα,α|β +
(µ+ p) v(v)αv(v)α
v ,αχ −Ψ
Ċ(t)αβ −
v ,α|βχ +
Ψ(v)(α|β)
vχ,α|β +
. (234)
The vector-type perturbation is described by Eq. (68) which gives
a4 (µ+ p) v(v)α
(µ+ p)α,α +
δp,α +Π
−αδp+
(µ+ p)
v ,βχ −Ψ
vχ,β −Ψ
α,αδµ
+(µ+ p)
vχ − 3Hα
v(v)α −
(µ+ p)
−v ,βχ +Ψ
v(v)β − v ,βχ +Ψ
− ϕ,βΠ
α + ϕ,αΠ
+ 2C(t)βγΠαβ|γ + C
(δµ+ δp) v(v)α +Π
≡ µcAα. (235)
We have [Aα] = L
−1. From this we have
(µ+ p)α+ δp+
Π = aµ∆−1∇ ·A, (236)
a4 (µ+ p) v(v)α
Π(v)α = µc
∆−1∇ ·A
. (237)
Equation for the tensor-type perturbation (gravitational waves) follows from Eqs. (66),(102).
C. Zero-pressure case
In the zero-pressure limit, Eqs. (236),(235) give
α = a∆−1∇ ·A, (238)
aAα ≡ −
v ,βχ −Ψ
vχ,β −Ψ
v ,βχ −Ψ
v ,βχ −Ψ
v(v)β , (239)
thus, α is purely second-order, and
v ,βχ −Ψ
vχ,β −Ψ
= ∆−1∇α
v ,βχ −Ψ
v ,βχ −Ψ
v(v)β
.(240)
Equations (233), (234), (237) give
δ̇ − κ =
v ,αχ −Ψ
(v)α − v(v)α
+Hv(v)α v
(v)α +
−ϕ,αv(v)α + 2C
(t)αβv
−3H∆−1∇α
v ,βχ −Ψ
v ,βχ −Ψ
v(v)β
, (241)
κ̇+ 2Hκ− 4πG̺δ =
v ,αχ −Ψ
v ,βχ −Ψ
+ Ċ(t)αβ
vχ,α|β −Ψ
+8πG̺v(v)αv(v)α −
3Ḣ + c2
∆−1∇α
v ,βχ −Ψ
v ,βχ −Ψ
v(v)β
,(242)
v̇(v)α +Hv
v ,βχ −Ψ
v ,βχ −Ψ
v(v)β
v ,βχ −Ψ
v ,βχ −Ψ
v(v)β
. (243)
D. Newtonian correspondence
In order to compare with Newtonian equations we continue identifying κ as in Eq. (120) to the second order. To
the linear order we identify
u ≡ −c
∇vχ − v
≡ ∇u+ u(v), (244)
u ≡ −cvχ, u
(v) ≡ cv(v). (245)
We introduce the following notations
Uα ≡ uα + cΨ
α , Ũα ≡ u,α + cΨ
α . (246)
As mentioned below Eq. (227), to the linear order, uα and Uα are related to the vorticity and the shear, respectively.
Equations (241)-(243) become
∇ · u = −
∇ · (δU) +
C(t)αβu
(v) · ∇ϕ+
(v) · u(v) + 3∆−1∇α
Uβ + u(v)βŨβ|α
,(247)
∇ · (u̇+Hu) + 4πG̺δ = −
∇ · (U · ∇U)− Ċ(t)αβ
Ũα|β
−u(v) · u(v) +
∆−1∇α
Uβ + u(v)βŨβ|α
, (248)
(v) +Hu(v) = −
U · ∇u−∇∆−1∇ · (U · ∇u)
. (249)
From Eq. (249) we notice that the tensor-type perturbation does not affect the vector-type perturbation to the second
order. The pure scalar-type perturbation also cannot generate the vector-type perturbation to the second-order; the
same is true in the Newtonian case, see below Eq. (15).
From Eq. (224), to the linear order, we have
v(v)α =
Ψ(v)α , (250)
where k is a comoving wavenumber with ∆ ≡ −k2, thus [k] = 1. Since (ck)/(aH) ∼ (visual-horizon)/(scale), we have
far inside horizon : v(v)α ≫ Ψ
α , U ≃ u = ∇u+ u
(v), Ũ ≃ ∇u,
far outside horizon : v(v)α ≪ Ψ
α , U ≃ Ũ = u,α + cΨ
α . (251)
Apparently, contributions of vector-type perturbation to the second order depend on the visual-horizon scale.
Far inside the horizon, we can ignore cΨ
α compared with u
α . In the matter dominated era we have
δv, (252)
where we used Eqs. (293),(329),(330) of [11]. Thus, the third term in the right-hand-side of Eq. (247) is (aH/kc)2-order
smaller than the first term. The fourth (and last) term in the right-hand-side of Eq. (247) is (aH/kc)[u(v)/(cδ)]-order
smaller than the first term. The third (and last) term in the right-hand-side of Eq. (248) is also (aH/kc)2-order
smaller than the first term. Thus, Eqs. (247)-(249) give
∇ · u = −
∇ · (δu) +
C(t)αβu
, (253)
∇ · (u̇+Hu) + 4πG̺δ = −
∇ · (u · ∇u)− Ċ(t)αβ
u,α|β + cΨ
, (254)
(v) +Hu(v) = −
u · ∇u−∇∆−1∇ · (u · ∇u)
. (255)
Thus, if we could ignore the tensor-type combination in Eqs. (253),(254), Eqs. (253)-(255) coincide exactly with the
Newtonian equations: see Eqs. (8),(12),(15) ignoring the pressure terms and the subindices i. The vector-tensor
combinations in Eq. (253),(254) are new relativistic contributions of the vector-type perturbations; compare these two
equations with Eqs. (17),(18) which are valid in the absence of the vector-type perturbations. Notice the form of last
term u,α|β + cΨ
in Eq. (254) which subtly differs from the expression uα|β in Eq. (18).
Contributions from the vector-type perturbation become more complicated near and outside the horizon scale.
The presence of vector-type metric perturbation Ψ
α , the scalar-type curvature perturbation ϕ, and the tensor-type
perturbation C
coupled with the vector-type perturbation give additional effects.
E. Pure vector-type perturbations
In Sec. VII-E of [11] we have considered a situation with pure vector-type perturbation. As the analysis was made
based on the fluid quantities in the normal frame, in the following we present the case based on the fluid quantities
in the energy frame. Considering only the vector-type perturbation of a fluid, Eq. (68) gives
a4(µ+ p)
(µ+ p) v(v)α +
(α|β)
v(v)β
c(∆ + 2K)Π
2a2(µ+ p)
v(v)β +Ψ(v)|β
v(v)βv
. (256)
Thus, for Π
α = 0 we have
a4(µ+ p)
a4 (µ+ p) v(v)α
v(v)β +Ψ(v)|β
v(v)βv
. (257)
This differs from Eq. (365) of [11] which is due to the difference in the frame choice. The momentum constraint
equation in Eq. (101) of [11] becomes
∆ + 2K
Ψ(v)α +
(µ+ p) v(v)α = −
(α|β)
v(v)β . (258)
Under the gauge transformation, from Eq. (61) we have
v̂(v)α = v
α − v
ξ(v)β,α − v
ξ(v)β . (259)
To the linear order, from Eq. (230) of [11] we have B̂
α = B
α + ξ
α . We consider a gauge transformation from the
C-gauge (C
α ≡ 0, without hat) to the B-gauge (B
α ≡ 0, with hat). We have B̂
α ≡ 0, and B
α |C−gauge = −ξ
Thus,
ξ(v)α = −
B(v)α |C−gaugedη = −a
2Ψ(v)α
∫ η dη
, (260)
where we used B
α |C−gauge = Ψ
α ∝ a
−2. Thus, Eq. (259) gives
v(v)α |B−gauge =
v(v)α −
Ψ(v)β,α − v
Ψ(v)β
∫ η dη
C−gauge
. (261)
X. EQUATIONS WITH FIELDS
A. A minimally coupled scalar field
Equations in the case of a minimally coupled scalar field are presented in Eqs. (112)-(114) of [11]. The equation
of motion in Eq. (112) and the full Einstein’s equations in Eqs. (99)-(105) expressed using the normal-frame fluid
quantities together with the normal-frame fluid quantities for the scalar field in Eqs. (114) all in [11] provide a
complete set of equations we need to the second-order. The fluid quantities in Eq. (114) of [11] are presented in the
normal-frame four-vector and it is convenient to know the conventionally used fluid quantities which are based on the
energy-frame four-vector. These latter quantities can be read from Eqs. (88),(114) of [11] and Eq. (79) as
QN(φ)α = −
φ̇δφ,α + δφ,α
δφ̇− φ̇A
µ(φ) + p(φ)
V (φ)α −Bα +ABα + 2V
(φ)βCαβ
δµ(φ) + δp(φ)
V (φ)α −Bα
µ(φ) + p(φ) + δµ(φ) + δp(φ)
−v(φ),α + v
(φ,v)
δµ(φ) = δµN(φ) −
δφ,αδφ,α, δp
(φ) = δpN(φ) −
δφ,αδφ,α,
δφ,αδφ,β −
δφ,γδφ,γ
. (262)
Thus, using Eq. (114) of [11] we have
v(φ) =
∆−1∇α
δφ̇−Aφ̇
v(φ,v)α =
δφ̇− φ̇A
δφ,α −∇α∆
δφ̇− φ̇A
δµ(φ) = φ̇δφ̇− φ̇2A+ V,φδφ+
δφ̇2 −
δφ,αδφ,α +
V,φφδφ
2 − 2φ̇δφ̇A+
φ̇δφ,αB
α + 2φ̇2A2 −
φ̇2BαBα,
δp(φ) = φ̇δφ̇− φ̇2A− V,φδφ+
δφ̇2 −
δφ,αδφ,α −
V,φφδφ
2 − 2φ̇δφ̇A+
φ̇δφ,αB
α + 2φ̇2A2 −
φ̇2BαBα,
= 0. (263)
Notice that no anisotropic stress is caused by a minimally coupled scalar field even to the second order in perturbations.
The uniform-field gauge takes δφ ≡ 0 as a temporal gauge (slicing) condition to the second-order in perturbation.
The uniform-field gauge gives v(φ) = 0 which is the comoving gauge, and vice versa. Thus,
δφ = 0 ↔ v(φ) = 0. (264)
We also have
δµ(φ) − δp(φ) = 2V,φδφ+ V,φφδφ
2, (265)
and under the uniform-field gauge we have
= −φ̇2
A (1− 2A) +
. (266)
Equation (263) apparently shows that the vector-type perturbation v
(φ,v)
α does not vanish to the second-order.
However, the second-order quantities in right-hand-side depend on the temporal gauge condition for the scalar-type
perturbations, and trivially vanish for the uniform-field gauge. We can also show that it vanishes for the uniform-
density gauge or the uniform-pressure gauge where we have δφ̇ − φ̇A ∝ δφ to the linear order. In fact, we can show
that the vector-type perturbation is not sourced by the scalar field to the second order by evaluating the rotational
tensor ω̃αβ in Eq. (60) for the scalar field: i.e., for a minimally coupled scalar field we have
= 0, (267)
to the second order. Thus, we conclude that a minimally coupled scalar field does not contribute to the rotational
perturbation to the second order in perturbations. In fact, we can show that a single scalar field do not support
vector-type perturbations to all orders in perturbations. As we take the energy frame, thus q̃a ≡ 0, from Eq. (23) of
[11] we can show
ũa =
−φ̃,cφ̃,c
. (268)
Using the definition of the vorticity tensor in Eq. (33) we can show that ω̃ab = 0.
Using the fluid quantities in Eq. (263) we can handle the scalar field using our non-ideal fluid formulation. The
fluid equations in the energy frame, like Eqs. (62)-(68), remain valid in the case of scalar field with the fluid quantities
expressed as in Eq. (263). The anisotropic stress vanishes and the entropic perturbation e is given as
e(φ) ≡ δp(φ) − c2(φ)δµ
(φ), c2(φ) ≡
ṗ(φ)
µ̇(φ)
φ̈− V,φ
φ̈+ V,φ
. (269)
Under the comoving gauge v ≡ 0, using Eq. (266) we have
δp(φ)v = δµ
v , e
1− c2(φ)
δµ(φ)v , (270)
which is a well known relation, now valid to the second order in perturbations. A fluid formulation of the scalar field
to the linear order is presented in [24, 29]. Using Eq. (270) together with vanishing anisotropic stress, Eqs. (62)-(68)
or (88)-(94) provide the fluid formulation for a minimally coupled scalar field to the second order in perturbations.
The perturbed equation of motion of the scalar field is presented in Eq. (112) of [11].
B. Minimally coupled scalar fields
In the case of multiple minimally coupled scalar fields, the equation of motions in Eq. (119) and the full Einstein’s
equations in Eqs. (99)-(105) expressed using the normal-frame fluid quantities together with the normal-frame fluid
quantities for the scalar field in Eqs. (121) all in [11] provide a complete set of equations we need to the second-order.
The fluid quantities in Eq. (121) of [11] are presented in the normal-frame four-vector and it is convenient to know
the conventionally used fluid quantities which are based on the energy-frame four-vector. These latter quantities can
be read from Eqs. (121) of [11] and Eq. (58) as
v(φ) =
φ̇(k)δφ(k) +∆
δφ̇(k) − φ̇(k)A
δφ(k) − 2φ̇(k)
φ̇(l)δφ(l)
v(φ,v)α = −
δφ̇(k) − φ̇(k)A
δφ(k) − 2φ̇(k)
φ̇(l)δφ(l)
δφ̇(k) − φ̇(k)A
δφ(k) − 2φ̇(k)
φ̇(l)δφ(l)
δµ(φ) = δµN(φ) −
φ̇(k)δφ
φ̇(l)δφ(l),α
δp(φ) = δpN(φ) −
φ̇(k)δφ
φ̇(l)δφ(l),α
φ̇(k)δφ(k),α
φ̇(l)δφ(l),β
φ̇(k)δφ
φ̇(l)δφ(l),γ
(271)
where the normal-frame fluid quantities are presented in Eq. (121) of [11]. Thus, by moving into the energy-frame
fluid quantities from the normal-frame ones we have rather complicated terms which do not cancel out nicely any term
in the normal-frame quantities. In single field case we had such cancelations, for example we have Π
= 0 in Eq.
(263). But, we do not have such a luxury in the multi-component situation. Apparently Π
in the energy-frame does
not vanish and looks more complicated. Thus, in the multi-field situation we had better use both the fluid quantities
and Einstein’s equations all expressed in the normal frame: these are Eqs. (99)-(105),(121) in [11].
As the temporal gauge condition we can set any one field perturbation, say the specific ℓ-th one δφ(ℓ), equal to zero
which might be called the uniform-φ(ℓ) gauge to the second order. This apparently differs from the comoving gauge
which sets v(φ) ≡ 0. In the multi-component situation we cannot take a gauge condition which makes v
(φ,v)
α = 0.
Thus, the multiple scalar fields source the vector-type perturbation to the second order in perturbations. The scalar-,
vector-, and tensor-type decomposition of the anisotropic stress Π
can be read by using decomposition formulae in
Eq. (177) of [11].
In multiple-field situation, it is ad hoc and cumbersome (if not impossible) to introduce individual fluid quantity
for each field variable even to the background and linear order perturbations, see [30].
C. Generalized gravity case
In Sec. IV.D of [11]we presented the equation of motion and effective fluid quantities in a class of generalized gravity
theories together with additional presence of fluids and fields to the second order. The equation of motion is in Eq.
(128) of [11], and the full Einstein’s equations in Eqs. (99)-(105) expressed using the normal-frame fluid quantities
together with the normal-frame effective fluid quantities in Eq. (130) of [11] provide a complete set of equations
we need to the second-order. The effective fluid quantities in Eq. (130) of [11] are presented in the normal-frame
four-vector and using Eq. (88) in [11] we can easily derive the effective fluid quantities based on the energy-frame
four-vector. As in the multiple field case in a previous subsection, by moving into the energy-frame, the effective fluid
quantities become more complicated compared with the ones in the normal-frame. Thus, in these class of generalized
gravity theories we had better use both the fluid quantities and Einstein’s equations all expressed in the normal frame:
these are Eqs. (99)-(105),(130) in [11].
XI. CURVATURE PERTURBATIONS AND LARGE-SCALE CONSERVATIONS
In the large-scale limit the spatial curvature perturbation ϕ in several different gauge conditions is known to remain
constant in expanding phase. Often the conservation properties are shown based on the first time derivative of the
curvature perturbation. In order to show the conservation properties properly we have to construct the closed form
second-order differential equations for the curvature perturbation. In the following we will derive such first-order and
second-order differential equations for ϕ̇v, ϕ̇χ, ϕ̇κ, and ϕ̇δ. First we will derive equations for the linear perturbation
including the background curvature and non-ideal fluid properties. Then we will derive equations for the second
order perturbation assuming a flat background; including the background curvature is trivial, though. We consider a
single-component fluid.
A. Linear-order equations
We introduce a combination
Φ ≡ ϕv −
4πG (µ+ p)
ϕχ. (272)
This combination was first introduced by Field and Shepley in [28]. From Eqs. (88),(90),(92), Eqs. (88)-(90),(92),(94),
Eqs. (88),(93), and Eqs. (88),(89),(91), respectively, we can derive
4πG (µ+ p)
Φ− 8πGHΠ, (273)
4πG (µ+ p)
, (274)
ϕ̇δ =
, (275)
ϕ̇κ = −
3Ḣ +∆/a2
1 + 3c2s
) ∆+ 3K
ϕκ − 12πGe
χκ. (276)
These equations were presented in [24, 31]
We can derive closed form second-order differential equations for Φ, ϕχ, ϕδ, and ϕκ
(µ+ p) a
8πGH2
= c2s
ϕχ − 4πG
, (277)
H2c2s
(µ+ p) a3
(µ+ p) a3
H2c2s
= c2s
, (278)
3Ḣ +∆/a2
(µ+ p) a3
(µ+ p) a3
3Ḣ +∆/a2
ϕ̇δ +
4πG (µ+ p) + c2s (∆ + 3K) /a
12πG (µ+ p)− (∆ + 3K) /a2
3Ḣ +
, (279)
ϕ̇κ +
3Ḣ +∆/a2
1 + 3c2s
) ∆+ 3K
ϕκ − 12πGe
3Ḣ +∆/a2
1 + 3c2s
) ∆+ 3K
ϕκ − 12πGe
+ 8πGΠ
. (280)
Equations (277),(278) follow by combining Eqs. (273),(274). Equations (279) and (280) follow from Eqs.
(88),(89),(91),(92), and Eqs. (88),(89),(91),(92), respectively.
From Eqs. (89),(90) we can derive
∆+ 3K
12πG (µ+ p) a2
ϕκ = ϕv −
∆+ 3K
12πG (µ+ p)a2
ϕχ = Φ−
12πG (µ+ p) a2
ϕχ. (281)
These relations were presented in [17]. In the large-scale limit and in near flat background, thus ignoring ∆ and K
terms, we have
Φ ≃ ϕv ≃ ϕδ ≃ ϕκ, (282)
to the leading order in the large-scale expansion.
In the large-scale limit, ignoring ∆ terms, in near flat background and for an ideal fluid case, thus setting K = 0
and e = 0 = Π, Eqs. (277)-(280) give
(µ+ p) a
, (283)
ϕ̇v ∝
H2c2s
(µ+ p) a3
, (284)
ϕ̇δ ∝ ϕ̇κ ∝
. (285)
Notice that if we simply ignore the ∆ terms in Eqs. (274)-(276) we simply have ϕ̇v = ϕ̇δ = ϕ̇κ = 0. In such a way
we cannot recover the terms in the right-hand-side of Eqs. (284),(285); these terms lead to decaying solutions (in an
expanding era) in the large-scale limit and are higher order in the large-scale expansion compared with the decaying
solution of ϕχ, see below. From Eqs. (283)-(285) we have general large-scale asymptotic solutions
ϕχ = 4πGC(x)
∫ t a (µ+ p)
dt+ d(x)
, (286)
ϕv = C(x) +
∫ t c2sH
(µ+ p) a3
dt, (287)
ϕδ = ϕκ = C(x) +
∫ t dt
, (288)
where C(x) and d(x) are integration constants which correspond to the relatively growing and decaying modes,
respectively, in an expanding phase; in a collapsing phase the roles are reversed. The coefficients are fixed using
the relations in Eqs. (273),(274),(281). Notice that for the C-mode the relation in Eq. (282) is satisfied, and simply
remain constant. For the d-mode, ϕv, ϕδ, and ϕκ are ∆/(aH)
2-order higher compared with the d-mode of ϕχ. The
ϕv is one of the well known conserved quantity in the large-scale even in the context of generalized gravity theories
[24, 32].
In order to evaluate the solutions to the second order in the next section, we need complete sets of linear order
solutions for different gauge conditions. For an ideal fluid, and for a minimally coupled scalar field such complete sets
of solutions are presented in tabular forms in [29, 31]. In the following we summarize such sets of solutions in an ideal
fluid case for four different gauge conditions. From Table 8 of [31] we have
ϕχ = −αχ = C
, δχ = −
= −2C
adt, vχ = −C
ϕv = C, Hχv = C
adt, δv = −
1 + w
αv = −
ϕδ = C, Hχδ = C
= 3αδ = −
C, vδ =
ϕκ = C, Hχκ = C
adt, δκ = −3
1 + w
1 + 3c2s
ακ = −
C, vκ = −
adt. (289)
For corresponding sets of solutions for a minimally coupled scalar field, see Table 1 of [29]. Compared with the notation
used in [31] we have γΨ = −8πG(µ+p)av. The lower bounds of integration of solutions in Eq. (289) give behaviors of
d-modes. For solutions without integration, the d-modes are ∆/(aH)2-order higher than the non-vanishing d-mode,
for example, see the solutions in Table 2 of [31]. Thus, the d-modes are
ϕχ = −αχ = d
, δχ = −
, vχ = d
∫ t c2sH
(µ+ p) a3
dt, Hχv = −
d, δv = −
1 + w
αv = −
∫ t dt
, Hχδ = −
∫ t dt
, αδ =
∫ t dt
, vδ =
∫ t dt
, Hχκ = −
d, δκ = −3
1 + w
1 + 3c2s
ακ = −
∫ t dt
, vκ =
d. (290)
B. Second-order equations
We assume K = 0. From Eqs. (88),(90),(92), Eqs. (88)-(90),(94), Eqs. (88),(93), and Eqs. (88),(89),(91), respec-
tively, we can derive
(ϕ−Hχ)
4πG (µ+ p)
(ϕ− aHv)− 8πGHΠ+
(n0 − n2)−Hn4, (291)
(ϕ− aHv)
4πG (µ+ p)
(ϕ−Hχ)−
n1 +Hn2
(n0 − n2)− aHn6, (292)
ϕ̇δ =
, (293)
ϕ̇κ = −
3Ḣ +∆/a2
1 + 3c2s
ϕκ − 12πGe−
1 + 3c2s
n1 − n3
n0. (294)
The perturbed order variables in Eqs. (293),(294) are evaluated in the uniform-density gauge (δ ≡ 0), and the
uniform-expansion gauge (κ ≡ 0), respectively. Equation (293) also follows from Eq. (41) evaluated to the second
order.
We can derive closed form second-order differential equations for ϕv, ϕχ, ϕδ, and ϕκ
(ϕ−Hχ)
+ 8πGHΠ−
(n0 − n2) +Hn4
= c2s
(ϕ−Hχ)− 4πG
+ c2s
n1 +Hn2
4πG (µ+ p)
(n0 − n2)− aHn6
, (295)
H2c2s
4πG (µ+ p) a3
4πG (µ+ p) a3
H2c2s
(ϕ− aHv)
(n0 − n2) + aHn6
n1 −Hn2
= c2s
ϕ− aHv −
Hc2s∆
4πG (µ+ p) a2
(n0 − n2)−Hn4
, (296)
1 + ∆/(3a2Ḣ)
1 + ∆/(3a2Ḣ)
ϕ̇δ +
3Ḣ +∆/a2
ṅ1 +
n1 −Hn3
1− c2s∆/(a
3 + ∆/(a2Ḣ)
3Ḣ +
9Ḣa2
− n3 −
3Ḣ +
, (297)
ϕ̇κ +
3Ḣ +∆/a2
1 + 3c2s
− 12πGe− n3
−ϕκ +
3Ḣ +∆/a2
1 + 3c2s
− 12πGe− n3
− 8πGΠ− n4
. (298)
Equations (295),(296) follow by combining Eqs. (291),(292). Equation (297) follows from Eqs. (88),(89),(91),(92).
Equation (298) follows from Eqs. (88),(89),(91),(92). The perturbed order variables in Eqs. (297),(298) are evaluated
in the uniform-density gauge (δ ≡ 0), and the uniform-expansion gauge (κ ≡ 0), respectively.
C. Large-scale solutions
Now, we assume an ideal fluid, thus set e = 0 = Π. In the large-scale limit, thus ignoring the ∆/(aH)2-order higher
terms, Eqs. (296)-(298) give
ϕ̇v −
(n0 − n2) + aHn6 +
4πG (µ+ p)
n1 +Hn2
H2c2s
4πG (µ+ p) a3
, (299)
ϕ̇δ +
ṅ1 +
Hn1 −Hn3
, (300)
ϕ̇κ −
1 + 3c2s
n1 + n3
, (301)
where the perturbed order variables in Eq. (299) are evaluated in the comoving gauge (v ≡ 0). We already used the
behavior of linear order solutions in Eqs. (289),(290) in order to show that the right-hand-side of Eqs. (296)-(298)
vanish. Using the solutions in Eqs. (289),(290) we can show that
ϕv − ϕ
+O(∆C2,∆2d2) ∝
H2c2s
(µ+ p)a3
, (302)
ϕδ − ϕ
+O(∆C2,∆2d2) ∝
ϕκ − ϕ
+O(∆C2,∆2d2) ∝
. (303)
Thus, we have general large-scale asymptotic solutions
ϕv − ϕ
v = C(x) +
∫ t c2sH
(µ+ p) a3
dt, (304)
ϕδ − ϕ
δ = ϕκ − ϕ
κ = C(x) +
∫ t dt
, (305)
where C(x) and d(x) are integration constants now including the second-order contributions, i.e., C = C(1) + C(2),
etc. Ignoring the transient solutions in an expanding phase we have
ϕv = ϕδ = ϕκ = C(x), (306)
even to the second order in perturbations in the large-scale limit.
XII. DISCUSSION
In this work we presented pure general relativistic effects of second-order perturbations in Friedmann cosmological
world model. In our previous work we have shown that to the second-order perturbations, the density and velocity
perturbation equations of general relativistic zero-pressure, irrotational, single-component fluid in a flat background
coincide exactly with the ones known in Newton’s theory, [12]. We also have shown the effect of gravitational waves to
the second-order, and pure general relativistic correction terms appearing in the third-order perturbations, [12, 13].
Here, we presented results of second-order perturbations relaxing all the assumptions made in our previous work
in [12]. We derived the general relativistic correction terms arising due to (i) pressure, (ii) multi-component, (iii)
background curvature, and (iv) rotation. We also presented a general proof of large-scale conserved behaviors of
curvature perturbation variable in several gauge conditions, now to the second order.
Effects of pressure can be found in Eqs. (123)-(125). As we emphasized, the effect of pressure is generically
relativistic even in the background world model and the linear order perturbations. Still, our equations show the
pure general relativistic effects of pressure (including stresses) appearing in the second-order perturbations. Effects
of multi-component fluids can be found in Eqs. (195)-(200). Although these equations apparently show deviations
from Newtonian situation, in Sec. VII D we showed that if we ignore purely decaying terms in an expanding phase the
equations are effectively the same as in the Newtonian situation. Effects of background spatial curvature K can be
read from Eqs. (217),(218) or Eqs. (211)-(214). Effects of vector-type perturbation can be read from Eqs. (247)-(249).
In the small-scale limit we showed that, if we ignore the tensor-type perturbation, the equations coincide with the
Newtonian ones.
Our results may have important practical implications in cosmology and the large-scale structure formation. Our
new result showing relativistic/Newtonian correspondence in the zero-pressure irrotational multi-component fluids
is practically relevant in currently favored cosmology where baryon and dark matter are two important ingredients
of the current matter content in addition to the cosmological constant. All equations in our work are valid in the
presence of the cosmological constant. A related important result is the relativistic/Newtonian correspondence valid
in the presence of rotational perturbation far inside horizon. Thus, inside the horizon scale, even in the presence of
rotational perturbations we can still rely on the Newtonian equations to handle quasi-linear evolution of large-scale
structures. As the spatial curvature in the present cosmological era is known to be small [33], the possible presence of
small spatial curvature may not be important in the second-order perturbations. Still, while the Newtonian equation
is exactly valid to the linear order even in the presence of the spatial curvature, we have nontrivial general relativistic
correction terms present to the second order in perturbations. Our second-order perturbation equations in the presence
of pressure may have an interesting role as we approach early stage of universe where the effect of radiation becomes
important. The importance of pressure to the second-order perturbations, of course, depends on whether nonlinear
effects are significant in the early evolution stage of the large-scale structure during the radiation era and in the early
matter dominated era. Realistic estimations of the diverse pure general relativistic contributions using the complete
set of equations presented in this work are left for future investigations.
In an accompanying paper we will investigate the effects of third-order perturbations of zero-pressure irrotational
multi-component fluids in a flat background. This is one obvious remaining issue in our series of investigation of
nonlinear cosmological perturbations where nontrivial general relativistic effects are expected. In the case of a single
fluid we presented the pure general relativistic effects appearing in the third order in [13]. Corresponding results in
the case of multi-component will be presented in [34].
Acknowledgments
H.N. was supported by grant No. C00022 from the Korea Research Foundation.
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	Contents
	Introduction
	Newtonian nonlinear perturbations
	Summary of previous work
	Relativistic fully nonlinear equations
	Covariant equations
	ADM equations
	Second-order perturbations
	Basic equations in the energy-frame
	Decomposition
	Comoving gauge and irrotational condition
	Effects of pressure
	Irrotational case
	Comoving gauge
	Newtonian correspondence
	Linear-order relativistic pressure corrections
	Effects of multi-component
	Irrotational case
	Linear perturbations
	Exact solutions
	Comoving gauge
	Newtonian correspondence
	Effects of curvature
	Newtonian correspondence
	Effects of vector-type perturbation
	Linear perturbations
	Second-order perturbations
	Zero-pressure case
	Newtonian correspondence
	Pure vector-type perturbations
	Equations with fields
	A minimally coupled scalar field
	Minimally coupled scalar fields
	Generalized gravity case
	Curvature perturbations and large-scale conservations
	Linear-order equations
	Second-order equations
	Large-scale solutions
	Discussion
	Acknowledgments
	References

  We present general relativistic correction terms appearing in Newton's
gravity to the second-order perturbations of cosmological fluids. In our
previous work we have shown that to the second-order perturbations, the density
and velocity perturbation equations of general relativistic zero-pressure,
irrotational, single-component fluid in a flat background coincide exactly with
the ones known in Newton's theory. Here, we present the general relativistic
second-order correction terms arising due to (i) pressure, (ii)
multi-component, (iii) background curvature, and (iv) rotation. In case of
multi-component zero-pressure, irrotational fluids under the flat background,
we effectively do not have relativistic correction terms, thus the relativistic
result again coincides with the Newtonian ones. In the other three cases we
generally have pure general relativistic correction terms. In case of pressure,
the relativistic corrections appear even in the level of background and linear
perturbation equations. In the presence of background curvature, or rotation,
pure relativistic correction terms directly appear in the Newtonian equations
of motion of density and velocity perturbations to the second order. In the
small-scale limit (far inside the horizon), relativistic equations including
the rotation coincide with the ones in Newton's gravity.

Introduction 2
II. Newtonian nonlinear perturbations 3
III. Summary of previous work 5
IV. Relativistic fully nonlinear equations 6
A. Covariant equations 6
B. ADM equations 8
V. Second-order perturbations 8
A. Basic equations in the energy-frame 10
B. Decomposition 14
C. Comoving gauge and irrotational condition 16
VI. Effects of pressure 17
A. Irrotational case 17
∗Electronic address: jchan@knu.ac.kr
†Electronic address: hr@kasi.re.kr
http://arxiv.org/abs/0704.1927v1
mailto:jchan@knu.ac.kr
mailto:hr@kasi.re.kr
B. Comoving gauge 17
C. Newtonian correspondence 19
D. Linear-order relativistic pressure corrections 20
VII. Effects of multi-component 20
A. Irrotational case 21
B. Linear perturbations 22
1. Exact solutions 24
C. Comoving gauge 25
D. Newtonian correspondence 26
VIII. Effects of curvature 28
A. Newtonian correspondence 28
IX. Effects of vector-type perturbation 29
A. Linear perturbations 29
B. Second-order perturbations 30
C. Zero-pressure case 31
D. Newtonian correspondence 31
E. Pure vector-type perturbations 33
X. Equations with fields 33
A. A minimally coupled scalar field 33
B. Minimally coupled scalar fields 35
C. Generalized gravity case 35
XI. Curvature perturbations and large-scale conservations 36
A. Linear-order equations 36
B. Second-order equations 38
C. Large-scale solutions 39
XII. Discussion 39
Acknowledgments 40
References 40
I. INTRODUCTION
Large amount of cosmological data on the large-scale structures and motions of galaxies [1, 2], and the temperature
and polarization anisotropies of cosmic microwave background radiation [3, 4] have been accumulating recently. In
current standard cosmological scenario such structures are explained as small (linear) or large (nonlinear) deviations
from spatially homogeneous and isotropic Friedmann background world model. In order to explain these data theo-
retically, researchers rely on the linear perturbation theory based on relativistic gravity, and quasi-linear perturbation
theories and nonlinear simulations based on Newton’s gravity. To the linear order in perturbation the general rel-
ativistic result was first derived by Lifshitz in 1946 [5], and later shown to coincide with the Newtonian result in a
zero-pressure medium [6]. The same is also known to be true for the background world model. That is, the general
relativistic result was first derived by Friedmann in 1922 [7], and later shown to coincide with Newtonian result in a
zero-pressure medium [8].
The observed large-scale distribution of galaxies shows that in the largest observed scale (say, larger than several
hundred mega-parsec scale) the distribution may not be inconsistent with the linear assumption around the Friedmann
background. However, as the scale becomes smaller the distribution apparently shows quasi-linear to fully nonlinear
structures. The fully nonlinear processes occur in small scale where the relativistic effects characterized by GM/rc2 ∼
v2/c2 are quite small. If we could ignore such relativistic effects, Newton’s gravity would be sufficient to handle
the relevant nonlinear processes. If we need to consider the weakly relativistic correction terms in fully nonlinear
stage, instead of the relativistic perturbation approach which can handle the fully relativistic processes under weakly
nonlinear assumption, we can use the post-Newtonian approximation developed in the context of cosmology in [9].
For structures in the quasi-linear evolution phase, previous researches were based on Newton’s gravity especially
assuming the single component zero-pressure fluid without rotational perturbation [10]. In our previous works in
[11, 12] we have shown that, in the single component zero-pressure fluid without rotational perturbation, cosmological
scalar-type perturbation equations in a spatially flat background coincide exactly with the Newtonian ones up to the
second order in perturbation. In [11, 12] we also have shown the contribution of gravitational wave perturbations to
the hydrodynamic parts in the second-order perturbations. In Newton’s gravity the hydrodynamic equations of zero-
pressure fluid contain only the quadratic order nonlinearity. In [13] we presented pure general relativistic correction
terms appearing in the third-order perturbation, and showed that all third-order correction terms are 10−5 times
smaller than the second-order relativistic/Newtonian terms, and independent of the horizon scale.
In this work we will take into account of the pure general relativistic effects appearing in the second-order perturba-
tions which were ignored in our previous work in [12]. We will consider general relativistic effects of (i) pressure, (ii)
multi-component, (iii) background curvature, (iv) rotation in cosmological fluids to the second-order perturbations.
As results we will show that, although in [12] we have shown the exact relativistic/Newtonian correspondence to the
second-order perturbations by ignoring the above four conditions and the gravitational waves, as we take these four
effects into account we often encounter pure general relativistic effects appearing in the corresponding Newtonian equa-
tions even to the second order in perturbations. Our results will show that the relativistic/Newtonian correspondence
continues even in the multi-component situation assuming zero-pressure irrotational fluid in a flat background, but
in the presence of the cosmological constant. This is a practically useful result because the matter content of present
universe is dominated by collisionless dark matter and baryon both of which practically have zero-pressure. In the
other three cases, relaxing any of the assumptions about pressure, rotation, and background curvature generally leads
to pure general relativistic correction terms to the second order. We will present such correction terms in the context
of Newtonian hydrodynamics. One additional relativistic/Newtonian correspondence occurs in the case of rotation in
small-scale (sub-horizon-scale) limit which is another practically important result. This correspondence allows us to
use the Newtonian equations safely in such a small-scale limit even in the presence of rotational perturbation to the
second order.
In Sec. II we summarize Newtonian hydrodynamic perturbation equations valid to fully nonlinear order. In Sec.
III we briefly summarize our previous result of relativistic/Newtonian correspondence to the second order, and pure
general relativistic correction terms appearing in the third order. In Sec. IV we present parts of the covariant and
the ADM (Arnowitt-Deser-Misner) equations which are valid in multi-component situation. In Sec. V we present
the basic perturbation equations valid to second order. In [11] the basic set of equations was presented using fluid
quantities based on the normal-frame four-vector. The fluid quantities in the present work are based on the energy-
frame four-vector, and in this section we present the basic equations using such fluid quantities. In Secs. VI-IX we
analyse the effects of the pressure, the multi-components, the background curvature, and the rotational perturbation,
respectively. In Sec. X we present equations in the scalar fields and generalized gravity theories using the energy-frame
fluid quantities. In Sec. XI we properly derive conservation properties of curvature perturbation in various temporal
gauge (hypersurface) conditions to the second order in perturbations. Section XII is a discussion. In this work we
follow notations used in [11, 12]. We set c ≡ 1, but when we compare with Newtonian case we often recover the speed
of light c.
II. NEWTONIAN NONLINEAR PERTURBATIONS
In order to compare properly the relativistic results with the Newtonian ones, in this section we summarize the
Newtonian cosmological perturbation theory in fully nonlinear context. We consider multi-component fluids in the
presence of isotropic pressure. In case of n-fluids with the mass densities ̺i, the pressures pi, the velocities vi
(i = 1, 2, . . . n), and the gravitational potential Φ, we have
˙̺i +∇ · (̺ivi) = 0, (1)
v̇i + vi · ∇vi = −
∇pi −∇Φ, (2)
∇2Φ = 4πG
̺j . (3)
Assuming the presence of spatially homogeneous and isotropic but temporally dynamic background, we introduce
fully nonlinear perturbations as
̺i = ¯̺i + δ̺i, pi = p̄i + δpi, vi = Hr+ ui, Φ = Φ̄ + δΦ, (4)
where H ≡ ȧ/a, and a(t) is a cosmic scale factor. We move to the comoving coordinate x where
r ≡ a(t)x, (5)
∇ = ∇
−Hx · ∇
. (6)
In the following we neglect the subindex x. To the background order we have
˙̺i + 3H̺i = 0,
̺j , H
, (7)
where E is an integration constant which can be interpreted as the specific total energy in Newton’s gravity; in
Einstein’s gravity we have 2E = −Kc2 where K can be normalized to be the sign of spatial curvature. To the
perturbed order we have [14]
δ̇i +
∇ · ui = −
∇ · (δiui) , (8)
u̇i +Hui +
ui · ∇ui = −
a ¯̺i
1 + δi
∇δΦ, (9)
∇2δΦ = 4πG
¯̺jδj . (10)
By introducing the expansion θi and the rotation
−→ω i of each component as
∇ · ui,
−→ω i ≡
∇× ui, (11)
Eq. (9) gives
θ̇i + 2Hθi + 4πG
¯̺jδj = −
∇ · (ui · ∇ui)−
a2 ¯̺i
1 + δi
, (12)
−̇→ω i + 2H
−→ω i = −
∇× (ui · ∇ui) +
a2 ¯̺i
(∇δi)×∇δpi
(1 + δi)2
. (13)
By introducing decomposition of perturbed velocity into the potential- and transverse parts as
ui ≡ ∇ui + u
i , ∇ · u
i ≡ 0; θi =
−→ω i =
i , (14)
instead of Eq. (13) we have
i +Hu
i = −
ui · ∇ui +
1 + δi
−∇∆−1∇ ·
ui · ∇ui +
1 + δi
. (15)
We note that the pure ui contributions in the right-hand-side of Eq. (13) or Eq. (15) vanish. Thus, under van-
ishing pressure, pure irrotational perturbation cannot generate the rotational perturbation. Equation (13) shows
that presence of pressure perturbation oblique (i.e., non-parallel) to the density perturbation can generate rotational
perturbation. Combining Eqs. (8),(12) we can derive
δ̈i + 2Hδ̇i − 4πG
¯̺jδj = −
[a∇ · (δiui)]
∇ · (ui · ∇ui) +
a2 ¯̺i
1 + δi
. (16)
Equations (8)-(16) are valid to fully nonlinear order. Notice that for vanishing pressure these equations have only
quadratic order nonlinearity in perturbations.
III. SUMMARY OF PREVIOUS WORK
In [11, 12] we have derived second-order perturbation equations valid for the single component, irrotational, and
zero-pressure medium in zero-curvature background. These are
∇ · u = −
∇ · (δu) , (17)
∇ · (u̇+Hu) + 4πG̺δ = −
∇ · (u · ∇u)− Ċ(t)αβ
uα|β + Ċ
, (18)
δ̈ + 2Hδ̇ − 4πG̺δ =
∇ · (u · ∇u)−
[a∇ · (δu)]
+ Ċ(t)αβ
uα|β + Ċ
. (19)
Except for the presence of tensor-type perturbation, Eqs. (17)-(19) are exactly the same as the ones known in the
Newtonian theory. We note that these equations are valid in the presence of Λ. To the linear order, these are valid in
the presence of general K, see Sec. VIII. We have correctly identified the relativistic density and velocity perturbation
variables which correspond to the Newtonian counterparts to the second order. In the relativistic context, our δ and
u are the density perturbation and (related to) the perturbed expansion scalar, respectively, in the comoving gauge;
the variables are equivalently gauge-invariant. However, we were not able to identify relativistic variable which
corresponds to the Newtonian gravitational potential to the second order; this is understandable if we consider the
factor of two difference between Einstein’s (post-Newtonian) and Newton’s gravity theories in predicting the light
bending under the gravitational field. To the linear order the spatial curvature perturbation in the zero-shear gauge
can be identified as the perturbed Newtonian potential [15, 16]. Equations (17)-(19) include effects of gravitational
waves to the density and velocity perturbations. Equations of the gravitational waves can be found in [12].
To the third order, we have [13]
∇ · u = −
∇ · (δu) +
2ϕu−∇
· ∇δ, (20)
+ 4πGµδ = −
∇ · (u · ∇u)
ϕu · ∇ (∇ · u) +
u · ∇u−
u∇ · u
u · ∇
u · ∇X +
X∇ · u,(21)
δ̈ + 2
δ̇ − 4πGµδ = −
[a∇ · (δu)] +
∇ · (u · ∇u) +
2ϕu−∇
ϕu · ∇ (∇ · u)−
u · ∇u−
u∇ · u
u · ∇
u · ∇X −
X∇ · u,(22)
where
X ≡ 2ϕ∇ · u− u · ∇ϕ+
∆−1∇ · [u · ∇ (∇ϕ) + u∆ϕ] . (23)
In these equations we ignored the role of tensor-type perturbation; for a complete set of equations, see [13]. The
variable ϕ is a perturbed-order metric (spatial curvature) variable in the comoving gauge condition, see later.
All the third-order correction terms in Eqs. (20)-(22) are simply of ϕ-order higher than the second-order rela-
tivistic/Newtonian terms. Thus, the pure general relativistic effects are at least ϕ-order higher than the relativis-
tic/Newtonian ones in the second order equations. Thus, we only need the behavior of ϕ to the linear order which is
related to the other hydrodynamic variables as
ϕ = −δΦ+ ȧ∆−1∇ · u. (24)
It also satisfies [17]
ϕ̇ = 0, (25)
thus ϕ = C(x) with no decaying mode; this is true considering the presence of the cosmological constant, see [17].
IV. RELATIVISTIC FULLY NONLINEAR EQUATIONS
In this section, for convenience, we present some additional covariant or ADM equations not available in [11]. In
the multi-component situation, we have
T̃ab ≡
T̃(j)ab. (26)
The energy-momentum conservation gives
T̃ b(i)a;b = Ĩ(i)a,
Ĩ(j)a = 0. (27)
Tildes indicate the covariant quantities.
A. Covariant equations
We introduce the fluid quantities as
T̃(i)ab ≡ µ̃(i)ũ(i)aũ(i)b + p̃(i)
g̃ab + ũ(i)aũ(i)b
+ q̃(i)aũ(i)b + q̃(i)bũ(i)a + π̃(i)ab, (28)
where
ũa(i)ũ(i)a ≡ −1, ũ
(i)q̃(i)a ≡ 0 ≡ ũ
(i)π̃(i)ab, π̃
(i)a ≡ 0. (29)
The fluid quantities of each component are based on the fluid four-vector ũ(i)a as
µ̃(i) ≡ T̃(i)abũ
(i)ũ
(i), p̃(i) ≡
T̃(i)abh̃
(i), q̃(i)a ≡ −T̃(i)cdũ
(i)h̃
(i)a, π̃(i)ab ≡ T̃(i)cdh̃
(i)ah̃
(i)b − p̃(i)h̃(i)ab, (30)
where h̃(i)ab ≡ g̃ab + ũ(i)aũ(i)b. Equation (27) gives
µ(i) +
µ̃(i) + p̃(i)
θ̃(i) + q̃
(i);a + q̃
(i)ã(i)a + π̃
(i)σ̃(i)ab = −ũ
(i)Ĩ(i)a, (31)
µ̃(i) + p̃(i)
ã(i)a + h̃
p̃(i),b +
q(i)b + π̃
(i)b;c
+ q̃b(i)
ω̃(i)ab + σ̃(i)ab +
θ̃(i)h̃(i)ab
= h̃ b(i)aĨ(i)b, (32)
where the kinematic quantities are also based on the fluid four-vector ũ(i)a as
h̃ c(i)ah̃
(i)b ũ(i)c;d = h̃
(i)[ah̃
(i)b]ũ(i)c;d + h̃
(i)(ah̃
(i)b)ũ(i)c;d ≡ ω̃(i)ab + θ̃(i)ab = ũ(i)a;b + ã(i)aũ(i)b,
θ̃(i) ≡ u
(i);a, σ̃(i)ab ≡ θ̃(i)ab −
θ̃(i)h̃(i)ab, ã(i)a ≡ ũ(i)a;bũ
(i) ≡
u(i)a,
µ(i) ≡ µ̃(i),aũ
(i). (33)
In the multi-component situation, we can derive the corresponding equation of Raychaudhury equation for the indi-
vidual component. From ũ(i)a;bc − ũ(i)a;cb ≡ ũ(i)dR̃
abc we can derive
θ(i) +
θ̃2(i) − ã
(i);a + σ̃
(i)σ̃(i)ab − ω̃
(i)ω̃(i)ab = 4πG
µ̃− 3p̃− 2T̃abũ
(i)ũ
+ Λ. (34)
In the energy-frame we take q̃(i)a ≡ 0 for each component of the fluids without losing any physical degree of freedom.
In a single component situation, taking the energy-frame, the energy conservation equation, the momentum con-
servation equation, and the Raychaudhury equation are [18]
µ+ (µ̃+ p̃) θ̃ + π̃abσ̃ab = 0, (35)
(µ̃+ p̃) ãa + h̃
p̃,b + π̃
= 0, (36)
θ̃2 − ãa;a + σ̃
abσ̃ab − ω̃
abω̃ab + 4πG (µ̃+ 3p̃)− Λ = 0. (37)
By combining Eqs. (35)-(37) we can derive
µ+ π̃abσ̃ab
µ̃+ p̃
µ+ π̃abσ̃ab
µ̃+ p̃
= 4πG (µ̃+ 3p̃)− Λ + σ̃abσ̃ab − ω̃
abω̃ab +
h̃ab(p̃,b + π̃
µ̃+ p̃
. (38)
This equation was derived in Eq. (88) of [19], see also [20]. In the multi-component case, in the energy-frame,
combining Eqs. (31)-(34) we can derive
µ(i) + π̃
σ̃(i)ab + ũ
Ĩ(i)a
µ̃(i) + p̃(i)
µ(i) + π̃
σ̃(i)ab + ũ
Ĩ(i)a
µ̃(i) + p̃(i)
= −4πG
µ̃− 3p̃− 2T̃abũ
(i)ũ
− Λ + σ̃ab(i)σ̃(i)ab − ω̃
(i)ω̃(i)ab +
(p̃(i),b + π̃
(i)b;c
− Ĩ(i)b)
µ̃(i) + p̃(i)
. (39)
In [21] Langlois and Vernizzi derived a simple covariant relation which leads to one of the conserved variable in the
large-scale limit. These authors introduced
ζ̃a ≡ h̃
α̃,b +
3(µ̃+ p̃)
α ≡ α̃,aũ
θ̃. (40)
Using only the energy conservation in Eq. (35) we can derive
£ũζ̃a = −
3(µ̃+ p̃)
p̃,a +
(µ̃+ p̃)θ̃
ũaπ̃
bcσ̃bc
3(µ̃+ p̃)
π̃bcσ̃bc
3(µ̃+ p̃)
µ̃,aπ̃
bcσ̃bc
3(µ̃+ p̃)2
, (41)
where £ũ is a Lie derivative along ũa with £ũζ̃a ≡ ζ̃a;bũ
b + ζ̃bũb;a. Thus, for vanishing anisotropic pressure, we have
the Langlois-Vernizzi relation [21]
£ũζ̃a = −
3(µ̃+ p̃)
p̃,b −
. (42)
These equations are valid in a single component fluid, or in multiple component fluids for the collective fluid variables.
We can easily extend the relation to the individual fluid component as follows. We introduce
ζ̃(i)a ≡ h̃
α̃(i),b +
µ̃(i),b
3(µ̃(i) + p̃(i))
α(i) ≡ α̃(i),aũ
(i) ≡
θ̃(i). (43)
Using only Eq. (31) we can derive
£ũ(i) ζ̃(i)a = −
θ̃(i)
3(µ̃(i) + p̃(i))
p̃(i),a +
(µ̃(i) + p̃(i))θ̃(i)
µ̃(i),a
ũ(i)a
σ̃(i)bc + ũ
Ĩ(i)b
3(µ̃(i) + p̃(i))
σ̃(i)bc + ũ
Ĩ(i)b
3(µ̃(i) + p̃(i))
µ̃(i),a
σ̃(i)bc + ũ
Ĩ(i)b
3(µ̃(i) + p̃(i))
. (44)
Thus, for vanishing anisotropic pressure and direct interactions among fluids, i.e., π̃(i)ab = 0 = Ĩ(i)a, we have
£ũ(i) ζ̃(i)a = −
θ̃(i)
3(µ̃+ p̃)
h̃ b(i)a
p̃(i),b −
µ̃(i),b
. (45)
Application of these compact relations to large-scale conservation properties to the second order will be studied in
Sec. XI.
B. ADM equations
The ADM formulation [22] is presented in Eqs. (2)-(13),(47),(48) of [11]. Interpretation of the ADM fluid quantities
in Eqs. (45),(46) of [11] was based on the normal-frame fluid quantities; for relations to the energy-frame fluid
quantities, see Eq. (57) below. The ADM fluid quantities of individual component are introduced as
E(i) ≡ ñañbT̃
(i) = N
2T̃ 00(i) , J(i)α ≡ −ñbT̃
(i)α = NT̃
(i)α,
S(i)αβ ≡ T̃(i)αβ , S(i) ≡ h
αβS(i)αβ , S̄(i)αβ ≡ S(i)αβ −
hαβS(i). (46)
Equation (27) gives Eqs. (12),(13),(47),(48) in [11]. Equations (10),(12),(47) in [11] can be arranged as
(∂0 −N
α∂α)K
(Kαα )
= 4πG (E + S)− Λ + K̄αβK̄αβ −
N :αα, (47)
Kαα =
)−1 [
(∂0 −N
α∂α)E +
− S̄αβK̄αβ
, (48)
Kαα =
E(i) +
)−1 [
(∂0 −N
α∂α)E(i) +
N2Jα(i)
K̄αβ +
Ĩ(i)0 − Ĩ(i)αN
. (49)
Momentum conservation equations for the collective and individual components can be found in Eqs. (13),(48) of
[11]. By combining Eqs. (47)-(49) we can derive the ADM counterpart of the density perturbation equations in Eqs.
(38),(39).
V. SECOND-ORDER PERTURBATIONS
We use a metric convention in Eq. (49) of [11]
g̃00 ≡ −a
2 (1 + 2A) , g̃0α ≡ −a
2Bα, g̃αβ ≡ a
+ 2Cαβ
. (50)
The subindex 0 indicates the conformal time η with adη ≡ cdt. To the second-order in perturbation we introduce the
fluid four-vector of individual component as
ũα(i) ≡
V α(i), ũ
(i) =
1− A+
V α(i)V(i)α −B
αV(i)α
ũ(i)α = a
V(i)α −Bα +ABα + 2V
, ũ(i)0 = −a
1 +A−
V α(i)V(i)α
. (51)
In this definition of fluid four-vector we follow the notation in Eq. (53) of [11]. If we introduce ũα
≡ V̄ α
, we have
V α(i) = (1−A)V̄
(i). The fluid quantities of individual component are introduced as
µ̃(i) ≡ µ(i) + δµ(i), p̃(i) ≡ p(i) + δp(i), π̃(i)αβ ≡ a
2Π(i)αβ , π̃(i)0α = −a
2Π(i)αβV
, π̃(i)00 = 0, (52)
where from π̃ a
≡ 0 we have
Π α(i)α − 2C
αβΠ(i)αβ = 0. (53)
The energy-momentum tensor of individual component in the energy-frame follows from Eq. (28) as
T̃ 0(i)0 = −µ(i) − δµ(i) −
µ(i) + p(i)
V α(i) −B
V(i)α,
T̃ 0(i)α =
µ(i) + p(i)
V(i)α −Bα −AV(i)α + 2ABα + 2V
δµ(i) + δp(i)
V(i)α −Bα
+Π(i)αβ
T̃ α(i)β =
p(i) + δp(i)
δαβ +
µ(i) + p(i)
V α(i)
V(i)β −Bβ
+Π α(i)β − 2C
αγΠ(i)βγ . (54)
Using T̃ ab =
, and the total fluid quantities in Eq. (82) of [11] we have
µ(j), p =
p(j), (55)
for the background order fluid quantities, and
δµ(j) +
µ(j) + p(j)
V α(j) −B
V(j)α − Vα
δp(j) +
µ(j) + p(j)
V α(j) −B
V(j)α − Vα
(µ+ p)Vα =
µ(j) + p(j)
V(j)α +
δµ(j) + δp(j)
V(j)α − Vα
V(j)β − Vβ
Παβ =
Π α(j)β +
µ(j) + p(j)
vα(j) −B
V(j)β − Vβ
V(j)γ − Vγ
, (56)
for perturbed order fluid quantities to the second-order. From Eq. (46) we have the ADM fluid quantities based on
the energy-frame fluid quantities
E(i) = µ(i) + δµ(i) +
µ(i) + p(i)
V α(i) −B
V(i)α −Bα
J(i)α = a
µ(i) + p(i)
V(i)α −Bα +ABα + 2V
δµ(i) + δp(i)
V(i)α −Bα
+ Π(i)αβ
S(i) = 3
p(i) + δp(i)
µ(i) + p(i)
V α(i) −B
V(i)α −Bα
S̄(i)αβ = a
Π(i)αβ +
µ(i) + p(i)
V(i)α −Bα
V(i)β −Bβ
V(i)γ −Bγ
. (57)
We can compare Eq. (57) with the ADM fluid quantities based on the normal-frame fluid quantities in Eq. (76) of
[11]; in a single component case we simply delete (i) subindices.
In [11] the fluid quantities are based on the normal-frame vector. By taking ũ(i)α ≡ 0, ũ(i)a becomes the normal-
frame vector ña, see Eq. (54) in [11]. Based on the normal-frame vector, to the second order, the fluid quantities have
contributions due to the frame choice: for example, even in the zero-pressure fluid, the perturbed pressure based on
the normal-frame does not necessarily vanish to the second-order, see Eq. (58) below. By comparing Eq. (57) with
Eq. (76) of [11] we have
δµN(i) = δµ(i) +
µ(i) + p(i)
V α(i) −B
V(i)α −Bα
δpN(i) = δp(i) +
µ(i) + p(i)
V α(i) −B
V(i)α −Bα
Q(i)α =
µ(i) + p(i)
V(i)α −Bα +ABα + 2V
δµ(i) + δp(i)
V(i)α −Bα
+Π(i)αβ
ΠN(i)αβ = Π(i)αβ +
µ(i) + p(i)
V(i)α −Bα
V(i)β −Bβ
V(i)γ −Bγ
. (58)
For the total fluid quantities the relations between the two frames are presented in Eq. (87) of [11]. Thus, by replacing
all fluid quantities in Eqs. (99)-(107) of [11] using Eq. (58) and Eq. (87) of [11] we have the equations in the energy
frame. Using Q(i)α in Eq. (58), Eq. (54) gives
T̃ 0(i)α = (1−A)Q(i)α. (59)
As the fluid four-velocity of i-th component we can use either Q(i)α or V(i)α −Bα related by Eq. (58).
The kinematic quantities in the energy-frame are presented in Eqs. (63)-(66) of [11]. In Eq. (33) we introduced
kinematic quantities for the individual component. To the second order we can show
θ̃(i) = 3
Cα′α +
V α(i)|α +
A2 − 3
BαV(i)α −
ACα′α +
V α(i)
A,α + C
V α(i)V(i)α +
V α(i) −B
V(i)α −Bα
CαβC′αβ ,
σ̃(i)αβ = a
V(i)(α|β) + C
αβ −AC
V(i)(α −B(α
V(i)β) −Bβ)
+ V(i)(αA,β) + V
Cαβ|γ + 2V
(i)|(α
C(β)γ
Cγ′γ + V
(i)|γ
(i)|γ
+ Cγ′γ −AC
V(i)γ −Bγ
A,γ + V
Cδδ|γ − 2C
γδC′γδ
ω̃(i)αβ = a
V(i)[α −B[α +AB[α + 2V
V(i)[α −B[α
V(i)β] −Bβ]
+A,β]
ã(i)α = A,α +
V(i)α −Bα +ABα + 2V
− 2AA,α −A
V(i)α − Bα
V(i)α −Bα
+Bβ|α
σ̃(i)α0 = −V
σ̃(i)αβ , σ̃(i)00 = 0; ω̃(i)α0 = −V
ω̃(i)αβ , ω̃(i)00 = 0; ã(i)0 = −V
(i)ã(i)α, (60)
where a prime indicates the time derivative based on η.
The gauge transformation properties of the fluid quantities are presented in Eqs. (232)-(235) of [11] for the normal-
frame, and Eqs. (238) of [11] for the energy-frame. A prescription to get the gauge transformation properties for
individual fluid quantities is also presented below Eq. (235) of [11]. Under the gauge transformation we have
δµ̂(i) = δµ(i) −
µ′(i) + δµ
ξ0 − δµ(i),αξ
µ′′(i)ξ
0ξ0 + µ′(i)
ξ0ξ0′ + ξαξ0,α
δp̂(i) = δp(i) −
p′(i) + δp
ξ0 − δp(i),αξ
p′′(i)ξ
0ξ0 + p′(i)
ξ0ξ0′ + ξαξ0,α
V̂(i)α − B̂α + ÂB̂α + 2V̂
Ĉαβ = V(i)α −Bα +ABα + 2V
Cαβ + ξ
V(i)α −Bα
V(i)α −Bα
V(i)β −Bβ
ξβ,α −
V(i)α −Bα
A− ξ0′ −
ξ0,α − ξ
,β − ξ
0ξ0′,α − ξ
βξ0,αβ ,
Π̂(i)αβ = Π(i)αβ −
Π′(i)αβ + 2
Π(i)αβ
ξ0 −Π(i)αβ,γξ
γ − 2Π(i)γ(αξ
. (61)
A. Basic equations in the energy-frame
The basic set of equations with fluid quantities based on the normal-frame is presented in Eqs. (99)-(107) of [11].
By using Eq. (58) we can recover the equations with fluid quantities based on the energy frame. For convenience, in
the following we present the complete set of equations with fluid quantities in the energy frame. These equations are
written without taking any gauge conditions yet, thus in a sort of gauge-ready form. To the linear order this method
was suggested by Bardeen [23, 24].
Definition of δK:
K̄ + 3H + δK − 3HA+ Ċαα +
HA− Ċαα −
HBαBα +
+ 2Cαβ
Ċαβ +
≡ n0. (62)
Energy constraint equation:
16πGµ+ 2Λ− 6H2 −
R(3) + 16πGδµ+ 4HδK −
R(3)Cαα
δK2 − 16πG (µ+ p) (V α −Bα) (Vα −Bα)−
Ċαβ +
B(α|β)
Ċαβ +
Ċαα +
R(3)Cαγ C
− Cα|βα
+ Cαβ|γ
3Cαβ|γ − 2Cαγ|β
≡ n1. (63)
Momentum constraint equation:
Ċβα +
+B |βα
Ċγγ +
δK,α + 8πGa (µ+ p)
Vα −Bα +ABα + 2V
AδK,α − 8πGa
(δµ+ δp) (Vα −Bα) + (µ+ p)A (Vα −Bα) + Παβ
V β −Bβ
Ċβα +
+ B |βα
Ċγα +
B |γα +B
+ 2Cβγ
Ċαγ +
B(α|γ)
+ Cγ|βα − C
Ċβγ +
Ċγγ +
+ 2Cγδ
Ċγδ +
≡ n2α. (64)
Trace of the ADM propagation equation:
3Ḣ + 3H2 + 4πG (µ+ 3p)− Λ
+ δK̇ + 2HδK − 4πG (δµ+ 3δp) +
3Ḣ +
= AδK̇ −
δK,αB
δK2 + 8πG (µ+ p) (V α −Bα) (Vα −Bα)
3A2 −BαBα
2A∆A+A,αA,α −
∆ (BαBα) +A
+ 2CαβA,α|β
Ċαβ +
B(α|β)
Ċαβ +
Ċαα +
≡ n3. (65)
Tracefree ADM propagation equation:
Ċαβ +
Bα|β +B
Ċαβ +
Bα|β +B
Ċγγ +
Ċγγ +
A|γ γ
R(3)Cαβ −
R(3)Cγγ
− 8πGΠαβ
Ċαβ +
Bα|β +B
A+ 2Cαγ
Ċβγ +
B(β|γ)
Ċαβ +
Bα|β +B
A+ 2Cαγ
Ċβγ +
B(β|γ)
Ċαβ +
Bα|β +B
Ċαβ +
Bα |β +B
Bγ + δK
Ċαβ +
Bα|β +B
−A2 +BγBγ
− 2CαγA,β|γ −
Ċγγ +
A+ 2Cγδ
Ċγδ +
Ċγγ +
A+ 2Cγδ
Ċγδ +
Ċγγ +
Ċγγ +
Bδ + δK
Ċγγ +
−AA|γ γ +
−A2 +BδBδ
− 2CγδA,γ|δ −
− Cγ|δγ
Ċαγ +
Bα|γ +B
Cαδ|βγ + C
− Cαβ|δγ − C
+ 2Cαγ
Cδγ|βδ + C
β|γδ − C
− Cδδ|γβ
R(3)Cαγ C
Cαδ |β + C
− Cγδ|βC
γδ|α + 2Cαγ|δ
Cβδ|γ − Cβγ|δ
Cǫγ|δǫ + C
γ|ǫδ − C
− Cǫǫ|γδ
R(3)CδγC
2Cǫδ|ǫ − C
− Cγ|δγ
+ Cγδ|ǫ
2Cγǫ|δ − 3Cγδ|ǫ
−16πGCαγΠβγ + 8πG (µ+ p)
(V α −Bα) (Vβ −Bβ)−
δαβ (V
γ −Bγ) (Vγ −Bγ)
≡ n α4β . (66)
Energy conservation equation:
[µ̇+ 3H (µ+ p)] + δµ̇+ 3H (δµ+ δp)− (µ+ p) (δK − 3HA) +
(µ+ p)
V α −Bα +ABα + 2V βCαβ
δµ,αB
α + (δµ+ δp) (δK − 3HA) + (µ+ p)AδK +
H (µ+ p)
A2 −BαBα
a4 (µ+ p) (V α −Bα) (Vα −Bα)
(δµ+ δp) (V α −Bα) + Παβ (Vβ −Bβ)
(µ+ p)
−A (V α −Bα)|α − 2A,α (V
α −Bα) + 2
Cαβ (Vβ −Bβ)
− Cαα|β
V β −Bβ
Ċαβ +
≡ n5. (67)
Momentum conservation equation:
a4 (µ+ p)
Vα −Bα +ABα + 2V
(µ+ p)A,α +
δp,α +Π
= (µ+ p) (δK − 3HA) (Vα −Bα)−
(δµ+ δp) (Vα −Bα) + Π
α (Vβ −Bβ)
δp,α +Π
A− (δµ+ δp)A,α
− (µ+ p)
−AA,α +Bβ|αV
β + (Vα −Bα)|β V
β + (Vα −Bα)
V β −Bβ
CβγΠαγ
Πγα + C
Πβγ −A,βΠ
≡ n6α. (68)
In the multi-component situation we additionally have the energy and the momentum conservation of individual
component. Using the energy-frame fluid quantities, Eqs. (106),(107) in [11] become
µ̇(i) + 3H
µ(i) + p(i)
I(i)0
+ δµ̇(i) + 3H
δµ(i) + δp(i)
µ(i) + p(i)
(δK − 3HA)
µ(i) + p(i)
V α(i) −B
α +ABα + 2V
δI(i)0
δµ(i),αB
δµ(i) + δp(i)
(δK − 3HA) +
µ(i) + p(i)
AδK +
µ(i) + p(i)
A2 −BαBα
µ(i) + p(i)
V α(i) −B
V(i)α −Bα
δµ(i) + δp(i)
V α(i) −B
V(i)β −Bβ
µ(i) + p(i)
V α(i) −B
− 2A,α
V α(i) −B
V(i)β −Bβ
− Cα|βα
V(i)β −Bβ
Ċαβ +
δI(i)αB
α ≡ n(i)5, (69)
µ(i) + p(i)
V(i)α −Bα +ABα + 2V
µ(i) + p(i)
A,α +
δp(i),α +Π
(i)α|β
− δI(i)α
µ(i) + p(i)
(δK − 3HA)
V(i)α −Bα
δµ(i) + δp(i)
V(i)α −Bα
V(i)β −Bβ
δp(i),α +Π
(i)α|β
− δI(i)α
δµ(i) + δp(i)
µ(i) + p(i)
−AA,α +Bβ|αV
V(i)α −Bα
V(i)α −Bα
CβγΠ(i)αγ
−A,βΠ
≡ n(i)6α. (70)
By removing indices indicating the components in Eqs. (69),(70) we recover equations for the collective component
which coincide with the equations in a single component situation in Eqs. (67),(68).
To the background order, from Eqs. (69),(65) we have
µ̇(i) + 3H
µ(i) + p(i)
I(i)0, (71)
µ̇+ 3H (µ+ p) = 0, (72)
3Ḣ + 3H2 = −
µ(j) + 3p(j)
+ Λc2, (73)
µ(j) −
, (74)
where we recovered the speed of light c. Dimensions are
[G̺] = T−2, [c] = LT−1, [η] = 1, [p] = [µ] = [̺c2], [a] = L, [K] = 1, [Λ] = L−2, [I(i)0] = [µ]. (75)
Equation (72) follows from the sum of Eq. (71) over components. Equation (74) follows from integrating Eq. (73)
where K-term can be regarded as an integration constant; in Einstein’s gravity K-term can be normalized as the
sign of spatial curvature. Compared with the Newtonian background equations in Eq. (7), ignoring the direct inter-
action terms in Eq. (71), the presence of pressure terms in Eqs. (71)-(74) is the pure general relativistic effect. The
cosmological constant Λ can be introduced by hand even in the Newtonian case.
B. Decomposition
We decompose the metric to three perturbation types
A ≡ α, Bα ≡ β,α +B
α , Cαβ ≡ ϕg
+ γ,α|β + C
(α|β)
, (76)
where superscripts (v) and (t) indicate the transverse vector-type, and transverse-tracefree tensor-type perturbations,
respectively. We introduce
χ ≡ a
β + c−1aγ̇
, Ψ(v)α ≡ B
α + c
−1aĊ(v)α , (77)
which are spatially gauge-invariant to the linear order. We set
Kαα ≡ −3H + κ. (78)
We will identify κ with Newtonian velocity variable which will be an important step in our analysis, see Eqs.
(120),(194),(215).
For the fluid quantities we decompose
ũ(i)α = a
V(i)α −Bα +ABα + 2V
≡ av(i)α ≡ a
−v(i),α + v
, (79)
Π(i)αβ ≡
Π(i),α|β −
∆Π(i)
(i)(α|β)
(i)αβ
, δI(i)α ≡ δI(i),α + δI
. (80)
The perturbed fluid velocity variables v(i) and v
subtly differ from the ones introduced in [11]; see Sec. VC. For
the collective fluid component or for a single component case, we simply delete (i) subindices. For isotropic pressure
we introduce
δp ≡ c2sδµ+ e, c
. (81)
The perturbation variable e is called an entropic perturbation. Defined in this way e is gauge-invariant only to the
linear order. To the second order, from Eq. (61) we can derive the following gauge-invariant combination
δpδµ ≡ e−
. (82)
In our notation, δpδµ is a gauge-invariant combination which is the same as δp in the δµ = 0 slicing (temporal gauge)
condition to the second order; as the spatial gauge we take γ ≡ 0 to the second order; for the derivation, see the
prescription below Eq. (266) of [11]. In the multi-component case, we similarly have
δp(i)δµ(i) ≡ e(i) −
δµ(i)
µ̇(i)
ė(i) +
c2(i)
δµ(i)
, (83)
where
δp(i) ≡ c
(i)δµ(i) + e(i), c
(i) ≡
ṗ(i)
µ̇(i)
. (84)
We have [25]
e = erel + eint, erel ≡
c2(j) − c
δµ(j), eint ≡
e(j). (85)
Equation (56) gives
δµ(j) +
µ(j) + p(j)
vα(j)
v(j)α − vα
δp(j) +
µ(j) + p(j)
vα(j)
v(j)α − vα
(µ+ p) vα =
µ(j) + p(j)
v(j)α +
δµ(j) + δp(j)
v(j)α − vα
v(j)β − vβ
Παβ =
Π α(j)β +
µ(j) + p(j)
vα(j)
v(j)β − vβ
δαβ v
v(j)γ − vγ
, (86)
By recovering c, dimensions of the variables are
[g̃ab] = [g̃
ab] = [ũa] = 1, [T̃ab] = [µ], [g
] = 1, [∇] = [∆] = 1, [w] = [c2s] = 1,
[A] = [Bα] = [Cαβ ] = 1, [α] = [ϕ] = [β] = [γ] = [B
α ] = [C
α ] = [Ψ
α ] = [C
] = 1, [χ] = L, [κ] = T−1,
[δµ] = [δp] = [e] = [Παβ ] = [Π
] = [µ], [δ] = [Vα] = 1, [v] = [v
α ] = 1, [Π] = L
2[µ], [Π(v)α ] = L[µ]. (87)
Scalar-type perturbation equations can be derived from Eqs. (62)-(70)
κ− 3Hα+ 3ϕ̇+
χ = n0, (88)
4πGδµ+Hκ+
∆+ 3K
n1, (89)
∆+ 3K
χ− 12πG (µ+ p) av = n2 ≡
∆−1∇αn2α, (90)
κ̇+ 2Hκ− 4πG (δµ+ 3δp) +
3Ḣ +
α = n3, (91)
χ̇+Hχ− ϕ− α− 8πGΠ = n4 ≡
a2 (∆ + 3K)
∆−1∇α∇βn
4α, (92)
δµ̇+ 3H (δµ+ δp)− (µ+ p)
κ− 3Hα+
= n5, (93)
a4 (µ+ p) v
a4 (µ+ p)
a (µ+ p)
∆+ 3K
= n6 ≡ −
∆−1∇αn6α, (94)
δµ̇(i) + 3H
δµ(i) + δp(i)
µ(i) + p(i)
κ− 3Hα+
δI(i)0 = n5(i), (95)
µ(i) + p(i)
µ(i) + p(i)
µ(i) + p(i)
δp(i) +
∆ + 3K
Π(i) − δI(i)
= n6(i) ≡ −
µ(i) + p(i)
∆−1∇αn6α(i). (96)
Equations for the vector-type perturbation follow from Eqs. (64),(66),(68),(70)
∆ + 2K
Ψ(v)α + 8πG(µ+ p)v
n2α −∇α∆
−1∇βn2β
2α , (97)
Ψ̇(v)α + 2HΨ
α − 8πGΠ
α = 2a (∆ + 2K)
4α −∇α∆
−1∇γ∇βn
4α , (98)
[a4(µ+ p)v
a4(µ+ p)
∆+ 2K
n6α −∇α∆
−1∇βn6β
6α , (99)
[a4(µ(i) + p(i))v
a4(µ(i) + p(i))
∆ + 2K
µ(i) + p(i)
µ(i) + p(i)
µ(i) + p(i)
n6(i)α −∇α∆
−1∇βn6(i)β
6(i)α
.(100)
Equations for the tensor-type perturbation follow from Eq. (66)
+ 3HĊ
∆− 2K
− 8πGΠ
= n4αβ −
∇α∇β −
(∆ + 3K)
∆−1∇γ∇δn
−2∇(α (∆ + 2K)
∇γn4β)γ −∇β)∆
−1∇γ∇δn
4αβ . (101)
In order to derive eqs. (92,98,101) it is convenient to show
∇α∇β −
(χ̇+Hχ− ϕ− α− 8πGΠ) +
(α|β)
− 8πG
(α|β)
+ 3HĊ
∆− 2K
− 8πGΠ
= n4αβ , (102)
which follows from Eq. (66). Quadratic combinations of linear-order perturbation variables of all three types of
perturbations contribute to all three types of perturbation to the second order.
C. Comoving gauge and irrotational condition
In Eq. (180) of [11] we introduced
Q(i)α ≡
µ(i) + p(i)
−v̄(i),α + v̄
, (103)
where we put overbars to v̄(i) and v̄
in order to distinguish these from our new notations to be used in this paper;
in [11] we didn’t have overbars. From Eq. (58) we have
−v̄(i),α + v̄
≡ V(i)α −Bα +ABα + 2V
Cαβ +
δµ(i) + δp(i)
µ(i) + p(i)
V(i)α −Bα
Π(i)αβ
µ(i) + p(i)
. (104)
It is more convenient to introduce the decomposition in Eq. (79). Thus, we have
−v̄(i),α + v̄
≡ −v(i),α + v
δµ(i) + δp(i)
µ(i) + p(i)
−v(i),α + v
Π(i)αβ
µ(i) + p(i)
. (105)
The variable v̄
introduced in [11] cannot be regarded as a proper vector-type perturbation. However, if we also
take the temporal comoving gauge in [11] which sets v̄ ≡ 0 together with v̄
= 0, we have v = 0 = v
; these
are the same as taking the proper irrotational condition (v
= 0) and the temporal comoving gauge (v = 0). Our
analyses in [12, 13] are, in fact, based on taking these two conditions together. In the irrotational fluids, the temporal
comoving gauge v ≡ 0 leads to ũα = 0, thus ũa coincides with the normal frame four-vector ña.
From Eq. (60) we have
ω̃(i)αβ = av
(i)[α|β]
µ(i) + p(i)
−v(i),[α + v
(i)[α
δp(i) +
∆ + 3K
Π(i) − δI(i)
∆+ 3K
(i)β]
(i)β]
(106)
where we used Eq. (70) to the linear order. For vanishing vector-type perturbation we set v
≡ 0, etc. In this case,
we have
ω̃(i)αβ = −
µ(i) + p(i)
v(i),[α
δp(i) +
∆+ 3K
Π(i) − δI(i)
, (107)
which vanishes for δp(i) = 0 = Π(i) and δI(i) = 0.
VI. EFFECTS OF PRESSURE
We consider a single component situation without rotational perturbations. Equations (65),(67),(68), and Eq. (64)
to the linear order provide a complete set of equations we need in the following.
A. Irrotational case
As an irrotational fluid we ignore all vector-type perturbations, thus v
α = B
α = C
α = Π
α = 0. Quadratic
combinations of linear-order perturbations of all three-types of perturbations contribute to each type of second-order
perturbations. Thus, concerning the second-order scalar-type and vector-type perturbations, ignoring the pure vector-
type perturbation corresponds to ignoring the vector-type perturbation only to the linear order; to the linear order the
vector-type perturbation only has a decaying solution in the expanding phase. Assuming the background equations
are valid Eqs. (67),(68),(65),(64) give
δ̇ + 3H
− (1 + w)
v − 3Hα
(κ− 3Hα) + (1 + w)
H (1 + w)
α2 − β,αβ,α
χ,α|β + Ċ
(1 + w) v,α|β
+ γ,α|β + C
(1 + w) v,α
v,α −
δp,α +Π
+ ϕ,α − [(∆ + 4K)γ],α
(δµ+ δp) v,α +Παβv,β
, (108)
a4 (µ+ p) v,α
(µ+ p)α,α + δp,α +Π
(µ+ p) (v − β)
(v − β)
+(µ+ p) v,α
v − 3Hα
(δµ+ δp) v,α +Π
− (δµ+ δp)α,α + α
(µ+ p)α,α − δp,α −Π
− α,βΠ
α + 2
CγβΠαγ
−ΠγαC
+ΠβγC
, (109)
κ̇+ 2Hκ− 4πG (δµ+ 3δp) +
3Ḣ +
α = −
,α + ακ̇+
3α2 − β,αβ,α
+ 8πG (µ+ p) v,αv,α
2∆α (α+ ϕ) + α,α [α− ϕ+ (∆ + 4K)γ]
+ 2α,α|β
γ,α|β + C
∆ (β,αβ,α)
χ,α|β + Ċ(t)αβ
χ,α|β + Ċ
, (110)
κ = −
∆+ 3K
χ+ 12πGa (µ+ p) v. (111)
Equation (111) is valid to the linear order.
B. Comoving gauge
As the temporal and spatial gauge conditions we set
v ≡ 0 ≡ γ, (112)
thus, β = χ/a. Under these gauge conditions Eq. (109) gives
α = −
χ,αχ,α −
1 + w
1 + 3c2s
2(1 + w)
+ αΠ, (113)
where the imperfect fluid contributions (stresses) are denoted by αΠ with
∆ + 3K
1 + w
∆ + 3K
∆−1∇α
1 + c2s
1 + w
e,α +Π
1 + w
δ,β +
e,β +Π
Πβα + 2
CβγΠαγ
−ΠγαC
+ΠβγC
(114)
Using this, Eqs. (108)-(111) give
δ̇ − 3Hwδ − (1 + w) κ = −
,α + κδ +
1 + w
δ2 + δΠ, (115)
κ̇+ 2Hκ− 4πGµδ −
1 + w
∆+ 3K
δ = −
2(1 + w)
4πGµ−
1 + 2c2s
1 + w
∆+ 3K
1 + w
2Hδκ+
ϕ,αδ,α − 2ϕ∆δ − 2δ
,α|βC
χ,α|β + Ċ(t)αβ
χ,α|β + Ċ
+κΠ, (116)
κ = −
∆+ 3K
χ, (117)
where Eq. (117) is valid only to the linear order, and
δΠ ≡ 2H
∆+ 3K
κ+ 3H
1 + w
χ,α|β + Ċ
∆+ 3K
3Hδ − (1 + w) κ+
∆ + 3K
∆−1∇α
1 + c2s
1 + w
e,α +Π
1 + w
δ,β +
e,β +Π
Πβα + 2
CβγΠαγ
−ΠγαC
+ΠβγC
κΠ ≡ 12πGe
1 + w
3Ḣ +
∆ + 3K
1 + w
∆ + 3K
∆−1∇α
1 + c2s
1 + w
e,α +Π
1 + w
δ,β +
e,β +Π
CβγΠαγ
−ΠγαC
+ΠβγC
−2Hκ+ 4πG
1 + 6c2s
µδ + 12πGe−
1 + w
∆ + 3K
∆+ 3K
∆ + 3K
∆+ 3K
1 + w
∆ + 3K
ϕ,α − 2
∆+ 3K
),α|β
. (118)
Combining Eqs. (115),(116) we can derive
1 + w
(µ+ p)a
− c2s
δ = (1 + w)
χ,α|βχ,α|β −
1 + w
a2κδ − δ,αχ
1 + w
4πGµ−
1 + 2c2s
1 + w
∆+ 3K
2a2Hδκ+ ϕ,αδ,α − 2ϕ∆δ − 2δ
,α|βC
+ (1 + w) Ċ(t)αβ
χ,α|β + Ċ
+(1 + w) κΠ +
1 + w
1 + w
δΠ,αχ
,α. (119)
Notice that the equations above are valid in the presence of general K and Λ.
C. Newtonian correspondence
Guided by our success in the zero-pressure case, we continue to identify
κ ≡ −
∇ · u, (120)
to the second order. Using Eq. (117), assuming K = 0, we can identify
κ = −c
χ = −
∇ · u, u ≡
∇χ = −c∇vχ, (121)
to the linear order. We have recovered the speed of light c. We have [u] = LT−1. Equation (113) becomes
α = −
1 + w
1 + 3c2s
2(1 + w)
+ αΠ. (122)
Equations (115), (116) give
δ̇ − 3wHδ + (1 + w)
∇ · u = −
∇ · (δu) +
1 + w
Hδ2 + δΠ, (123)
∇ · (u̇+Hu) +
1 + w
δ = −
∇ · (u · ∇u)− Ċ(t)αβ
uα|β + Ċ
1 + w
1 + 2c2s
1 + w
δ2 + 2Hδ
∇ · u+
2ϕ∆δ − (∇ϕ) · ∇δ + 2δ,α|βC
− κΠ.(124)
We have [αΠ] = 1, [δΠ] = T
−1, and [κΠ] = T
−2. Combining these equations or Eq. (119) gives
1 + w
(µ+ p)a
− c2sc
1 + w
∇ · (u · ∇u)−
1 + w
1 + w
∇ · (δu)−
1 + w
+ (1 + w) Ċ(t)αβ
uα|β + Ċ
1 + 2c2s
1 + w
δ2 − c2s
2aHδ∇ · u+ 2ϕc2∆δ − c2 (∇ϕ) · ∇δ + 2c2δ,α|βC
+(1 + w) κΠ +
1 + w
1 + w
u · ∇δΠ. (125)
We note that, to the linear order, Eq. (125) is valid in the presence of general K and Λ, see Eq. (119).
D. Linear-order relativistic pressure corrections
To the linear order, ignoring the entropic perturbation e and the anisotropic stress Π, Eqs. (123)-(125) become
δ̇ − 3wHδ + (1 + w)
∇ · u = 0, (126)
∇ · (u̇+Hu) + 4πG̺δ = −
1 + w
, (127)
1 + w
a(1 + w)̺
, (128)
where ignoring the specific internal energy density ǫ we used µ = ̺c2, thus δ = δ̺/̺; in general we have µ = ̺
c2 + ǫ
[18]. Equation (128) can be expanded to give
1 + w
a(1 + w)̺
= δ̈ +
2− 6w + 3c2s
Hδ̇ −
1 + 8w − 6c2s − 3w
4πG̺− 12
w − c2s
) Kc2
5w − 3c2s
δ, (129)
which is valid in the presence of K and Λ. Equation of density perturbation in the comoving gauge was first derived
by Nariai in [26].
In a single component case the Newtonian equations in Eqs. (8),(12),(16) give
∇ · u = 0, (130)
∇ · (u̇+Hu) + 4πG̺δ = −
, (131)
δ̈ + 2Hδ̇ − 4πG̺δ =
. (132)
Comparing Eqs. (126),(127) with Eqs. (130),(131), we notice the presence of w ≡ p/(̺c2) term in three places in
the relativistic equations. Even to the linear order the presence of these terms should be regarded as pure general
relativistic effect of the isotropic pressure. The effects of pressures to the second order compared with the Newtonian
equations can be found in Eqs. (123)-(125) which should be compared with Newtonian equations in Eqs. (8),(12),(16).
Pressure has the genuine relativistic role in cosmology even in the background level.
VII. EFFECTS OF MULTI-COMPONENT
We assume zero-pressure medium, thus set
p(i) ≡ 0, δp(i) ≡ 0 ≡ Π(i)αβ . (133)
From Eq. (86) we notice that to the second order the collective fluid quantities differ from simple sum of individual
fluid quantities. Even in the zero-pressure mediums, we have
δµ(j) + µ(j)v
v(j)α − vα
µ(j)v
v(j)α − vα
µvα =
µ(j)v(j)α + δµ(j)
v(j)α − vα
Παβ =
vα(j)
v(j)β − vβ
δαβ v
v(j)γ − vγ
, (134)
thus δp 6= 0 6= Παβ to the second order.
Equations (69),(70),(65),(67),(68),(64) give
δ̇(i) − δK + 3HA+
1 + δ(i)
vα(i)
δ(i),αB
α + δ(i) (δK − 3HA) +AδK +
A2 −BαBα
−v(i)α
2v̇α(i) +Hv
Avα(i)|α −
(i) +
Cαβv(i)β
Cα|βα v(i)β , (135)
1 + δ(i)
v(i)α
A,α = (δK − 3HA) v(i)α
AA,α − δ(i)A,α −B
βBβ|α −
v(i)αv
− v(i)α|βB
β − v(i)βB
, (136)
δK̇ + 2HδK − 4πG (δµ+ 3δp) +
3Ḣ +
A = AδK̇ −
δK,αB
δK2 +
3A2 −BαBα
2A∆A+A,αA,α −
∆ (BαBα) +A
+ 2CαβA,α|β
Ċαβ +
B(α|β)
Ċαβ +
Ċαα +
+ 8πGµvαvα, (137)
δ̇ − δK + 3HA+
[(1 + δ) vα]|α = −
α + δ (δK − 3HA) +AδK +
A2 −BαBα
−vα (2v̇
α +Hvα)−
Avα|α −
Cαβvβ
Cα|βα vβ − 3H
, (138)
[a (1 + δ) vα]
A,α = (δK − 3HA) vα
AA,α − δ(i)A,α −B
βBβ|α −
− vα|βB
β − vβB
δp,α +Π
, (139)
Ċβα +
+B |βα
Ċγγ +
δK,α = −8πGa
µ(j)v(j)α. (140)
where Eq. (140) is valid to the linear order.
A. Irrotational case
Assuming an irrotational condition we ignore all vector-type perturbations. As the spatial gauge condition we take
γ ≡ 0, (141)
thus, β ≡ χ/a. Equations (135)-(140) become
δ̇(i) − κ+ 3Hα−
1 + δ(i)
δ(i),αχ
,α + δ(i) (κ− 3Hα) + ακ+
χ,αχ,α
+Hv(i),αv
α∆v(i) +
ϕ,αv(i),α −
ϕ∆v(i) −
C(t)αβv(i),α|β , (142)
1 + δ(i)
= (κ− 3Hα) v(i),α +
−αα,α +
χ,βχ,β|α + δ(i)α,α +
v(i),αv
v(i),βχ
, (143)
κ̇+ 2Hκ− 4πG (δµ+ 3δp) +
3Ḣ +
α = ακ̇−
3α2 −
χ,αχ,α
2α∆α+ α,αα,α −
(χ,αχ,α)− α
,αϕ,α + 2ϕ∆α+ 2C
(t)αβα,α|β
Ċ(t)αβ +
χ,α|β
χ,α|β
+ 8πGµv,αv,α, (144)
δ̇ − κ+ 3Hα−
[(1 + δ) v,α]|α = −
,α + δ (κ− 3Hα) + ακ+
χ,αχ,α
+Hv,αv
α∆v +
ϕ,αv,α −
ϕ∆v −
C(t)αβv,α|β − 3H
, (145)
[a (1 + δ) v]
α,α = (κ− 3Hα) v,α
−αα,α +
χ,βχ,β|α + δα,α +
δp,α +Π
, (146)
∆ + 3K
χ+ κ = 12πGa
µ(j)v(j). (147)
Equation (147) is valid to the linear order.
B. Linear perturbations
To the linear order Eqs. (142)-(146) give
δ̇(i) − κ+ 3Hα− c
v(i) = 0, (148)
v̇(i) +Hv(i) −
α = 0, (149)
κ̇+ 2Hκ− 4πG
̺(j)δ(j) +
3Ḣ + c2
α = 0, (150)
∆+ 3K
χ+ κ =
̺(j)v(j), (151)
δ̇ − κ+ 3Hα− c
v = 0, (152)
v̇ +Hv −
α = 0, (153)
where we have recovered the speed of light c. The following additional equations can be found in Eqs. (195),(196),(199)
of [11]
κ− 3Hα+ 3ϕ̇+ c
χ = 0, (154)
4πG̺δ +Hκ+ c2
∆+ 3K
ϕ = 0, (155)
(χ̇+Hχ)− ϕ− α = 0. (156)
Equation (156) gives
αχ = −ϕχ. (157)
Equation (153) gives
αv = 0. (158)
Equations (151), (154) give
ϕ̇v =
χv. (159)
Thus, for K = 0 we have
ϕ̇v = 0, (160)
which is valid even in the presence of multi-components and the cosmological constant. Equations (149), (153) give
v̇χ +Hvχ +
ϕχ = 0, (161)
v̇(i)χ +Hv(i)χ +
ϕχ = 0. (162)
Equations (151), (155) give
∆+ 3K
ϕχ = −4πG̺δv = −4πG
̺(j)δ(j)v = −4πG
̺(j)δ(j)v(j) . (163)
We can derive density perturbation equation in many different temporal gauge (hypersurface) conditions all of which
naturally correspond to gauge-invariant variables. In a single component case, density perturbation in the comoving
gauge (v = 0) is known to give Newtonian result. In the multi-component situation we have many different comoving
gauge conditions. Here, we consider two such gauges for δ(i) variable: one based on v = 0 gauge, and the other based
on v(ℓ) = 0 gauge for a specific ℓ.
(I) Equation (148) evaluated in the v = 0 gauge, and using Eq. (151) we can derive
δ̇(i)v − c
v(i)χ − c
vχ = 0, (164)
where we used χv ≡ χ − av ≡ −avχ, vv(i) ≡ v − v(i), and αv ≡ α − c
−1 (av)
= 0. Using Eqs. (161),(162),(163), we
δ̈(i)v + 2Hδ̇(i)v − 4πG
̺(j)δ(j)v = 0. (165)
This coincides exactly with the Newtonian result in Eq. (16) to the linear order, even in the presence of K. Thus, we
may identify δ(i)v as the Newtonian density perturbation δi to the linear order even in the presence of K. For K = 0
we may also identify −c∇v(i)χ as the Newtonian velocity perturbation ui. However, in the presence of K, we cannot
identify the relativistic variables which correspond to the Newtonian velocity perturbation of individual component.
Therefore, to the linear order we have the following Newtonian correspondences
δi = δ(i)v, ui = −c∇v(i)χ, (166)
where the latter one is valid only for K = 0.
(II) Evaluating Eq. (148) in the v(ℓ) = 0 gauge for a specific ℓ, and using Eq. (151) we can derive
δ̇(i)v(ℓ) − c
∆+ 3K
v(ℓ)χ =
v(j) − v(ℓ)
v(i) − v(ℓ)
, (167)
where we used χv(ℓ) ≡ χ − av(ℓ) ≡ −av(ℓ)χ, v(i)v(ℓ) ≡ v(i) − v(ℓ), and αv(ℓ) ≡ α − c
av(ℓ)
= 0. Using Eqs.
(161),(162),(163), we have
δ̈(i)v(ℓ) + 2Hδ̇(i)v(ℓ) − 4πG
̺(j)δ(j)v(j) =
v(ℓ) − v(j)
. (168)
The terms in right-hand-sides of Eqs. (167),(168) look like relativistic correction terms present even to the linear
order based on the variable δ(i)v(ℓ) . Since no such correction terms appear in Eq. (165) based on the variable δ(i)v,
the relativistic correction terms in Eq. (168) can be regarded as being caused by a complicated hypersurface (gauge)
choice.
1. Exact solutions
Assuming K = 0, we can identify Newtonian perturbation variables as
δ ≡ δv, κv ≡ −
∇ · u, u ≡ −c∇vχ, δΦ ≡ −c
2ϕχ, δi ≡ δ(i)v, ui ≡ −c∇v(i)χ. (169)
Equations (152),(150),(148),(149),(163) become
δ̇ = −
∇ · u, (170)
u̇+Hu = −
∇δΦ, (171)
δ̇i = −
∇ · ui, (172)
u̇i +Hui = −
∇δΦ, (173)
δΦ = 4πG̺δ = 4πG
̺jδj . (174)
Under the identification in Eq. (169) these equations are valid in both Newton’s and Einstein’s gravity theories.
Equations (170), (171), (174), and Eqs. (172),(173),(174), respectively, give
δ̈ + 2Hδ̇ − 4πG̺δ =
= 0, (175)
δ̈i + 2Hδ̇i − 4πG
̺jδj = 0. (176)
Equation (175) has an exact solution
δ(x, t) = H
cg(x)
∫ t dt
+ cd(x)
, (177)
where cg and cd are integration constants which indicate the relatively growing and decaying solutions in expanding
phase; we do not consider the lower bound of integration which is absorbed to the cd mode. Equations (175), (176),
and the solution in Eq. (177) are valid considering general K and Λ in the background world model. Equation (174)
can be solved to give
δΦ = −G̺a2
δ(x′, t)
|x′ − x|
d3x′. (178)
From Eqs. (170),(171),(174) we can show [14]
u = −a
4πG̺a2
D(x), ∇ ·D ≡ 0, (179)
where the D term is the solution of the homogeneous part of Eq. (171); it decouples from the density inhomogeneity
and corresponds to the peculiar velocity in the background world model. Since the D term is not connected to the
density inhomogeneity and simply decays, we may ignore it to the linear order.
Now, for the individual component, from Eqs. (172),(173) we have
ui = −a
4πG̺a2
∇di(x) +
Di(x), ∇ ·Di ≡ 0, (180)
δi = δ + ci(x)−∆di(x)
∫ t dt′
a2(t′)
, (181)
̺jdj ≡ 0 ≡
̺jcj . (182)
The ci and di are the two isocurvature-type (δ = 0, thus δΦ = 0) solutions. It happens that the relatively decaying
isocurvature-type solution, i.e., di-mode, temporally behaves the same as the peculiar velocity in the background, i.e.,
Di-mode. The relatively growing isocurvature-type solution ci does not contribute to the ui, see Eq. (172). From
Eqs. (179),(180) we have
ui − u =
[∇di(x) +Di(x)−D(x)] , (183)
which simply decays; the Di and D solutions are divergence-free and decoupled from the density perturbation, and
are the peculiar velocity perturbation present in the background world model.
C. Comoving gauge
To the linear order, only the hypersurface condition v = 0 (the comoving temporal gauge) allows the density
perturbation equation presented in the Newtonian form. Thus, we take
v ≡ 0, (184)
even to the second order. We take γ ≡ 0 in Eq. (141) as the spatial gauge condition. Equation (146) gives
α = −
χ,αχ,α −
vα(j)v(j)α +∆
vα(j)v
(j)|β
. (185)
Using this, Eqs. (142)-(145) give
δ̇ − κ = −
,α + δκ+
µ(j)v
(j)v(j)α + 3H
∆−1∇α
vα(j)v
(j)|β
, (186)
κ̇+ 2Hκ− 4πG̺δ = −
Ċ(t)αβ +
χ,α|β
χ,α|β
3Ḣ + 8πG̺+ c2
µ(j)v
(j)v(j)α +
3Ḣ + c2
∆−1∇α
vα(j)v
(j)|β
, (187)
δ̇(i) − κ+ c
1 + δ(i)
vα(i)
δ(i),αχ
,α + δ(i)κ+Hv(i)αv
ϕ,αv(i)α − 2ϕv
(i)|α − 2v
µ(j)v
(j)v(j)α + 3H
∆−1∇α
vα(j)v
(j)|β
, (188)
1 + δ(i)
v(i)α
v(i)βχ
+ κv(i)α −
v(i)αv
µ(j)∇α
v(j)β
(j)|β
. (189)
From Eqs. ((186),187), and Eqs. (187)-(189) we can derive, respectively,
− 4πG̺δ (1 + δ) = −
Ċ(t)αβ +
χ,α|β
χ,α|β
2Ḣ + 4πG̺+
µ(j)v
(j)v(j)α +
6Ḣ + c2
∆−1∇α
vα(j)v
(j)|β
, (190)
δ̇(i) +
δ(i),αχ
− 4πG̺δ
1 + δ(i)
Ċ(t)αβ +
χ,α|β
χ,α|β
(κ+ ϕ̇)
v(i)α + 2 (κ− ϕ̇) v
(i)|α
v(i)αχ
vα(i)v
+ Ḣv(i)αv
(i) +
3Ḣ + 4πG̺
µ(j)v
(j)v(j)α + 6Ḣ
∆−1∇α
vα(j)v
(j)|β
. (191)
Notice the O(vα
v(j)α) correction terms are present in Eqs. (186),(187),(190) even in the single component situation.
Except for these O(vα
v(j)α) terms, the remaining parts of these equations coincide with the ones in the single
component situation.
D. Newtonian correspondence
For K = 0, to the linear order, we have
χv ≡ χ− av ≡ −avχ, κv = −c
χv = c
vχ, v(i)v ≡ v(i) − v = v(i)χ − vχ, ϕ̇v = 0. (192)
To the linear order we identify
u ≡ −c∇vχ, ui ≡ −c∇v(i)χ. (193)
Now, to the second order, we attempt identifying the Newtonian perturbation variables δ, δi, u, and ui as
κv ≡ −
∇ · u, χv ≡
u, u ≡ ∇u, v(i)v ≡
(ui − u) , δ ≡ δv, δi ≡ δ(i)v. (194)
Using these identifications Eqs. (186)-(191) can be written as
∇ · u = −
∇ · (δu) +H
|uj − u|
2 + 3∆−1∇ · [(uj − u)∇ · (uj − u)]
, (195)
∇ · (u̇+Hu) + 4πG̺δ = −
∇ · (u · ∇u)− Ċ(t)αβ
uα|β + Ċ
4πG̺− c2
|uj − u|
12πG̺− c2
∆−1∇ · [(uj − u)∇ · (uj − u)]
, (196)
δ̇i +
∇ · ui = −
∇ · (δiui) +
2ϕ∇ · (ui − u)− (ui − u) · ∇ϕ+ 2 (u
i − u
|ui − u|
2 + 3H
|uj − u|
2 +∆−1∇ · [(uj − u)∇ · (uj − u)]
, (197)
∇ · (u̇i +Hui) + 4πG̺δ = −
∇ · (ui · ∇ui)− Ċ
(t)αβ
uα|β + Ċ
|uj − u|
2 + 3∆−1∇ · [(uj − u)∇ · (uj − u)]
, (198)
− 4πG̺δ = −
[a∇ · (δu)]
∇ · (u · ∇u) + Ċ(t)αβ
uα|β + Ċ
4πG̺−
|uj − u|
24πG̺− c2
∆−1∇ · [(uj − u)∇ · (uj − u)]
, (199)
a2δ̇i
− 4πG̺δ = −
[a∇ · (δiui)]
∇ · (ui · ∇ui) + Ċ
i + Ċ
(t)αβ
{∆ [u · (ui − u)]−∇ · [(ui − u) · ∇u+ u · ∇ (ui − u)]}
|ui − u|
2 − 8πG
|uj − u|
2 + 3∆−1∇ · [(uj − u)∇ · (uj − u)]
. (200)
In Sec. VII B 1 we have shown that, to the linear order, (ui − u) simply decays (∝ a
−1) in an expanding phase.
From Eq. (183) we have
ui − u =
[∇di(x) +Di(x)−D(x)] ,
̺jdj ≡ 0, ∇ ·D ≡ 0 ≡ ∇ ·Di. (201)
Thus, ui−u simply decays in an expanding background. If we ignore these contributions from the velocity differences,
except for the presence of the tensor-type perturbation, Eq. (200) coincides exactly with the zero-pressure limit of
Newtonian result in Eq. (16). Terms in the last two lines of Eq. (200) are relativistic correction terms which vanish
for a single component case leading to Eq. (199); the second line in Eq. (199) also vanishes in the single component
case.
Ignoring quadratic combination of (ui − u) terms, we have
∇ · u = −
∇ · (δu) , (202)
∇ · (u̇+Hu) + 4πG̺δ = −
∇ · (u · ∇u)− Ċ(t)αβ
uα|β + Ċ
, (203)
− 4πG̺δ = −
[a∇ · (δu)]
∇ · (u · ∇u) + Ċ(t)αβ
uα|β + Ċ
, (204)
δ̇i +
∇ · ui = −
∇ · (δiui) +
2ϕ∇ · (ui − u)− (ui − u) · ∇ϕ+ 2 (u
i − u
, (205)
∇ · (u̇i +Hui) + 4πG̺δ = −
∇ · (ui · ∇ui)− Ċ
(t)αβ
uα|β + Ċ
, (206)
a2δ̇i
− 4πG̺δ = −
[a∇ · (δiui)]
∇ · (ui · ∇ui) + Ċ
i + Ċ
(t)αβ
{∆ [u · (ui − u)]−∇ · [(ui − u) · ∇u+ u · ∇ (ui − u)]} . (207)
Equations (202)-(204) coincide with the density and velocity perturbation equations of a single component medium
[12]; thus, except for the contribution from gravitational waves, these equations coincide with ones in the Newtonian
context.
If we further ignore (ui − u) terms appearing in the pure second-order combinations, Eqs. (205)-(207) become
δ̇i +
∇ · ui = −
∇ · (δiui) , (208)
∇ · (u̇i +Hui) + 4πG̺δ = −
∇ · (ui · ∇ui)− Ċ
(t)αβ
uα|β + Ċ
, (209)
a2δ̇i
− 4πG̺δ = −
[a∇ · (δiui)]
∇ · (ui · ∇ui) + Ċ
i + Ċ
(t)αβ
. (210)
Notice that, by ignoring i-indices, Eqs. (208)-(210) coincide with Eqs. (202)-(204). In this context, except for the
contribution from gravitational waves, the above equations coincide exactly with ones in the Newtonian context even
in the multi-component case; compare with Eqs. (8),(9),(16) without pressure. In the single component situation such
a relativistic/Newtonian correspondence to the second order was shown in [11, 12]. In the present case, the same
equation valid in the single component is now valid in the multi-component case for the collective fluid variables. This
justifies our identifications of Newtonian perturbation variables in Eq. (194).
VIII. EFFECTS OF CURVATURE
We consider a single zero-pressure, irrotational fluid. We take the temporal comoving gauge (v ≡ 0) and the spatial
γ ≡ 0 gauge. In the presence of background curvature the basic equations are presented in Eqs. (115)-(117) for
nonvanishing pressure, or Eqs. (188)-(191) for zero-pressure multiple component fluids. By setting pressures equal to
zero in Eqs. (115)-(117), or from Eqs. (187)-(190), we have
δ̇ − κ = −
,α + δκ, (211)
κ̇+ 2Hκ− 4πG̺δ = −
Ċ(t)αβ +
χ,α|β
χ,α|β
, (212)
δ̈ + 2Hδ̇ − 4πG̺δ = 4πG̺δ2 −
(δ,αχ
Ċ(t)αβ +
χ,α|β
χ,α|β
, (213)
∆+ 3K
χ+ κ = 0, (214)
where Eq. (214) is valid to the linear order. Compared with the situation with vanishing curvature, the effects of
curvature in the above perturbed set of equations appear only in the linear-order relation between κ and χ in Eq.
(214).
A. Newtonian correspondence
Considering the successful Newtonian correspondence to the linear order even in the presence of the background
curvature, we assume the identification in Eq. (120) is valid to the second order. Then, to the linear order, from Eq.
(214) we have
κ ≡ −
∇ · u ≡ −
u = −c
∆+ 3K
χ, (215)
where u ≡ ∇u and χ = χv = −avχ. Thus, to the linear order, formally we have
∆+ 3K
u. (216)
In the presence of curvature, the scalar-type perturbation can be handled by solving Eqs. (211),(212),(214) together
with the identifications made above. We can formally separate the effects of pure curvature contribution. Using Eqs.
(215),(216), Eqs. (211),(212),(214) become
∇ · u = −
∇ · (δu) +
(∇δ) · ∇
∆+ 3K
, (217)
∇ · (u̇+Hu) + 4πG̺δ = −
∇ · (u · ∇u)− Ċ(t)αβ
u,α|β
∆+ 3K
∆+ 3K
∇ · u
2∆+ 3K
∆+ 3K
∇ · u
∆+ 3K
),α|β
∆+ 3K
. (218)
From Eq. (74), the K term can be written as
(Ωt − 1) , Ωt ≡ Ω+ ΩΛ, Ω ≡
, ΩΛ ≡
. (219)
IX. EFFECTS OF VECTOR-TYPE PERTURBATION
The spatial C-gauge sets
γ ≡ 0 ≡ C(v)α . (220)
The remaining variables under this gauge condition are completely free of the spatial gauge modes and have unique
spatially gauge-invariant counterparts. If we simultaneously take any temporal gauge which also removes the temporal
gauge mode completely, all the remaining variables have corresponding unique gauge-invariant counterparts. The
above statements are true to all orders in perturbations, see Sec. VI of [11]. From Eq. (77) we have
χ, B(v)α ≡ Ψ
α . (221)
Thus, we have
χ,α +Ψ
α , Cαβ ≡ ϕg
. (222)
As the temporal comoving gauge we set
v ≡ 0. (223)
As we mentioned, the remaining variables under these gauge conditions are completely free of the gauge modes and
have unique gauge-invariant counterparts to all orders in perturbation, see [11].
A. Linear perturbations
To the linear order, the three types of perturbations decouple, and evolve independently. The rotational perturbation
is described by Eqs. (206)-(209) in [11]:
∆ + 2K
Ψ(v)α +
(µ+ p) v(v)α = 0, (224)
a4 (µ+ p) v
a4 (µ+ p)
∆+ 2K
, (225)
Ψ̇(v)α + 2HΨ
Π(v)α , (226)
where the last equation follows from the first two. Compared with Bardeen’s notation in [16], we have
B(v)α = B
(1)Q(1)α , C
α = −
α , Ψ
α = ΨQ
B(1) −
Q(1)α ,
v(v)α = vcQ
v(1) −B(1)
Q(1)α , v
α = v
v(1) −
Q(1)α , (227)
where Bardeen’s v
c and v
s , thus our v
α and v
α + Ψ
α , are related to the vorticity and the shear, respectively.
From Eq. (60), to the linear order, we have
ω̃αβ = av
[α|β]
σ̃αβ = a
∇α∇β −
. (228)
Bardeen called Ψ
α a ‘frame-dragging potential’. The difference between v
c and v
s is crucially important to show
Mach’s principle including the linear order rotational perturbation in [27].
In the absence of anisotropic stress which can act as a sink or source of the angular momentum, we have
Angular momentum ∝ a4 (µ+ p) v(v)α ∝ a
2Ψ(v)α ∝ constant in time. (229)
Thus, for vanishing anisotropic stress, we have
v(v)α ∝
a4(µ+ p)
, Ψ(v)α ∝
. (230)
In the zero-pressure limit we have
v(v)α ∝
, Ψ(v)α ∝
. (231)
B. Second-order perturbations
We consider a single component situation with general pressure. We set K ≡ 0. To the linear-order we use
χ ≡ χv ≡ −avχ, κ ≡ κv = c
vχ. (232)
The scalar-type perturbation is described by Eqs. (67),(65) which give
δ̇ + 3H
c2s − w
δ + 3H
− (1 + w) κ
= −3H (1 + w)
v ,αχ −Ψ
vχ,α −Ψ
v ,αχ − Ψ
vχ − 3Hα
+ (1 + w)αc
vχ + (1 + w) v
1− 3c2s
Hv(v)α +
Π(v)α
(1 + w)
− (2α+ ϕ)
v(v)α + 2C(t)αβv
Παα −
vχ,α|β +
(δµ+ δp)
v(v)α +
, (233)
κ̇+ 2Hκ−
(δµ+ 3δp) = −
3Ḣ + c2
v ,αχ −Ψ
vχ,α −Ψ
− 2Hαc
(δµ+ 3δp)α+
Ḣα2 + (α+ 2ϕ) c2
α,α (α− ϕ)
C(t)αβα,α|β +
(µ+ p) v(v)αv(v)α
v ,αχ −Ψ
Ċ(t)αβ −
v ,α|βχ +
Ψ(v)(α|β)
vχ,α|β +
. (234)
The vector-type perturbation is described by Eq. (68) which gives
a4 (µ+ p) v(v)α
(µ+ p)α,α +
δp,α +Π
−αδp+
(µ+ p)
v ,βχ −Ψ
vχ,β −Ψ
α,αδµ
+(µ+ p)
vχ − 3Hα
v(v)α −
(µ+ p)
−v ,βχ +Ψ
v(v)β − v ,βχ +Ψ
− ϕ,βΠ
α + ϕ,αΠ
+ 2C(t)βγΠαβ|γ + C
(δµ+ δp) v(v)α +Π
≡ µcAα. (235)
We have [Aα] = L
−1. From this we have
(µ+ p)α+ δp+
Π = aµ∆−1∇ ·A, (236)
a4 (µ+ p) v(v)α
Π(v)α = µc
∆−1∇ ·A
. (237)
Equation for the tensor-type perturbation (gravitational waves) follows from Eqs. (66),(102).
C. Zero-pressure case
In the zero-pressure limit, Eqs. (236),(235) give
α = a∆−1∇ ·A, (238)
aAα ≡ −
v ,βχ −Ψ
vχ,β −Ψ
v ,βχ −Ψ
v ,βχ −Ψ
v(v)β , (239)
thus, α is purely second-order, and
v ,βχ −Ψ
vχ,β −Ψ
= ∆−1∇α
v ,βχ −Ψ
v ,βχ −Ψ
v(v)β
.(240)
Equations (233), (234), (237) give
δ̇ − κ =
v ,αχ −Ψ
(v)α − v(v)α
+Hv(v)α v
(v)α +
−ϕ,αv(v)α + 2C
(t)αβv
−3H∆−1∇α
v ,βχ −Ψ
v ,βχ −Ψ
v(v)β
, (241)
κ̇+ 2Hκ− 4πG̺δ =
v ,αχ −Ψ
v ,βχ −Ψ
+ Ċ(t)αβ
vχ,α|β −Ψ
+8πG̺v(v)αv(v)α −
3Ḣ + c2
∆−1∇α
v ,βχ −Ψ
v ,βχ −Ψ
v(v)β
,(242)
v̇(v)α +Hv
v ,βχ −Ψ
v ,βχ −Ψ
v(v)β
v ,βχ −Ψ
v ,βχ −Ψ
v(v)β
. (243)
D. Newtonian correspondence
In order to compare with Newtonian equations we continue identifying κ as in Eq. (120) to the second order. To
the linear order we identify
u ≡ −c
∇vχ − v
≡ ∇u+ u(v), (244)
u ≡ −cvχ, u
(v) ≡ cv(v). (245)
We introduce the following notations
Uα ≡ uα + cΨ
α , Ũα ≡ u,α + cΨ
α . (246)
As mentioned below Eq. (227), to the linear order, uα and Uα are related to the vorticity and the shear, respectively.
Equations (241)-(243) become
∇ · u = −
∇ · (δU) +
C(t)αβu
(v) · ∇ϕ+
(v) · u(v) + 3∆−1∇α
Uβ + u(v)βŨβ|α
,(247)
∇ · (u̇+Hu) + 4πG̺δ = −
∇ · (U · ∇U)− Ċ(t)αβ
Ũα|β
−u(v) · u(v) +
∆−1∇α
Uβ + u(v)βŨβ|α
, (248)
(v) +Hu(v) = −
U · ∇u−∇∆−1∇ · (U · ∇u)
. (249)
From Eq. (249) we notice that the tensor-type perturbation does not affect the vector-type perturbation to the second
order. The pure scalar-type perturbation also cannot generate the vector-type perturbation to the second-order; the
same is true in the Newtonian case, see below Eq. (15).
From Eq. (224), to the linear order, we have
v(v)α =
Ψ(v)α , (250)
where k is a comoving wavenumber with ∆ ≡ −k2, thus [k] = 1. Since (ck)/(aH) ∼ (visual-horizon)/(scale), we have
far inside horizon : v(v)α ≫ Ψ
α , U ≃ u = ∇u+ u
(v), Ũ ≃ ∇u,
far outside horizon : v(v)α ≪ Ψ
α , U ≃ Ũ = u,α + cΨ
α . (251)
Apparently, contributions of vector-type perturbation to the second order depend on the visual-horizon scale.
Far inside the horizon, we can ignore cΨ
α compared with u
α . In the matter dominated era we have
δv, (252)
where we used Eqs. (293),(329),(330) of [11]. Thus, the third term in the right-hand-side of Eq. (247) is (aH/kc)2-order
smaller than the first term. The fourth (and last) term in the right-hand-side of Eq. (247) is (aH/kc)[u(v)/(cδ)]-order
smaller than the first term. The third (and last) term in the right-hand-side of Eq. (248) is also (aH/kc)2-order
smaller than the first term. Thus, Eqs. (247)-(249) give
∇ · u = −
∇ · (δu) +
C(t)αβu
, (253)
∇ · (u̇+Hu) + 4πG̺δ = −
∇ · (u · ∇u)− Ċ(t)αβ
u,α|β + cΨ
, (254)
(v) +Hu(v) = −
u · ∇u−∇∆−1∇ · (u · ∇u)
. (255)
Thus, if we could ignore the tensor-type combination in Eqs. (253),(254), Eqs. (253)-(255) coincide exactly with the
Newtonian equations: see Eqs. (8),(12),(15) ignoring the pressure terms and the subindices i. The vector-tensor
combinations in Eq. (253),(254) are new relativistic contributions of the vector-type perturbations; compare these two
equations with Eqs. (17),(18) which are valid in the absence of the vector-type perturbations. Notice the form of last
term u,α|β + cΨ
in Eq. (254) which subtly differs from the expression uα|β in Eq. (18).
Contributions from the vector-type perturbation become more complicated near and outside the horizon scale.
The presence of vector-type metric perturbation Ψ
α , the scalar-type curvature perturbation ϕ, and the tensor-type
perturbation C
coupled with the vector-type perturbation give additional effects.
E. Pure vector-type perturbations
In Sec. VII-E of [11] we have considered a situation with pure vector-type perturbation. As the analysis was made
based on the fluid quantities in the normal frame, in the following we present the case based on the fluid quantities
in the energy frame. Considering only the vector-type perturbation of a fluid, Eq. (68) gives
a4(µ+ p)
(µ+ p) v(v)α +
(α|β)
v(v)β
c(∆ + 2K)Π
2a2(µ+ p)
v(v)β +Ψ(v)|β
v(v)βv
. (256)
Thus, for Π
α = 0 we have
a4(µ+ p)
a4 (µ+ p) v(v)α
v(v)β +Ψ(v)|β
v(v)βv
. (257)
This differs from Eq. (365) of [11] which is due to the difference in the frame choice. The momentum constraint
equation in Eq. (101) of [11] becomes
∆ + 2K
Ψ(v)α +
(µ+ p) v(v)α = −
(α|β)
v(v)β . (258)
Under the gauge transformation, from Eq. (61) we have
v̂(v)α = v
α − v
ξ(v)β,α − v
ξ(v)β . (259)
To the linear order, from Eq. (230) of [11] we have B̂
α = B
α + ξ
α . We consider a gauge transformation from the
C-gauge (C
α ≡ 0, without hat) to the B-gauge (B
α ≡ 0, with hat). We have B̂
α ≡ 0, and B
α |C−gauge = −ξ
Thus,
ξ(v)α = −
B(v)α |C−gaugedη = −a
2Ψ(v)α
∫ η dη
, (260)
where we used B
α |C−gauge = Ψ
α ∝ a
−2. Thus, Eq. (259) gives
v(v)α |B−gauge =
v(v)α −
Ψ(v)β,α − v
Ψ(v)β
∫ η dη
C−gauge
. (261)
X. EQUATIONS WITH FIELDS
A. A minimally coupled scalar field
Equations in the case of a minimally coupled scalar field are presented in Eqs. (112)-(114) of [11]. The equation
of motion in Eq. (112) and the full Einstein’s equations in Eqs. (99)-(105) expressed using the normal-frame fluid
quantities together with the normal-frame fluid quantities for the scalar field in Eqs. (114) all in [11] provide a
complete set of equations we need to the second-order. The fluid quantities in Eq. (114) of [11] are presented in the
normal-frame four-vector and it is convenient to know the conventionally used fluid quantities which are based on the
energy-frame four-vector. These latter quantities can be read from Eqs. (88),(114) of [11] and Eq. (79) as
QN(φ)α = −
φ̇δφ,α + δφ,α
δφ̇− φ̇A
µ(φ) + p(φ)
V (φ)α −Bα +ABα + 2V
(φ)βCαβ
δµ(φ) + δp(φ)
V (φ)α −Bα
µ(φ) + p(φ) + δµ(φ) + δp(φ)
−v(φ),α + v
(φ,v)
δµ(φ) = δµN(φ) −
δφ,αδφ,α, δp
(φ) = δpN(φ) −
δφ,αδφ,α,
δφ,αδφ,β −
δφ,γδφ,γ
. (262)
Thus, using Eq. (114) of [11] we have
v(φ) =
∆−1∇α
δφ̇−Aφ̇
v(φ,v)α =
δφ̇− φ̇A
δφ,α −∇α∆
δφ̇− φ̇A
δµ(φ) = φ̇δφ̇− φ̇2A+ V,φδφ+
δφ̇2 −
δφ,αδφ,α +
V,φφδφ
2 − 2φ̇δφ̇A+
φ̇δφ,αB
α + 2φ̇2A2 −
φ̇2BαBα,
δp(φ) = φ̇δφ̇− φ̇2A− V,φδφ+
δφ̇2 −
δφ,αδφ,α −
V,φφδφ
2 − 2φ̇δφ̇A+
φ̇δφ,αB
α + 2φ̇2A2 −
φ̇2BαBα,
= 0. (263)
Notice that no anisotropic stress is caused by a minimally coupled scalar field even to the second order in perturbations.
The uniform-field gauge takes δφ ≡ 0 as a temporal gauge (slicing) condition to the second-order in perturbation.
The uniform-field gauge gives v(φ) = 0 which is the comoving gauge, and vice versa. Thus,
δφ = 0 ↔ v(φ) = 0. (264)
We also have
δµ(φ) − δp(φ) = 2V,φδφ+ V,φφδφ
2, (265)
and under the uniform-field gauge we have
= −φ̇2
A (1− 2A) +
. (266)
Equation (263) apparently shows that the vector-type perturbation v
(φ,v)
α does not vanish to the second-order.
However, the second-order quantities in right-hand-side depend on the temporal gauge condition for the scalar-type
perturbations, and trivially vanish for the uniform-field gauge. We can also show that it vanishes for the uniform-
density gauge or the uniform-pressure gauge where we have δφ̇ − φ̇A ∝ δφ to the linear order. In fact, we can show
that the vector-type perturbation is not sourced by the scalar field to the second order by evaluating the rotational
tensor ω̃αβ in Eq. (60) for the scalar field: i.e., for a minimally coupled scalar field we have
= 0, (267)
to the second order. Thus, we conclude that a minimally coupled scalar field does not contribute to the rotational
perturbation to the second order in perturbations. In fact, we can show that a single scalar field do not support
vector-type perturbations to all orders in perturbations. As we take the energy frame, thus q̃a ≡ 0, from Eq. (23) of
[11] we can show
ũa =
−φ̃,cφ̃,c
. (268)
Using the definition of the vorticity tensor in Eq. (33) we can show that ω̃ab = 0.
Using the fluid quantities in Eq. (263) we can handle the scalar field using our non-ideal fluid formulation. The
fluid equations in the energy frame, like Eqs. (62)-(68), remain valid in the case of scalar field with the fluid quantities
expressed as in Eq. (263). The anisotropic stress vanishes and the entropic perturbation e is given as
e(φ) ≡ δp(φ) − c2(φ)δµ
(φ), c2(φ) ≡
ṗ(φ)
µ̇(φ)
φ̈− V,φ
φ̈+ V,φ
. (269)
Under the comoving gauge v ≡ 0, using Eq. (266) we have
δp(φ)v = δµ
v , e
1− c2(φ)
δµ(φ)v , (270)
which is a well known relation, now valid to the second order in perturbations. A fluid formulation of the scalar field
to the linear order is presented in [24, 29]. Using Eq. (270) together with vanishing anisotropic stress, Eqs. (62)-(68)
or (88)-(94) provide the fluid formulation for a minimally coupled scalar field to the second order in perturbations.
The perturbed equation of motion of the scalar field is presented in Eq. (112) of [11].
B. Minimally coupled scalar fields
In the case of multiple minimally coupled scalar fields, the equation of motions in Eq. (119) and the full Einstein’s
equations in Eqs. (99)-(105) expressed using the normal-frame fluid quantities together with the normal-frame fluid
quantities for the scalar field in Eqs. (121) all in [11] provide a complete set of equations we need to the second-order.
The fluid quantities in Eq. (121) of [11] are presented in the normal-frame four-vector and it is convenient to know
the conventionally used fluid quantities which are based on the energy-frame four-vector. These latter quantities can
be read from Eqs. (121) of [11] and Eq. (58) as
v(φ) =
φ̇(k)δφ(k) +∆
δφ̇(k) − φ̇(k)A
δφ(k) − 2φ̇(k)
φ̇(l)δφ(l)
v(φ,v)α = −
δφ̇(k) − φ̇(k)A
δφ(k) − 2φ̇(k)
φ̇(l)δφ(l)
δφ̇(k) − φ̇(k)A
δφ(k) − 2φ̇(k)
φ̇(l)δφ(l)
δµ(φ) = δµN(φ) −
φ̇(k)δφ
φ̇(l)δφ(l),α
δp(φ) = δpN(φ) −
φ̇(k)δφ
φ̇(l)δφ(l),α
φ̇(k)δφ(k),α
φ̇(l)δφ(l),β
φ̇(k)δφ
φ̇(l)δφ(l),γ
(271)
where the normal-frame fluid quantities are presented in Eq. (121) of [11]. Thus, by moving into the energy-frame
fluid quantities from the normal-frame ones we have rather complicated terms which do not cancel out nicely any term
in the normal-frame quantities. In single field case we had such cancelations, for example we have Π
= 0 in Eq.
(263). But, we do not have such a luxury in the multi-component situation. Apparently Π
in the energy-frame does
not vanish and looks more complicated. Thus, in the multi-field situation we had better use both the fluid quantities
and Einstein’s equations all expressed in the normal frame: these are Eqs. (99)-(105),(121) in [11].
As the temporal gauge condition we can set any one field perturbation, say the specific ℓ-th one δφ(ℓ), equal to zero
which might be called the uniform-φ(ℓ) gauge to the second order. This apparently differs from the comoving gauge
which sets v(φ) ≡ 0. In the multi-component situation we cannot take a gauge condition which makes v
(φ,v)
α = 0.
Thus, the multiple scalar fields source the vector-type perturbation to the second order in perturbations. The scalar-,
vector-, and tensor-type decomposition of the anisotropic stress Π
can be read by using decomposition formulae in
Eq. (177) of [11].
In multiple-field situation, it is ad hoc and cumbersome (if not impossible) to introduce individual fluid quantity
for each field variable even to the background and linear order perturbations, see [30].
C. Generalized gravity case
In Sec. IV.D of [11]we presented the equation of motion and effective fluid quantities in a class of generalized gravity
theories together with additional presence of fluids and fields to the second order. The equation of motion is in Eq.
(128) of [11], and the full Einstein’s equations in Eqs. (99)-(105) expressed using the normal-frame fluid quantities
together with the normal-frame effective fluid quantities in Eq. (130) of [11] provide a complete set of equations
we need to the second-order. The effective fluid quantities in Eq. (130) of [11] are presented in the normal-frame
four-vector and using Eq. (88) in [11] we can easily derive the effective fluid quantities based on the energy-frame
four-vector. As in the multiple field case in a previous subsection, by moving into the energy-frame, the effective fluid
quantities become more complicated compared with the ones in the normal-frame. Thus, in these class of generalized
gravity theories we had better use both the fluid quantities and Einstein’s equations all expressed in the normal frame:
these are Eqs. (99)-(105),(130) in [11].
XI. CURVATURE PERTURBATIONS AND LARGE-SCALE CONSERVATIONS
In the large-scale limit the spatial curvature perturbation ϕ in several different gauge conditions is known to remain
constant in expanding phase. Often the conservation properties are shown based on the first time derivative of the
curvature perturbation. In order to show the conservation properties properly we have to construct the closed form
second-order differential equations for the curvature perturbation. In the following we will derive such first-order and
second-order differential equations for ϕ̇v, ϕ̇χ, ϕ̇κ, and ϕ̇δ. First we will derive equations for the linear perturbation
including the background curvature and non-ideal fluid properties. Then we will derive equations for the second
order perturbation assuming a flat background; including the background curvature is trivial, though. We consider a
single-component fluid.
A. Linear-order equations
We introduce a combination
Φ ≡ ϕv −
4πG (µ+ p)
ϕχ. (272)
This combination was first introduced by Field and Shepley in [28]. From Eqs. (88),(90),(92), Eqs. (88)-(90),(92),(94),
Eqs. (88),(93), and Eqs. (88),(89),(91), respectively, we can derive
4πG (µ+ p)
Φ− 8πGHΠ, (273)
4πG (µ+ p)
, (274)
ϕ̇δ =
, (275)
ϕ̇κ = −
3Ḣ +∆/a2
1 + 3c2s
) ∆+ 3K
ϕκ − 12πGe
χκ. (276)
These equations were presented in [24, 31]
We can derive closed form second-order differential equations for Φ, ϕχ, ϕδ, and ϕκ
(µ+ p) a
8πGH2
= c2s
ϕχ − 4πG
, (277)
H2c2s
(µ+ p) a3
(µ+ p) a3
H2c2s
= c2s
, (278)
3Ḣ +∆/a2
(µ+ p) a3
(µ+ p) a3
3Ḣ +∆/a2
ϕ̇δ +
4πG (µ+ p) + c2s (∆ + 3K) /a
12πG (µ+ p)− (∆ + 3K) /a2
3Ḣ +
, (279)
ϕ̇κ +
3Ḣ +∆/a2
1 + 3c2s
) ∆+ 3K
ϕκ − 12πGe
3Ḣ +∆/a2
1 + 3c2s
) ∆+ 3K
ϕκ − 12πGe
+ 8πGΠ
. (280)
Equations (277),(278) follow by combining Eqs. (273),(274). Equations (279) and (280) follow from Eqs.
(88),(89),(91),(92), and Eqs. (88),(89),(91),(92), respectively.
From Eqs. (89),(90) we can derive
∆+ 3K
12πG (µ+ p) a2
ϕκ = ϕv −
∆+ 3K
12πG (µ+ p)a2
ϕχ = Φ−
12πG (µ+ p) a2
ϕχ. (281)
These relations were presented in [17]. In the large-scale limit and in near flat background, thus ignoring ∆ and K
terms, we have
Φ ≃ ϕv ≃ ϕδ ≃ ϕκ, (282)
to the leading order in the large-scale expansion.
In the large-scale limit, ignoring ∆ terms, in near flat background and for an ideal fluid case, thus setting K = 0
and e = 0 = Π, Eqs. (277)-(280) give
(µ+ p) a
, (283)
ϕ̇v ∝
H2c2s
(µ+ p) a3
, (284)
ϕ̇δ ∝ ϕ̇κ ∝
. (285)
Notice that if we simply ignore the ∆ terms in Eqs. (274)-(276) we simply have ϕ̇v = ϕ̇δ = ϕ̇κ = 0. In such a way
we cannot recover the terms in the right-hand-side of Eqs. (284),(285); these terms lead to decaying solutions (in an
expanding era) in the large-scale limit and are higher order in the large-scale expansion compared with the decaying
solution of ϕχ, see below. From Eqs. (283)-(285) we have general large-scale asymptotic solutions
ϕχ = 4πGC(x)
∫ t a (µ+ p)
dt+ d(x)
, (286)
ϕv = C(x) +
∫ t c2sH
(µ+ p) a3
dt, (287)
ϕδ = ϕκ = C(x) +
∫ t dt
, (288)
where C(x) and d(x) are integration constants which correspond to the relatively growing and decaying modes,
respectively, in an expanding phase; in a collapsing phase the roles are reversed. The coefficients are fixed using
the relations in Eqs. (273),(274),(281). Notice that for the C-mode the relation in Eq. (282) is satisfied, and simply
remain constant. For the d-mode, ϕv, ϕδ, and ϕκ are ∆/(aH)
2-order higher compared with the d-mode of ϕχ. The
ϕv is one of the well known conserved quantity in the large-scale even in the context of generalized gravity theories
[24, 32].
In order to evaluate the solutions to the second order in the next section, we need complete sets of linear order
solutions for different gauge conditions. For an ideal fluid, and for a minimally coupled scalar field such complete sets
of solutions are presented in tabular forms in [29, 31]. In the following we summarize such sets of solutions in an ideal
fluid case for four different gauge conditions. From Table 8 of [31] we have
ϕχ = −αχ = C
, δχ = −
= −2C
adt, vχ = −C
ϕv = C, Hχv = C
adt, δv = −
1 + w
αv = −
ϕδ = C, Hχδ = C
= 3αδ = −
C, vδ =
ϕκ = C, Hχκ = C
adt, δκ = −3
1 + w
1 + 3c2s
ακ = −
C, vκ = −
adt. (289)
For corresponding sets of solutions for a minimally coupled scalar field, see Table 1 of [29]. Compared with the notation
used in [31] we have γΨ = −8πG(µ+p)av. The lower bounds of integration of solutions in Eq. (289) give behaviors of
d-modes. For solutions without integration, the d-modes are ∆/(aH)2-order higher than the non-vanishing d-mode,
for example, see the solutions in Table 2 of [31]. Thus, the d-modes are
ϕχ = −αχ = d
, δχ = −
, vχ = d
∫ t c2sH
(µ+ p) a3
dt, Hχv = −
d, δv = −
1 + w
αv = −
∫ t dt
, Hχδ = −
∫ t dt
, αδ =
∫ t dt
, vδ =
∫ t dt
, Hχκ = −
d, δκ = −3
1 + w
1 + 3c2s
ακ = −
∫ t dt
, vκ =
d. (290)
B. Second-order equations
We assume K = 0. From Eqs. (88),(90),(92), Eqs. (88)-(90),(94), Eqs. (88),(93), and Eqs. (88),(89),(91), respec-
tively, we can derive
(ϕ−Hχ)
4πG (µ+ p)
(ϕ− aHv)− 8πGHΠ+
(n0 − n2)−Hn4, (291)
(ϕ− aHv)
4πG (µ+ p)
(ϕ−Hχ)−
n1 +Hn2
(n0 − n2)− aHn6, (292)
ϕ̇δ =
, (293)
ϕ̇κ = −
3Ḣ +∆/a2
1 + 3c2s
ϕκ − 12πGe−
1 + 3c2s
n1 − n3
n0. (294)
The perturbed order variables in Eqs. (293),(294) are evaluated in the uniform-density gauge (δ ≡ 0), and the
uniform-expansion gauge (κ ≡ 0), respectively. Equation (293) also follows from Eq. (41) evaluated to the second
order.
We can derive closed form second-order differential equations for ϕv, ϕχ, ϕδ, and ϕκ
(ϕ−Hχ)
+ 8πGHΠ−
(n0 − n2) +Hn4
= c2s
(ϕ−Hχ)− 4πG
+ c2s
n1 +Hn2
4πG (µ+ p)
(n0 − n2)− aHn6
, (295)
H2c2s
4πG (µ+ p) a3
4πG (µ+ p) a3
H2c2s
(ϕ− aHv)
(n0 − n2) + aHn6
n1 −Hn2
= c2s
ϕ− aHv −
Hc2s∆
4πG (µ+ p) a2
(n0 − n2)−Hn4
, (296)
1 + ∆/(3a2Ḣ)
1 + ∆/(3a2Ḣ)
ϕ̇δ +
3Ḣ +∆/a2
ṅ1 +
n1 −Hn3
1− c2s∆/(a
3 + ∆/(a2Ḣ)
3Ḣ +
9Ḣa2
− n3 −
3Ḣ +
, (297)
ϕ̇κ +
3Ḣ +∆/a2
1 + 3c2s
− 12πGe− n3
−ϕκ +
3Ḣ +∆/a2
1 + 3c2s
− 12πGe− n3
− 8πGΠ− n4
. (298)
Equations (295),(296) follow by combining Eqs. (291),(292). Equation (297) follows from Eqs. (88),(89),(91),(92).
Equation (298) follows from Eqs. (88),(89),(91),(92). The perturbed order variables in Eqs. (297),(298) are evaluated
in the uniform-density gauge (δ ≡ 0), and the uniform-expansion gauge (κ ≡ 0), respectively.
C. Large-scale solutions
Now, we assume an ideal fluid, thus set e = 0 = Π. In the large-scale limit, thus ignoring the ∆/(aH)2-order higher
terms, Eqs. (296)-(298) give
ϕ̇v −
(n0 − n2) + aHn6 +
4πG (µ+ p)
n1 +Hn2
H2c2s
4πG (µ+ p) a3
, (299)
ϕ̇δ +
ṅ1 +
Hn1 −Hn3
, (300)
ϕ̇κ −
1 + 3c2s
n1 + n3
, (301)
where the perturbed order variables in Eq. (299) are evaluated in the comoving gauge (v ≡ 0). We already used the
behavior of linear order solutions in Eqs. (289),(290) in order to show that the right-hand-side of Eqs. (296)-(298)
vanish. Using the solutions in Eqs. (289),(290) we can show that
ϕv − ϕ
+O(∆C2,∆2d2) ∝
H2c2s
(µ+ p)a3
, (302)
ϕδ − ϕ
+O(∆C2,∆2d2) ∝
ϕκ − ϕ
+O(∆C2,∆2d2) ∝
. (303)
Thus, we have general large-scale asymptotic solutions
ϕv − ϕ
v = C(x) +
∫ t c2sH
(µ+ p) a3
dt, (304)
ϕδ − ϕ
δ = ϕκ − ϕ
κ = C(x) +
∫ t dt
, (305)
where C(x) and d(x) are integration constants now including the second-order contributions, i.e., C = C(1) + C(2),
etc. Ignoring the transient solutions in an expanding phase we have
ϕv = ϕδ = ϕκ = C(x), (306)
even to the second order in perturbations in the large-scale limit.
XII. DISCUSSION
In this work we presented pure general relativistic effects of second-order perturbations in Friedmann cosmological
world model. In our previous work we have shown that to the second-order perturbations, the density and velocity
perturbation equations of general relativistic zero-pressure, irrotational, single-component fluid in a flat background
coincide exactly with the ones known in Newton’s theory, [12]. We also have shown the effect of gravitational waves to
the second-order, and pure general relativistic correction terms appearing in the third-order perturbations, [12, 13].
Here, we presented results of second-order perturbations relaxing all the assumptions made in our previous work
in [12]. We derived the general relativistic correction terms arising due to (i) pressure, (ii) multi-component, (iii)
background curvature, and (iv) rotation. We also presented a general proof of large-scale conserved behaviors of
curvature perturbation variable in several gauge conditions, now to the second order.
Effects of pressure can be found in Eqs. (123)-(125). As we emphasized, the effect of pressure is generically
relativistic even in the background world model and the linear order perturbations. Still, our equations show the
pure general relativistic effects of pressure (including stresses) appearing in the second-order perturbations. Effects
of multi-component fluids can be found in Eqs. (195)-(200). Although these equations apparently show deviations
from Newtonian situation, in Sec. VII D we showed that if we ignore purely decaying terms in an expanding phase the
equations are effectively the same as in the Newtonian situation. Effects of background spatial curvature K can be
read from Eqs. (217),(218) or Eqs. (211)-(214). Effects of vector-type perturbation can be read from Eqs. (247)-(249).
In the small-scale limit we showed that, if we ignore the tensor-type perturbation, the equations coincide with the
Newtonian ones.
Our results may have important practical implications in cosmology and the large-scale structure formation. Our
new result showing relativistic/Newtonian correspondence in the zero-pressure irrotational multi-component fluids
is practically relevant in currently favored cosmology where baryon and dark matter are two important ingredients
of the current matter content in addition to the cosmological constant. All equations in our work are valid in the
presence of the cosmological constant. A related important result is the relativistic/Newtonian correspondence valid
in the presence of rotational perturbation far inside horizon. Thus, inside the horizon scale, even in the presence of
rotational perturbations we can still rely on the Newtonian equations to handle quasi-linear evolution of large-scale
structures. As the spatial curvature in the present cosmological era is known to be small [33], the possible presence of
small spatial curvature may not be important in the second-order perturbations. Still, while the Newtonian equation
is exactly valid to the linear order even in the presence of the spatial curvature, we have nontrivial general relativistic
correction terms present to the second order in perturbations. Our second-order perturbation equations in the presence
of pressure may have an interesting role as we approach early stage of universe where the effect of radiation becomes
important. The importance of pressure to the second-order perturbations, of course, depends on whether nonlinear
effects are significant in the early evolution stage of the large-scale structure during the radiation era and in the early
matter dominated era. Realistic estimations of the diverse pure general relativistic contributions using the complete
set of equations presented in this work are left for future investigations.
In an accompanying paper we will investigate the effects of third-order perturbations of zero-pressure irrotational
multi-component fluids in a flat background. This is one obvious remaining issue in our series of investigation of
nonlinear cosmological perturbations where nontrivial general relativistic effects are expected. In the case of a single
fluid we presented the pure general relativistic effects appearing in the third order in [13]. Corresponding results in
the case of multi-component will be presented in [34].
Acknowledgments
H.N. was supported by grant No. C00022 from the Korea Research Foundation.
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	Contents
	Introduction
	Newtonian nonlinear perturbations
	Summary of previous work
	Relativistic fully nonlinear equations
	Covariant equations
	ADM equations
	Second-order perturbations
	Basic equations in the energy-frame
	Decomposition
	Comoving gauge and irrotational condition
	Effects of pressure
	Irrotational case
	Comoving gauge
	Newtonian correspondence
	Linear-order relativistic pressure corrections
	Effects of multi-component
	Irrotational case
	Linear perturbations
	Exact solutions
	Comoving gauge
	Newtonian correspondence
	Effects of curvature
	Newtonian correspondence
	Effects of vector-type perturbation
	Linear perturbations
	Second-order perturbations
	Zero-pressure case
	Newtonian correspondence
	Pure vector-type perturbations
	Equations with fields
	A minimally coupled scalar field
	Minimally coupled scalar fields
	Generalized gravity case
	Curvature perturbations and large-scale conservations
	Linear-order equations
	Second-order equations
	Large-scale solutions
	Discussion
	Acknowledgments
	References
