Baumann's Cosmology (6)

For relativistic fluids (like photons neutrons) and on scales larger than the Hubble radius the correct description requires a general-relativistic treatment which we will develop in this chapter. Fig. 6.1 illustrates the interactions of all known matter species in the universe.
   After neutrino decoupling and nucleosynthesis-at low energy-the only significant forces in cosmological evolution were gravity and electromagnetism. By equivalence principle, all species interact through a universal gravitational force. In addition, photon, electrons and protons were strongly coupled by their electromagnetic interactions and could be treated as a single photon-baryon fluid. After recombination, the electromagnetic coupling became insignificant and the evolution was determined by gravity alone. There are two types of gravitational effects: the homogeneous density of the species determines the expansion rate of the universe, while their perturbations source the local gravitational potential. Both of these effects feed back into the into the evolution of the species. This feedback is nonlinear and can lead to rather complex dynamics.
   In structure formation matter-radiation equality, 𝑧_eq ≈ 3400, and photon decoupling, 𝑧_dec ≈ 1100 are two important times. Before equality, the growth structure is suppressed by the large radiation pressure of photons. The clustering of dark matter perturbations only start to become significant when the universe becomes matter dominated. Before decoupling, baryons are tightly coupled to the photons and their density fluctuations oscillate. These oscillation arise because of two opposed forces: gravity and radiation pressure. As gravity makes the perturbations collapse, the radiation pressure increases and causes the perturbation to re-expand. The pressure then drops, so that gravity wins the battle again and the process repeats, leading to sustained oscillations or "sound waves' in the primordial plasma. The frequency depends on wavelength of the fluctuations. At decoupling, modes with different wavelengths are captured at different phases, giving rise to a characteristic pattern in CMB. After decoupling, the baryons lose the pressure support of the photons and fall into the gravitational potential wells created by the dark matter. Eventually, stars and galaxies will form.

          6.1 Linear Perturbations  Conceptually perturbation theory is GR is the same as in Newtonian gravity. We write the metric and energy momentum tensor as³     (6.1)   𝑔_𝜇𝜈(𝑡, 𝐱) = 𝑔̄_𝜇𝜈 + 𝛿𝑔_𝜇𝜈(𝑡, 𝐱),     𝛵_𝜇𝜈(𝑡, 𝐱) = 𝛵̄_𝜇𝜈 + 𝛿𝛵_𝜇𝜈(𝑡, 𝐱), and expand ∇^𝜇𝛵_𝜇𝜈 = 0 and 𝐺 = 8π𝐺𝛵_𝜇𝜈 to linear order in perturbations. In practice two features make it more complicated than the Newtonian analysis. First, the algebra is much more involved. To address this challenge we will perform all algebraic manipulations very explicitly and a Mathematica notebook will be provided on  the book's website which make us to perform the calculation without any effort. Second, on large scales, there is an ambiguity in what we call matter perturbations and what are metric perturbations-by change of coordinates we can transform on into other. This subtlety did not arise in the Newtonian treatment.           6.1.1 Metric Perturbations 
    We will take the background metric 𝑔̄_𝜇𝜈 to be the flat FRW metric. The perturbed spacetime can be written as
    (6.2)   𝑑𝑠² = 𝑎(𝜂)[-(1 + 2𝛢)𝑑𝜂² + 2𝛣_𝑖𝑑𝑥^𝑖𝑑𝜂 + (𝛿_𝑖𝑗 + 2𝐸_𝑖𝑗)𝑑𝑥^𝑖𝑑𝑥^𝑗],
where 𝛢, 𝛣_𝑖 and 𝐸_𝑖𝑗 are functions of space and (conformal) time. The factors of 2 are introduced for later convenience.
   It will be extremely useful to perform a scalar-vector-Tensor(SVT) decomposition of the perturbations. We can split any 3-vector into the gradient of a scalar and a divergenceless vector
    (6.3)   𝛣_𝑖 = ∂_𝑖𝛣 + 𝑩_𝑖,      ∂^𝑖𝑩_𝑖 = 0
In Fourier space, (6.3) becomes⁴ 𝛣_𝑖 = 𝑖𝑘̂𝛣 + 𝑩_𝑖, where 𝐤̂ ≡ 𝐤/∣𝐤∣. The vector was split into a piece that is paralleled to 𝐤 and a piece orthogonal to it. Any rank-2 symmetric tensor can be written
    (6.4)   𝐸_𝑖𝑗 = 𝐶𝛿_𝑖𝑗 + ∂_⟨𝑖∂_𝑗⟩𝐸 + ∂_(𝑖𝑬_𝑗) + 𝑬_𝑖𝑗,
where 𝐶𝛿_𝑖𝑗 + ∂_⟨𝑖∂_𝑗⟩𝐸 is scalar, ∂_⟨𝑖𝑬_𝑗⟩ is vector and 𝑬_𝑖𝑗 is tensor.
    (6.5)   ∂_⟨𝑖∂_𝑗⟩𝐸 ≡ (∂_𝑖∂_𝑗 - 1/3 𝛿_𝑖𝑗∇²)𝐸,
    (6.6)   ∂_(𝑖𝑬_𝑗) ≡ 1/2 (∂_𝑖𝑬_𝑗 + ∂_𝑗𝑬_𝑖).
The first term in (6,4) contains the trace of spatial perturbation, 𝐸^𝑖_𝑖 = 𝐶𝛿^𝑖_𝑖 = 3𝐶, while the remaining terms are traceless. As before the hatted quantities have vanishing divergence, ∂^𝑖𝑬_𝑖 = ∂^𝑖𝑬_𝑖𝑗 = 0, and so correspond to transverse vector and tensor perturbations. The tensor perturbations describe gravitational waves (see Section 6.5) and following Georgi's dictum⁵ they are denoted as 𝘩_𝑖𝑗 ≡ 2𝑬_𝑖𝑗. In Fourier space we write (6.5) as ∂_⟨𝑖∂_𝑗⟩𝐸 → -(𝑘̂_𝑖𝑘̂_𝑗 - 1/3 𝛿_𝑖𝑗)𝐸 ≡ -𝑘̂_⟨𝑖𝑘̂_𝑗⟩𝐸.
   The 10 degrees of freedom of metric was decomposed into 4+4+2 SVT degrees of freedom:
 • scalars: 𝛢, 𝛣, 𝐶, 𝐸      • vectors: 𝑩_𝑖, 𝑬_𝑖       • tensors: 𝑬_𝑖𝑗
What makes the SVT decomposition so powerful is the fact That the Einstein equation for scalars, vectors and tensors don't mix at linear order and can be treated separately. We will mostly be interested in scalar fluctuations and the associated density perturbations , and at linear order it is consistent to set vectors and tensors to zero. Vector perturbations aren't produced by inflation and even if they were, they would decay quickly with the expansion of the universe. Tensor perturbations are an important prediction of inflation and we will discuss them at various point.

          Gauge problem
The metric perturbations in (6.2) aren't uniquely defined, bit depends on our choice of coordinate or the gauge choice. Making a different choice of coordinate can change the values of the perturbation variables. It may even "fictitious perturbations." that can arise from an inconvenient choice of coordinates even if the background is perfectly homogeneous.
   Consider, for example, a homogeneous FRW spacetime and making the change of the spatial coordinates, 𝑥^𝑖 ↦ 𝑥̂^𝑖 = 𝑥^𝑖 + 𝜉^𝑖(𝜂, 𝐱). We assume that 𝜉^𝑖 is small, so that it can also be treated as a perturbation. Using 𝑑𝑥^𝑖 = 𝑑𝑥̂^𝑖 - ∂_𝜂𝜉^𝑖𝑑𝜂 - ∂_𝑘𝜉^𝑖𝑑𝑥̂^𝑘, the line element becomes
    (6.7)  𝑑𝑠² = 𝑎(𝜂)[-𝑑𝜂² + 𝛿_𝑖𝑗𝑑𝑥^𝑖𝑑𝑥^𝑗] = 𝑎(𝜂)[-𝑑𝜂² - 2𝜉_𝑖ʹ𝑑𝑥̂^𝑖𝑑𝜂 + (𝛿_𝑖𝑗 - 2∂_(𝑖𝜉_𝑗))𝑑𝑥̂^𝑖𝑑𝑥̂^𝑗],  
where we dropped the quadratic terms in 𝜉^𝑖 and defined 𝜉_𝑖ʹ ≡ ∂_𝜂𝜉_𝑖. We apparently introduced the metric perturbations 𝛣_𝑖 = -𝜉_𝑖ʹ and 𝐸̂_𝑖 = -𝜉_𝑖. These are just fictitious gauge modes that can be removed by going back to the old coordinates.
   As another example, consider a change in the time slicing, 𝜂 ↦ 𝜂̂ = 𝜂 + 𝜉⁰(𝜂, 𝐱). The homogeneous density of the universe then gets perturbed,
    (6.8)  𝜌(𝜂) = 𝜌[𝜂̂ - 𝜉⁰(𝜂, 𝐱)] = 𝜌̄(𝜂̂) - 𝜌̄ʹ𝜉⁰.  
A local change of the time coordinate can introduce a fictitious density perturbation 𝛿𝜌 = -𝜌̄ʹ𝜉⁰. Conversely, we can remove a real perturbation in the energy density by choosing hypersurface of constant time to coincide with that of constant energy density. We then have 𝛿𝜌 = 0 although there are real inhomogeneities. The fluctuations haven't disappear completely, but will manifest themselves as new perturbations in the metric.
   So when the perturbations don't change under a change of coordinates, we can identify true perturbations more conveniently.

          Coordinate transformations
Consider the following coordinate transformation
    (6.9)  𝑥^𝜇(𝑞) ↦  𝑥̂^𝜇(𝑞) ≡ 𝑥^𝜇(𝑞) + 𝜉^𝜇(𝑞),    where      𝜉⁰ ≡ 𝛵,   𝜉^𝑖 ≡ 𝐿^𝑖 = ∂^𝑖𝐿 + 𝑳^𝑖.
The old and new coordinates are evaluated as the same physical point. 𝜉^𝜇 can be treated as perturbation. 𝛵(𝜂, 𝐱) defines the hypersurfaces of constant time in the new coordinates, while 𝐿^𝑖 determines the spatial coordinates there. We can split the spatial shift 𝐿^𝑖 into a scalar 𝐿 and a divergenceless vector 𝑳^𝑖.
   Consider the transformation of a (Lorentz) scalar field 𝜙(𝜂, 𝐱) which may be the inflaton or the density of a fluid. Since we have only relabelled the coordinate, the value of the field at each point 𝑞 must be unchanged
    (6.10)  𝜙(𝑥^𝜇) = 𝜙̂(𝑥̂^𝜇).
In terms of the background value of the field and its perturbation, we get
    (6.11)  𝜙̄(𝜂) + 𝛿𝜙(𝑥^𝜇) = 𝜙̄(𝜂̂) + 𝛿𝜙̂(𝑥̂^𝜇) = 𝜙̄(𝜂 + 𝛵) + 𝛿𝜙̂(𝑥̂^𝜇)
Doing Taylor expanding on the right-hand side, we find   [Taylor expanding: 𝑓(𝛵 + 𝜂) = 𝑓(𝜂) + 𝛵𝑓ʹ(𝜂) + 1/2 𝛵²𝑓ʺ(𝜂) + ∙ ∙ ∙ ]
    (6.12-4)  𝜙̄(𝜂) + 𝛿𝜙(𝑥^𝜇) = 𝜙̄(𝜂) + 𝜙̄ʹ𝛵 + 𝛿𝜙̂(𝑥^𝜇).    ⇒    𝛿𝜙̂(𝑥^𝜇) = 𝛿𝜙(𝑥^𝜇) - 𝜙̄ʹ𝛵(𝑥^𝜇).    ⇒    𝛿𝜙̂ = 𝛿𝜙 - 𝜙̄ʹ𝛵.
Note that the both sides are evaluated at the same value 𝑥^𝜇 which in general do not correspond to the same physical point and the perturbations only depends on the choice of temporal gauge.
   To determine the transformation of the metric, we use the invariant spacetime interval:
    (6.15)  𝑑𝑠² = 𝑔_𝜇𝜈(𝑥)𝑑𝑥^𝜇𝑑𝑥^𝜈 = 𝑔̂_𝛼𝛽(𝑥̂)𝑑𝑥̂^𝛼𝑑𝑥̂^𝛽.
Writing 𝑑𝑥̂^𝛼 = (∂𝑥̂^𝛼/∂𝑥^𝜇)𝑑𝑥^𝜇 and similarly for 𝑑𝑥̂^𝛽, we find
    (6.16)  𝑔_𝜇𝜈(𝑥) = ∂𝑥̂^𝛼/∂𝑥^𝜇 ∂𝑥̂^𝛽/∂𝑥^𝜈 𝑔̂_𝛼𝛽(𝑥̂).
This relates the metric in the old coordinates 𝑔_𝜇𝜈 to the new 𝑔̂_𝛼𝛽. Given the transformation (6.9) we have in 2 × 2 matrix
    (6.17)  ∂𝑥̂^𝛼/∂𝑥^𝜇 = ⌈ ∂𝜂̂/∂𝜂   ∂𝜂̂/∂𝑥^𝑗 ⌉  =  ⌈ 1 + 𝛵ʹ       ∂_𝑖𝛵      ⌉ 
                                 ⌊ ∂𝑥̂^𝑖/∂𝜂  ∂𝑥̂^𝑖/∂𝑥^𝑗⌋      ⌊   𝐿^𝑖ʹ      𝛿^𝑖_𝑗 + ∂_𝑗𝐿^𝑖 ⌋   
    where 𝛼 labels the rows and 𝜇 the columns. In terms of the SVT decomposition, (6.16) implies the followings for the transformation of the metric perturbations in (6.2).
    (6.18-9)  𝛢 ↦ 𝛢˜ = 𝛢 - 𝛵ʹ - 𝓗𝛵,     𝛣 ↦ 𝛣˜ = 𝛣 + 𝛵 - 𝐿ʹ,   𝑩_𝑖 ↦ 𝑩_𝑖˜ = 𝑩_𝑖 - 𝑳_𝑖ʹ
    (6.20-1)  𝐶 ↦ 𝐶˜ = 𝐶 - 𝓗𝛵 - 1/3 ∇²𝐿,    𝐸 ↦ 𝐸˜ = 𝐸 - 𝐿,    𝑬_𝑖 ↦ 𝑬_𝑖˜ = 𝑬_𝑖 - 𝑳_𝑖,      𝑬_𝑖𝑗 ↦ 𝑬_𝑖𝑗.

Example   Consider 𝜇 = 𝜈 = 0 in (6.16):
    (6.22)  𝑔₀₀(𝑥) = ∂𝑥̂^𝛼/∂𝜂 ∂𝑥̂^𝛽/∂𝜂 𝑔̂_𝛼𝛽(𝑥̂).
   Consider 𝛼 = 0 and 𝛽 = 𝑖. The off-diagonal component of the metric 𝑔̂_0𝑖 is proportional to 𝑩_𝑖˜, so it's a first order perturbation. But ∂𝑥̂^𝑖/∂𝜂 is second order and can be neglected. A similar  argument holds for 𝛼 = 𝑖 and 𝛽 = 𝑗. So (6.22) reduces to
    (6.23)  𝑔₀₀(𝑥) = (∂𝜂̂/∂𝜂)² = 𝑔̂₀₀(𝑥̂).
Substituting (6.2), (6.9) and (6.17), we get
    (6.24-5)  𝑎²(𝜂)(1 + 2𝛢) = (1 + 𝛵ʹ)²𝑎²(𝜂 + 𝛵)(1 + 2𝛢˜) = (1 + 2𝛵ʹ + ∙ ∙ ∙ )(𝑎(𝜂) + 𝑎ʹ𝛵 + ∙ ∙ ∙ )²(1 + 2𝛢˜)
                           = 𝑎²(𝜂)(1 + 2𝓗𝛵 + 2𝛵ʹ + 2𝛢˜ + ∙ ∙ ∙ ).    ⇒     𝛢 ↦ 𝛢˜ = 𝛢 - 𝛵ʹ - 𝓗𝛵. ▮                      

Exercise 6.1   Show that the other metric components transform as
    (6.26)  𝛣_𝑖 ↦ 𝛣_𝑖˜ = 𝛣_𝑖 + ∂_𝑖𝛵 - 𝐿_𝑖ʹ,
    (6.27)  𝐸_𝑖𝑗 ↦ 𝐸_𝑖𝑗˜ = 𝐸_𝑖𝑗 - 𝓗𝛵𝛿_𝑖𝑗 - ∂_(𝑖𝐿_𝑗).
[Solution]   The transformation of the 0_𝑖-component of the metric is,   [RE (6.2) (6.9) (6.16) (6.17)]    (a)   𝑔_0𝑖(𝑥) = ∂𝑥̂^𝛼/∂𝜂 ∂𝑥̂^𝛽/∂𝑥^𝑖 𝑔̂_𝛼𝛽(𝑥̂) = ∂𝜂̂/∂𝜂 [∂𝜂̂/∂𝑥^𝑖 𝑔̂₀₀(𝑥̂) + ∂𝑥̂^𝑗/∂𝑥^𝑖 𝑔̂_0𝑗(𝑥̂)] + ∂𝑥̂^𝑗/∂𝜂 [∂𝜂̂/∂𝑥^𝑖 𝑔̂_𝑗0(𝑥̂) + ∂𝑥̂^𝑘/∂𝑥^𝑖 𝑔̂_𝑗𝑘(𝑥̂)]
              = (1 + 𝛵ʹ)[∂_𝑖𝛵𝑔̂₀₀(𝑥̂) + (𝛿^𝑗_𝑖 + ∂_𝑗𝐿^𝑖)𝑔̂_0𝑗(𝑥̂)] + 𝐿^𝑗ʹ[∂_𝑖𝛵𝑔̂_𝑗0(𝑥̂) + (𝛿^𝑘_𝑖 + ∂_𝑖𝐿^𝑘)𝑔̂_𝑗𝑘(𝑥̂)] = 𝑔̂_0𝑖(𝑥̂) + ∂_𝑖𝛵𝑔̂₀₀(𝑥̂) + 𝐿^𝑖ʹ𝑔̂_𝑖𝑗(𝑥̂) + 𝑂(2).
Substituting the perturbed metric components on both sides, we get   [𝑔_0𝑖(𝑥) = 𝛣_𝑖, 𝑔̂_0𝑖(𝑥̂) = 𝛣˜_𝑖, 𝑔̂₀₀ = -1, 𝑔̂_𝑖𝑗 = 1]
   (b)   𝛣_𝑖 = 𝛣˜_𝑖 - ∂_𝑖𝛵 + 𝐿^𝑖ʹ    ⇒     𝛣_𝑖 ↦ 𝛣_𝑖˜ = 𝛣_𝑖 + ∂_𝑖𝛵 - 𝐿_𝑖ʹ    ⇒     𝛣 ↦ 𝛣˜ = 𝛣 + 𝛵 - 𝐿ʹ,    𝑩_𝑖 ↦ 𝑩˜_𝑖 = 𝑩_𝑖 - 𝑳_𝑖. ▮
The transformation of 𝑖𝑗-component of metric is
   (c)   𝑔_𝑖𝑗(𝑥) = ∂𝑥̂^𝑘/∂𝑥^𝑖 ∂𝑥̂^𝑙/∂𝑥^𝑗 𝑔̂_𝑘𝑙(𝑥̂) = ∂𝑥̂^𝛼/∂𝑥^𝑖 ∂𝑥̂^𝛽/∂𝑥^𝑗 𝑔̂_𝛼𝛽(𝑥̂) + 𝑂(2) = (𝛿^𝑘_𝑖 + ∂_𝑖𝐿^𝑘)(𝛿^𝑖_𝑗 + ∂_𝑗𝐿^𝑙) 𝑔̂_𝑘𝑙(𝑥̂) = 𝑔̂_𝑖𝑗(𝑥̂) + ∂_𝑖𝐿^𝑘𝑔̂_𝑘𝑗(𝑥̂) + ∂_𝑗𝐿^𝑙𝑔̂_𝑖𝑙(𝑥̂). Substituting the perturbed metric component on both sides, we get   [RE (6.2)(6.9) (6.11-2)(c)]
   (d)   𝑎²(𝛿_𝑖𝑗 + 2𝐸_𝑖𝑗) = 𝑎²(𝜂 + 𝛵)(𝛿_𝑖𝑗 + 2𝐸˜_𝑖𝑗) + 𝑎(𝜂)(∂_𝑖𝐿^𝑘𝛿_𝑘𝑗 + ∂_𝑗𝐿^𝑙𝛿_𝑖𝑙) = 𝑎²(1 + 2𝑎ʹ/𝑎𝛵)(𝛿_𝑖𝑗 + 2𝐸˜_𝑖𝑗) + 2𝑎²∂_(𝑖𝐿_𝑗) =
                     = 𝑎²(𝛿_𝑖𝑗 + 2𝐸˜_𝑖𝑗) + 2𝑎²𝓗𝛵𝛿_𝑖𝑗 + 2𝑎²∂_(𝑖𝐿_𝑗) + 𝑂(2).    ⇒     𝐸_𝑖𝑗 ↦ 𝐸˜_𝑖𝑗 = 𝐸_𝑖𝑗 - 𝓗𝛵𝛿_𝑖𝑗 - ∂_(𝑖𝐿_𝑗). ▮  
In terms of the SVT decomposition of 𝐸_𝑖𝑗, this implies [RE (6.4-6)]
   (e)  𝐶 ↦ 𝐶˜ = 𝐶 - 𝓗𝛵 - 1/3 ∇²𝐿,     𝐸 ↦ 𝐸˜ =  𝛣 + 𝛵 - 𝐿ʹ,     𝑬_𝑖 ↦ 𝑬˜_𝑖 =  𝑬_𝑖 - 𝑳_𝑖,     𝑬_𝑖𝑗 ↦ 𝑬_𝑖𝑗.  ▮       

             Gauge-invariant variables
One way to avoid the gauge problem is define special combinations of the metric perturbation that do not transform under a change of coordinates. These are so-called Bardeen variables [1]
    (6.28)  𝛹 ≡ 𝛢 + 𝓗(𝛣 - 𝐸ʹ) + (𝛣 - 𝐸ʹ)ʹ,     𝜱_𝑖 ≡ 𝑩_𝑖 - 𝑬_𝑖ʹ,      𝑬_𝑖𝑗,      𝛷 ≡ -𝐶 + 1/3 ∇²𝐸 - 𝓗(𝛣 - 𝐸ʹ).
These gauge-invariant variables are the 'real' spacetime perturbations since they cannot be removed by a gauge transformation.

Exercise 6.2   Show that 𝛹, 𝛷 and 𝜱_𝑖 are gauge invariant. Explain why the tensor perturbation 𝐸̂_𝑖𝑗 is gauge invariant at linear order.
[Solution]   Under a coordinate transformation, scalar metric perturbation change    [RE (6.18-21)]
   (a)   𝛹 ≡ 𝛢 + 𝓗(𝛣 - 𝐸ʹ) + (𝛣 - 𝐸ʹ)ʹ  ↦ 𝛢 - 𝛵ʹ - 𝓗𝛵 + 𝓗[(𝛣 + 𝛵 - 𝐿ʹ) - (𝐸ʹ - 𝐿ʹ)] + (𝛣 + 𝛵 - 𝐸ʹ)ʹ = 𝛢 + 𝓗(𝛣 - 𝐸ʹ) + (𝛣 - 𝐸ʹ)ʹ = 𝛹.
   (b)   𝛷 ≡ -𝐶 + 1/3 ∇²𝐸 - 𝓗(𝛣 - 𝐸ʹ)  ↦ -𝐶 + 𝓗𝛵 + 1/3 ∇²𝐿 + 1/3 ∇²(𝐸 - 𝐿) - 𝓗(𝛣 + 𝛵 - 𝐸ʹ) = -𝐶 + 1/3 ∇²𝐸 - 𝓗(𝛣 - 𝐸ʹ) = 𝛷.
   (c)   𝜱_𝑖 ≡ 𝑩_𝑖 - 𝑬_𝑖ʹ  ↦  (𝑩_𝑖 - 𝑳_𝑖) - 𝑬_𝑖 - 𝑳_𝑖 = 𝑩_𝑖 - 𝑬_𝑖ʹ = 𝜱_𝑖.
Because the gauge transform only involves two scalar modes and one vector mode, tensor mode doen't transform at linear order. A tensor mode in the gauge transform would only arise at second order.  ▮

          Gauge fixing
An alternative solution to gauge problem is to fix the gauge and keep track of all perturbations in both the metric and the matter. A convenient choice of gauge often simplify the analysis. The following are popular gauge:

• Newtonian gauge   We can use the freedom in function 𝛵 and 𝐿 in (6.9) to set two of four scalar metric perturbations to zero. The Newtonian gauge is defined by the choice and so (6.2) becomes
    (6.29-30)  𝛣 = 𝐸 = 0,    ⇒     𝑑𝑠² = 𝑎(𝜂)[-(1 + 2𝛹)𝑑𝜂² + (1 - 2𝛷)𝛿_𝑖𝑗𝑑𝑥^𝑖𝑑𝑥^𝑗], where 𝛢 ≡ 𝛹 and 𝐶 ≡ -𝛷 to make contact with the Bardeen potential in (6.28). In this metric hypersurfaces of constant time are orthogonal to the worldlines of observers at rest (since 𝛣 = 0) and the induced geometry is isotropic (since 𝐸 = 0). Notice that the similarity to the usual weak-field limit of GR (Appendix A) with 𝛹 as the gravitational potential. Newtonian gauge will be our preferred one for structure formation.

• Spatially flat gauge   This is a convenient gauge for computing inflationary perturbations, by inflaton field 𝛿𝜙 (see Chapter 8),  with
    (6.31)  𝐶 = 𝐸 = 0, 

• Synchronous gauge   It is a historically important gauge introduced by Lifshitz in his pioneering work on perturbation [4] -, with
    (6.32)  𝛢 = 𝛣 = 0.
But this gauge still contain spurious gauge of freedom. These mode caused some confusion in the early history of cosmological perturbation theory and indeed were the motivation for Bardeen's gauge-invariant approach [1].

          6.1.2 Matter Perturbations 
We consider perturbations of the energy-momentum tensor.
    (6.33)  𝛵⁰₀ ≡ -(𝜌̄ + 𝛿𝜌),     𝛵⁰_𝑖 ≡ (𝜌̄ + 𝑃̄)𝑣^𝑖,     𝛵^𝑖_𝑗 ≡ (𝑃̄ + 𝛿𝑃)𝛿^𝑖_𝑗 + 𝛱^𝑖_𝑗,     𝛱^𝑖_𝑖 ≡ 0,
where 𝑣_𝑖 is the bulk velocity and 𝛱^𝑖_𝑖 is the anisotropic stress. Note that 𝛵⁰_𝑖 and 𝛵^𝑖₀ differ by sign. This is necessary in order for 𝛵_0𝑖 = 𝛵_𝑖0 to hold. We will write the momentum density as 𝑞_𝑖 ≡ (𝜌̄ + 𝑃̄)𝑣_𝑖. If there are several contributions to the energy-momentum tensor (e.g. photons,, baryon, dark matter etc.), they are added 𝛵_𝜇𝜈 = ∑_𝑎𝛵_𝜇𝜈 ^(𝑎), so that
    (6.34)  𝛿𝜌 = ∑_𝑎𝛿𝜌_𝑎,     𝛿𝑃 = ∑_𝑎𝛿𝑃_𝑎,     𝑞^𝑖 = ∑_𝑎𝑞^𝑖_(𝑎),     𝛱^𝑖𝑗 = ∑_𝑎^𝑖𝑗_(𝑎).
Note that only the velocities do not add, but momentum densities do.

Exercise 6.3   Given 𝑈̄^𝜇 = 𝑎^-1(1, 0, 0, 0) and 𝑔_𝜇𝜈𝑈^𝜇𝑈^𝜈 = -1, show that the perturbed 4-velocity must take the form
    (6.35-6)  𝑈^𝜇 = 𝑎^-1(1- 𝛢, 𝑣^𝑖),     𝑈_𝜇 = 𝑎^-1[-(1 + 𝛢), 𝑣_𝑖 + 𝛣_𝑖],
where 𝛢 and 𝛣_𝑖 are defined in (6.2) and 𝑣^𝑖 is the bulk velocity. Substituting these results into the perturbed energy-momentum tensor of a fluid,
    (6.37)  𝛿𝛵^𝜇_𝜈 = (𝛿𝜌 + 𝛿𝛲)𝑈̄^𝜇𝑈̄_𝜈 + (𝜌̄ + 𝑃̄)(𝛿𝑈̄^𝜇𝑈̄_𝜈 + 𝑈̄^𝜇𝛿𝑈̄_𝜈) + 𝛿𝛲𝛿^𝜇_𝜈 + 𝛱^𝜇_𝜈,
with 𝑈^𝜇𝛱_𝜇𝜈 = 0, show that this leads to the same result as in (6.33). Show also that 𝛵⁰_𝑖 = (𝜌̄ + 𝑃̄)(𝑣_𝑖 + 𝛣_𝑖).
[Solution]   From 𝑔_𝜇𝜈𝑈^𝜇𝑈^𝜈 = 𝑈_𝜈𝑈^𝜈 = -1, so we have 𝑈̄_𝜇 = 𝑎(-1, 0, 0, 0).
   (a)   -1 = 𝑔_𝜇𝜈𝑈^𝜇𝑈^𝜈 = 𝑔̄_𝜇𝜈𝑈̄^𝜇𝑈̄^𝜈 + 𝛿𝑔_𝜇𝜈𝑈̄^𝜇𝑈̄^𝜈 + 2𝑔̄_𝜇𝜈𝑈̄^𝜇𝛿𝑈^𝜈 + 𝑂(2) = -1 + 𝑎^-2𝛿𝑔₀₀ + 2𝑎^-1𝑔̄₀₀𝛿𝑈⁰   ⇒    𝛿𝑈⁰ = -1/2 𝑎^-1 𝛿𝑔₀₀/𝑔̄₀₀ = -𝑎^-1𝛢.
Since 𝛿𝑈^𝑖 is unconstrained, so we define 𝛿𝑈^𝑖 ≡ 𝑎^-1𝑣^𝑖. The perturbed 4-velocity is
   (b)   𝑈̄^𝜇 = 𝑎^-1(1 - 𝛢, 𝑣^𝑖).
   (c)   𝑈̄_𝜇 = 𝑔_𝜇𝜈𝑈^𝜈 = 𝑔̄_𝜇𝜈𝑈̄^𝜈 + 𝛿𝑔_𝜇𝜈𝑈̄^𝜈 + 𝑔̄_𝜇𝜈𝛿𝑈^𝜈 + 𝑂(2) = 𝑎^-1𝑔̄_𝜇0 + 𝑎^-1𝛿𝑔_𝜇0 - 𝑎^-1𝑔̄_𝜇0𝛢 + 𝑎^-1𝑔̄_𝜇𝑗𝑣^𝑗 = -𝑎𝛿_𝜇0 - 2𝑎𝛢𝛿_𝜇0 + 𝑎𝛢𝛿_𝜇0 + 𝑎𝛣_𝑖𝛿_𝜇𝑖 + 𝑎𝑣_𝑖𝛿_𝜇𝑖
   (d)   𝑈̄₀ = -𝑎(1 + 𝛢),     𝑈̄_𝑖 = 𝑎(𝛣_𝑖 + 𝑣_𝑖),     𝑈̄_𝜇 = 𝑎[-(1 + 𝛢), 𝑣_𝑖 + 𝛣_𝑖].
   (e)   𝛿𝛵⁰₀ = -(𝛿𝜌 + 𝛿𝛲) + (𝜌̄ + 𝑃̄)(𝛢 - 𝛢) + 𝛿𝛲 + 0 = -𝛿𝜌,     𝛿𝛵^𝑖₀ = 0 + (𝜌̄ + 𝑃̄)(𝛿𝑈̄^𝑖𝑈̄₀ + 0) + 0 + 0 = -(𝜌̄ + 𝑃̄)𝑣^𝑖,    
        𝛿𝛵⁰_𝑖 = 0 + (𝜌̄ + 𝑃̄)(0 + 𝑈̄⁰𝛿𝑈̄_𝑖) + 0 + 0 = (𝜌̄ + 𝑃̄)(𝑣_𝑖 + 𝛣_𝑖),     𝛿𝛵^𝑖_𝑗 = 0 + (𝜌̄ + 𝑃̄)(0 + 0) + 𝛿𝛲𝛿^𝑖_𝑗 + 𝛱^𝑖_𝑗 = 𝛿𝛲𝛿^𝑖_𝑗 + 𝛱^𝑖_𝑗,  ▮           

   As for the metric, we apply a SVT decomposition here: 𝛿𝜌 and 𝛿𝑃 have scalar part only, while 𝑣_𝑖 and 𝑞_𝑖 have scalar and vector parts,
    (6.38)  𝑣_𝑖 = ∂𝑖𝑣 + 𝐯_𝑖,     𝑞_𝑖 = ∂𝑖𝑞 + 𝐪_𝑖.
The scalar part of the velocity is sometimes written in terms of the velocity divergence, 𝜃 ≡ ∂𝑖𝑣^𝑖 = ∇²𝑣.^{[verification needed]} The anisotropic stress 𝛱^𝑖𝑗 has
    (6.39)  𝛱^𝑖𝑗 = ∂_⟨𝑖∂_𝑗⟩𝛱 + ∂_(𝑖𝜫_𝑗) + 𝜫^𝑖𝑗.
Sometimes it is useful to work with the rescaled anisotropic stress [2], 𝜎 ≡ 2/3 (𝜌̄ + 𝑃̄)^-1𝛱. Finally it also convenient to use the density contrast, 𝛿 ≡ 𝛿𝜌/𝜌. Perturbation theory applies when 𝛿 < 1.

          Coordinate transformations
Under a coordinate transformation, 𝑥^𝜇 ↦ 𝑥̂^𝜇, the tensor 𝛵^𝜇_𝜈 transform as
    (6.40)  𝛵^𝜇_𝜈(𝑥) = ∂𝑥^𝜇/∂𝑥̂^𝛼 ∂𝑥̂^𝛽/∂𝑥^𝜈 𝛵˜^𝛼_𝛽(𝑥̂).
The origin of the transformation law is explained in Section A.3.3. To evaluate it we require ∂𝑥^𝜇/∂𝑥̂^𝛼 which is the inverse matrix of ∂𝑥̂^𝛼/∂𝑥^𝜇 since
    (6.41)  ∂𝑥^𝜇/∂𝑥̂^𝛼 ∂𝑥̂^𝛼/∂𝑥^𝜈 = 𝛿^𝜇_𝜈.
For the transformation in (6.9) the matrix ∂𝑥̂^𝛼/∂𝑥^𝜇 was given by (6.17):
    (6.42)  ∂𝑥̂^𝛼/∂𝑥^𝜇 = ⌈ ∂𝜂̂/∂𝜂   ∂𝜂̂/∂𝑥^𝑗 ⌉  =  ⌈ 1 + 𝛵ʹ       ∂_𝑖𝛵      ⌉
                                 ⌊ ∂𝑥̂^𝑖/∂𝜂  ∂𝑥̂^𝑖/∂𝑥^𝑗⌋      ⌊   𝐿^𝑖ʹ      𝛿^𝑖_𝑗 + ∂_𝑗𝐿^𝑖 ⌋    
To determine the corresponding matrix, we make use of the fact that the inverse of a matrix of the form 𝟏 + 𝜀, where 𝟏 is the identity and 𝜀 is a small perturbation, is 𝟏 - 𝜀 to first order in 𝜀. It follows that
    (6.43)  ∂𝑥^𝜇/∂𝑥̂^𝛼 = ⌈ ∂𝜂/∂𝜂̂   ∂𝜂/∂𝑥̂^𝑗 ⌉  =  ⌈ 1 - 𝛵ʹ       -∂_𝑖𝛵     ⌉  
                                 ⌊ ∂𝑥^𝑖/∂𝜂  ∂𝑥^𝑖/∂𝑥̂^𝑗⌋      ⌊  -𝐿^𝑖ʹ      𝛿^𝑖_𝑗 - ∂_𝑗𝐿^𝑖 ⌋     
Substituting these result into (6.40), we can determine how the perturbation of the energy-momentum tensor transform.
    (6.44-48)    𝛿𝜌 ↦ 𝛿𝜌 - 𝜌̄𝛵,     𝛿𝑃 ↦ 𝛿𝑃 - 𝑃̄𝛵,      𝑞_𝑖 ↦ (𝜌̄ + 𝑃̄)𝐿^𝑖ʹ,      𝑣_𝑖 ↦ 𝑣_𝑖 + 𝐿^𝑖ʹ,      𝛱^𝑖𝑗 ↦ 𝛱_𝑖𝑗.
Note that the density and pressure perturbations transform in the same way as the scalar fields in (6.14), since 𝛿𝜌 and 𝛿𝑃 are both Lorentz scalars

Example   Applying (6.40) to the temporal component of the energy-momentum tensor, we get
    (6.49)   𝛵⁰₀(𝑥) = ∂𝜂/∂𝑥̂^𝛼 ∂𝑥̂^𝛽/∂𝜂 𝛵˜^𝛼_𝛽(𝑥̂) = ∂𝜂/∂𝜂̂ ∂𝜂̂/∂𝜂 𝛵˜⁰₀(𝑥̂) + 𝛰(2).
Substituting 𝛵⁰₀(𝑥) = -(𝜌̄ + 𝛿𝜌) on both sides, we get
    (6.50)   𝜌̄ + 𝛿𝜌 = (1 - 𝛵ʹ)(1 + 𝛵ʹ)[𝜌̄(𝜂 + 𝛵) + 𝛿𝜌̄(𝑥)] = 𝜌̄(𝜂) + 𝜌̄ʹ𝛵 + 𝛿𝜌̄(𝑥) + 𝛰(2).
Solving this for 𝛿𝜌̄ gives
    (6.51)   𝛿𝜌̄ = 𝛿𝜌 - 𝜌̄ʹ𝛵,
which agrees with the result in (6.44).

Exercise 6.4   Derive (6.45)-(6.48)
[Solution]   Applying (6.40) to the 𝑖𝑗-components, we get for the first term
   (a)   𝛵^𝑖_𝑗(𝑥) = ∂𝑥^𝑖/∂𝑥̂^𝛼 ∂𝑥̂^𝛽/∂𝑥^𝑗 𝛵˜^𝛼_𝛽(𝑥̂) = ∂𝑥^𝑖/∂𝑥̂^𝑘 ∂𝑥̂^𝑙/∂𝑥^𝑗 𝛵˜^𝑘_𝑙(𝑥̂) = (𝛿^𝑖_𝑘 - ∂_𝑘𝐿^𝑖)(𝛿^𝑘_𝑗 + ∂_𝑗𝐿^𝑘) 𝛵˜^𝑘_𝑙(𝑥̂),
Substituting 𝛵^𝑖_𝑗 = -(𝑃̄ + 𝛿𝑃̄)^𝑖_𝑗 on both sides, we get
   (b)   (𝑃̄ + 𝛿𝑃)𝛿^𝑖_𝑗 = (𝛿^𝑖_𝑘 - ∂_𝑘𝐿^𝑖)(𝛿^𝑘_𝑗 + ∂_𝑗𝐿^𝑘)[𝑃̄(𝜂 + 𝛵) + 𝛿𝑃̄]𝛿^𝑘_𝑙 = [𝑃̄(𝜂) +𝑃̄ʹ𝛵 + 𝛿𝑃̄]𝛿^𝑖_𝑗 + 𝛰(2).     ⇒
   (c)  𝛿𝑃 ↦ 𝛿𝑃 - 𝑃̄ʹ𝛵.
Applying (6.40) to the 𝑖𝑗-components, we get for the second term
   (d)    𝛵^𝑖_𝑗(𝑥) = 𝛱^𝑖_𝑗   ⇒   𝛱^𝑖_𝑗 = (𝛿^𝑖_𝑘 - ∂_𝑘𝐿^𝑖)(𝛿^𝑘_𝑗 + ∂_𝑗𝐿^𝑘) 𝛱̂^𝑘_𝑙 = 𝛱^𝑖_𝑗 + 𝛰(2),   𝛱^𝑖_𝑗 ↦ 𝛱^𝑖_𝑗.
Applying (6.40) to the 𝑖0-components, we get
   (e)   𝛵^𝑖₀(𝑥) = ∂𝑥^𝑖/∂𝑥̂^𝛼 ∂𝑥̂^𝛽/∂𝑥⁰ 𝛵˜^𝛼_𝛽(𝑥̂) = ∂𝑥^𝑖/∂𝜂̂ (∂𝜂̂/∂𝜂 𝛵˜⁰₀ + ∂𝑥̂^𝑘/∂𝜂 𝛵˜⁰_𝑘) + ∂𝑥^𝑖/∂𝑥̂^𝑗 (∂𝜂̂/∂𝜂 𝛵˜^𝑗₀ + ∂𝑥̂^𝑘/∂𝜂 𝛵˜^𝑗_𝑘)
               = (1 - 𝛵)[(1 + 𝛵)𝛵˜⁰₀ + 𝐿^𝑘ʹ 𝛵˜⁰_𝑘] + (𝛿^𝑖_𝑗 - ∂_𝑗𝐿^𝑖)[(1 + 𝛵)𝛵˜^𝑗₀ + 𝐿^𝑘ʹ𝛵˜^𝑗_𝑘] = -𝐿^𝑖ʹ𝛵˜⁰₀ + 𝛵˜^𝑖₀ + 𝐿^𝑗ʹ𝛵˜^𝑗_𝑗 + 𝛰(2)
                   = 𝐿^𝑖ʹ𝜌̄ + 𝛵˜^𝑖 ₀ + 𝐿^𝑖ʹ𝑃̄ + 𝛰(2) = 𝛵˜^𝑖 ₀ + (𝜌̄ + 𝑃̄)𝐿^𝑖ʹ
Since 𝛵˜^𝑖 ₀ = -𝑞̂^𝑖 and 𝑞^𝑖 = (𝜌̄ + 𝑃̄)𝑣^𝑖,
   (f)   𝑞^𝑖 ↦ 𝑞^𝑖 + (𝜌̄ + 𝑃̄)𝐿^𝑖ʹ,     𝑣^𝑖 ↦ 𝑣^𝑖 + 𝐿^𝑖ʹ.  ▮

          Gauge-invariant variables
We can define specific combinations of variables for which these transforms cancel. There are various gauge-invariant quantities that can be formed from the metric and matter variables.

•  One useful combination is
    (6.52)   𝜌̄∆ ≡ 𝛿𝜌 + 𝜌̄ʹ(𝑣 + 𝛣).
   The quantity ∆ is called the comoving density contrast because it reduces to the density contrast in comving guage, 𝑣 = 𝛣 = 0. In terms of the comoving density contrast the relativistic generalization of Poisson equation takes the same form as in the Newtonian approximation
    (6.53)   ∇²𝛷 = 4π𝐺𝑎² 𝜌̄∆,    with 𝛷 is the Bardeen potential defined in (6.28).

•  Two additional important gauge-invariant quantities are
    (6.54-5)   𝛇 = -𝐶 + 1/3 ∇²𝐸 + 𝓗 𝛿𝜌/𝜌̄ʹ,     𝓡 = -𝐶 + 1/3 ∇²𝐸 - 𝓗(𝑣 + 𝛣).
   These are called curvature perturbations because they reduce to the intrinsic curvature of spatial slices in uniform density and comoving gauge respectively.

Exercise 6.5   Consider the induced metric on surfaces of constant time
    (6.56)   𝛾_𝑖𝑗 = 𝑎²[(1 + 2𝐶)𝛿_𝑖𝑗 + 2𝐸_𝑖𝑗].
   Show that the intrinsic curvature on the spatial slice is
    (6.57)   𝑎²𝑅₍₃₎[𝛾_𝑖𝑗] = -4∇²𝐶 + 2∂_𝑖∂_𝑗𝐸^𝑖𝑗 = 4∇²(-𝐶 + 1/3 ∇²𝐸).
  This explains why -𝐶 + 1/3 ∇²𝐸 is called the "curvature perturbation."
[Solution]   To get Ricci scalar of the spatial slice, 𝑅₍₃₎ we must find Christoffel symbol
   (a)   𝛤^𝑖_𝑗𝑘 = 1/2 𝛾^𝑖𝑙(∂_𝑗𝛾_𝑘𝑙 + ∂_𝑘𝛾_𝑗𝑙 - ∂_𝑙𝛾_𝑗𝑘), Since at zeroth order 𝛾^𝑖𝑗 = 𝑎^-2𝛿^𝑖𝑗 and we find the spatial derivatives
   (b)   𝛤^𝑖_𝑗𝑘 = 𝛿^𝑖𝑙∂_𝑗(𝐶𝛿_𝑘𝑙 + 𝐸_𝑘𝑙) + 𝛿^𝑖𝑙∂_𝑘(𝐶𝛿_𝑗𝑙 + 𝐸_𝑗𝑙) - 𝛿^𝑖𝑙∂_𝑙(𝐶𝛿_𝑗𝑘 + 𝐸_𝑗𝑘) = (2𝛿^𝑖_(𝑗∂_𝑘)𝐶 - 𝛿_𝑗𝑘∂^𝑖𝐶) + (2∂_(𝑗𝐸_𝑘))^𝑖 - ∂^𝑖𝐸_𝑗𝑘).
   (c)   𝑅₍₃₎ = 𝛾^𝑖𝑘𝑅_𝑖𝑘 = 𝛾^𝑖𝑘(∂_𝑙𝛤^𝑙_𝑖𝑘 - ∂_𝑘𝛤^𝑙_𝑖𝑙 + 𝛤^𝑙_𝑖𝑘𝛤^𝑚_𝑙𝑚 - 𝛤^𝑚_𝑖𝑙𝛤^𝑙_𝑘𝑚),  
This reduces to at first order
   (d)   𝑎²𝑅₍₃₎ = 𝛿^𝑖𝑘∂_𝑙𝛤^𝑙_𝑖𝑘 - 𝛿^𝑖𝑘∂_𝑘𝛤^𝑙_𝑖𝑙.
   (f)   𝛿^𝑖𝑘∂_𝑙𝛤^𝑙_𝑖𝑘 = 𝛿^𝑖𝑘∂_𝑙(2𝛿^𝑖_(𝑗∂_𝑘)𝐶 - 𝛿_𝑗𝑘∂^𝑖𝐶) + 𝛿^𝑖𝑘∂_𝑙(2∂_(𝑗𝐸_𝑘))^𝑖 - ∂^𝑖𝐸_𝑗𝑘) = -∇²𝐶 + 2∂_𝑙∂_𝑘𝐸^𝑘𝑙
    (g)   𝛿^𝑖𝑘∂_𝑘𝛤^𝑙_𝑖𝑙 = 3∇²𝐶
    (h)   𝑎²𝑅₍₃₎ = -∇²𝐶 + 2∂_𝑖∂_𝑗𝐸^𝑖𝑗 - 3∇²𝐶 = -4∇²𝐶 + 2∂_𝑖∂_𝑗𝐸^𝑖𝑗.
Since only scalar perturbations contribute to ∂^𝑖∂^𝑗𝐸^𝑖𝑗, [RE (6.4)] we get     (i)   ∂^𝑖∂^𝑗𝐸^𝑖𝑗 = ∂_𝑖∂_𝑗(∂^𝑖∂^𝑗𝐸 -1/3 𝛿^𝑖𝑗∇²𝐸) = 2/3 ∇⁴𝐸.
Hence, we get
    (j)   𝑎²𝑅₍₃₎ = 4∇²(-𝐶 + 1/3 ∇²𝐸).  ▮

   The three gauge-invariant perturbations ∆, 𝜁 and 𝘙 have the following relation
    (6.58)   𝛇 = 𝓡 - 𝓗/𝜌̄ʹ 𝜌̄∆.
We see from the Poisson equation (6.53) that the comoving densiy contrast vanishes on superhorizon scales, so that the two curvature perturbations becomes equal
    (6.59)   𝛇  →(𝑘 ≪ 𝓗)→   𝓡.
On large scales we can therefore treat 𝜁 and 𝚁 interchangably. Since the curvature perturbations are constant on superhorizon scales if the matter perturbations are "adiabatic" (see Section 6.2.2), 𝛇 or 𝓡 is nice to describe the initial conditions.

          More gauges
Above we used our gauge freedom to set the two of the metric perturbations to zero. Alternatively, we can also define the gauge in the matter sector:

• Uniform density gauge [Constant density gauge]   We can use the freedom in the time slicing to set the total density perturbation to zero.
    (6.60)   𝛿𝜌 = 0.   [in summary: 𝛿𝜌 = 𝛣 = 0.]
   The main scalar perturbation here is the curvature perturbation, 𝛿𝑔_𝑖𝑗 = 𝑎²(1 - 2𝛇)𝛿_𝑖𝑗.

• Comoving gauge   Similarly, we can ask for the scalar momentum density to vanish,  
    (6.61)   𝑞 = 0.   [in summary: 𝑣 = 𝛣 = 0.]
   The main scalar perturbation here is the curvature perturbation, 𝛿𝑔_𝑖𝑗 = 𝑎²(1 - 2𝓡)𝛿_𝑖𝑗. This is another convenient gauge for the computation of the inflationary quantum fluctuations [5].
There are different versions of uniform density and comoving gauge depending on which of the metric fluctuation is set to zero.

          6.1.3 Conservation Equations 
Our next task is to derive the evaluation for the perturbations The evolution of the metric is governed by the Einstein equation 𝐺_𝜇𝜈 = 8π𝐺 𝛵_𝜇𝜈, while the evolution of the matter perturbations follows from the conservation of the energy-momentum tensor, ∇^𝜇𝛵_𝜇𝜈 = 0. Expanding these equations to linear order in the perturbations will gives us the linearized equations of the motion for the matter and metric perturbations.
   It will be convenient to use the Newtonian gauge where metric is
    (6.62)   𝑔_𝜇𝜈 = diag[-𝑎²(1 + 2𝛹), 𝑎²(1 - 2𝛷)𝛿̇_𝑖𝑗].
We will derive the equation of motion for the matter perturbations by linearizing ∇^𝜇𝛵_𝜇𝜈 = 0. Note that if there is no energy and momentum tranasfer between the different components, then the species are separately conserved and we have ∇^𝜇𝛵^𝑎_𝜇𝜈 = 0 with all species separately. So we have
    (6.63)   ∇^𝜇𝛵_𝜇𝜈 = 0 = ∂_𝜇𝛵^𝜇_𝜈 + 𝛤^𝜇_𝜇𝛼𝛵^𝛼_𝜈 - 𝛤^𝛼_𝜇𝜈𝛵^𝜇_𝛼.
To evaluate this, we need the purturbated connection coefficients. Using the metric metric (6.62), it is a straightforward exercise to shoe that
    (6.64-66)   𝛤⁰₀₀ = 𝓗 + 𝛹ʹ,     𝛤⁰_𝑖0 = ∂_𝑖𝛹,     𝛤^𝑖₀₀ = ∂^𝑖𝛹,
    (6.67-69)   𝛤⁰_𝑖𝑗 = 𝓗𝛿̇_𝑖𝑗 - [𝛷ʹ + 2𝓗(𝛷 + 𝛹)]𝛿̇_𝑖𝑗,     𝛤^𝑖_𝑗0 = (𝓗 - 𝛷ʹ)𝛿̇^𝑖_𝑗,     𝛤^𝑖_𝑗𝑘 = 2𝛿̇^𝑖_(𝑗∂_𝑘)𝛷 + 𝛿̇_𝑗𝑘∂^𝑖𝛷.

Example   Recall the general expression expression for the connection coefficients
    (6.70)   𝛤^𝛼_𝛽𝛾 = 1/2 𝑔^𝛼𝜌(∂_𝛽𝑔_𝜌𝛾 + ∂_𝛾𝑔_𝛽𝜌 - ∂_𝜌𝑔_𝛽𝛾). To evaluate this, we need the inverse of the metric (6.62)
    (6.71)   𝑔^𝜇𝜈 = diag[-𝑎^-2(1 - 2𝛷), 𝑎^-2(1 + 2𝛹)𝛿̇_𝑖𝑗].
where we have used that to linear order (1 + 2𝛹)^-1 ≈ (1 - 2𝛹). Applying this to 𝛤⁰₀₀, we then have
    (6.72)   𝛤⁰₀₀ = 1/2 𝑔^0𝜌(∂₀𝑔_𝜌0 + ∂₀𝑔_0𝜌 - ∂_𝜌𝑔₀₀) = 1/2 𝑔⁰⁰(∂₀𝑔₀₀ + ∂₀𝑔₀₀ - ∂_𝜌𝑔₀₀) = 1/2 𝑔⁰⁰∂₀𝑔₀₀
                   = 1/2 𝑎^-2(1 - 2𝛹)∂₀[𝑎²(1 + 2𝛹)] = 𝑎^-2[𝑎𝑎ʹ(1 + 2𝛹) + 𝑎²𝛹ʹ] ≈ 𝓗 + 𝛹ʹ.   ▮

Exercise 6.6   Derive (6.65)-(6.69).

   Substituting the perturbed connection efficient into (6.63), we can derive the linearized evolution equations  for the matter perturbations. The algebra is a bit involved and the results are summarized below.

•Continuity equation  Setting 𝜈 = 0 in (6.63), we find
    (6.73)   𝛿𝜌̄ʹ = -3𝓗(𝛿𝜌 + 𝛿𝑃) - ∂_𝑖𝑞^𝑖 + 3𝛷ʹ(𝜌̄ + 𝑃̄),
which describe the evolution of the density perturbation. On the right-hand side the first term is the dilution due to the background expansion, the second term accounts for the local fluid flow, and the last term is a purely relativistic effect corresponding to the density changes caused by perturbations to the local expansion rate. This arises because we can think of (1 - 𝛷)𝑎 is the "local scale factor" in the spatial part of the metric in Newtonian gauge. The first and last terms are therefore related. Substituting 𝛿𝜌 = 𝜌̄𝛿 and 𝑞^𝑖 = (𝜌̄ + 𝑃̄)𝑣^𝑖 into (6.73), we get evolution equation for the density contrast
    (6.74)   𝛿ʹ = -(1 + 𝑃̄/𝜌̄)(𝛁 ⋅ 𝐯 - 3𝛷ʹ) - 3𝓗(𝛿𝑃/𝛿𝜌 - 𝑃̄/𝜌̄)𝛿 .
In the limit of 𝑃 ≪ 𝜌, we recover the Newtonian equation (5.138), written conformal time, 𝛿ʹ = -𝛁 ⋅ 𝐯 + 3𝛷ʹ, but with a relativistic correction coming from the perturbed expansion rate. This correction is small on sub-Hubble scales and we reproduce the Newtonian limit.

•Euler equation  Setting 𝜈 = 𝑖 in (6.63), we get
    (6.75)   𝑞_𝑖ʹ = -4𝓗𝑞_𝑖 - (𝜌̄ + 𝑃̄)∂_𝑖𝛿𝑃 - ∂^𝑗𝜫_𝑖𝑗,
which is the relativistic version of the Euler equation.The first term on the right-hand side enforces the expected scaling of the momentum density, 𝑞 ∝ 𝑎^-4. The remaining terms are force terms due to gravity, pressure and anisotropic stress. substituting 𝑞^𝑖 = (𝜌̄ + 𝑃̄)𝑣^𝑖 into (6.75), we get an evolution of bulk velocity
    (6.76)   𝑣_𝑖ʹ = -[𝓗 + 𝑃̄ʹ/(𝜌̄ + 𝑃̄)]𝑣_𝑖 - 1/(𝜌̄ + 𝑃̄) (∂_𝑖𝛿𝑃 + ∂^𝑗𝛱_𝑖𝑗) - ∂_𝑖𝛹,
which should be compared to the Newtonian equation (5.139).
Derivation Consider first the 𝜈 = 0 component of (6.63)   [RE (6.33)]
    (6.77)    ∂₀𝛵⁰₀ + ∂_𝑖𝛵^𝑖₀ + ^𝑂(2)𝛤^𝜇_𝜇𝑖𝛵^𝑖₀ - 𝛤⁰₀₀𝛵⁰₀ - ^𝑂(2)𝛤⁰_𝑖0𝛵^𝑖₀ - ^𝑂(2)𝛤^𝑖₀₀𝛵⁰_𝑖 - 𝛤^𝑖_𝑗0𝛵^𝑗_𝑖 = 0.
Substituting the perturbed connection coefficients gives
    (6.78)    -(𝜌̄ + 𝛿𝜌)ʹ - ∂_𝑖𝑞^𝑖 - (𝓗 + 𝛹ʹ + 3𝓗 - 3𝛷ʹ)(𝜌̄ + 𝛿𝜌) + (𝓗 + 𝛹ʹ)(𝜌̄ + 𝛿𝜌) - (𝓗 - 𝛷ʹ)𝛿^𝑖_𝑗(𝑃̄ + 𝛿𝑃)𝛿^𝑗_𝑖 = 0,      and hence
    (6.79)    𝜌̄ʹ + 𝛿𝜌ʹ + ∂_𝑖𝑞^𝑖 + 3𝓗(𝜌̄ + 𝛿𝜌) - 3𝜌̄𝛷ʹ + 3𝓗(𝑃̄ + 𝛿𝑃) - 3𝑃̄𝛷ʹ = 0. Writing the zeroth-order and first-order parts separately, we get
    (6.80-1)    𝜌̄ʹ = -3𝓗(𝜌̄ + 𝑃̄).     𝛿𝜌ʹ = -3𝓗(𝛿𝜌 + 𝛿𝑃) - ∂_𝑖𝑞^𝑖 + 3𝜌𝛷ʹ(𝛿𝜌 + 𝛿𝑃).
(6.80) is simply the conservation of energy in the homogeneous background. (6.81) is the continuity equation for the density perturbation 𝛿𝜌. next consider 𝜈 = 𝑖 component of (6.63)
    (6.82)    ∂₀𝛵⁰_𝑖 + ∂_𝑗𝛵^𝑗_𝑖 + 𝛤^𝜇_𝜇0𝛵⁰_𝑖 + 𝛤^𝜇_𝜇𝑗𝛵^𝑗_𝑖 - 𝛤⁰_0𝑖𝛵⁰₀ - 𝛤⁰_𝑗𝑖𝛵^𝑗₀ - 𝛤^𝑖_0𝑖𝛵⁰_𝑗 - 𝛤^𝑗_𝑘𝑖𝛵^𝑘_𝑗 = 0.
Substituting the perturbed energy-momentum tensor, with 𝛵⁰_𝑖 = +𝑞_𝑖 and 𝛵^𝑖₀ = -𝑞_𝑖, and the perturbed connection coefficients gives
    (6.83)    𝑞_𝑖ʹ + ∂_𝑗[(𝑃̄ + 𝛿𝑃)𝛿^𝑗_𝑖 + 𝛱^𝑗_𝑖] + 4𝓗𝑞_𝑖 + (∂_𝑗𝛹 - 3∂_𝑗𝛷)𝑃̄𝛿^𝑗_𝑖 + ∂_𝑖𝛹𝜌̄ + 𝓗𝛿_𝑗𝑖𝑞^𝑖 - 𝓗𝛿^𝑗_𝑖𝑞_𝑗 - (-2𝛿^𝑗_(𝑘∂_𝑖)𝛷 + 𝛿_𝑘𝑖∂^𝑗𝛷)𝑃̄𝛿^𝑗_(𝑘 = 0.     Cleaning this up, we find
    (6.84)    𝑞_𝑖ʹ = -4𝓗𝑞_𝑖 - (𝜌̄ + 𝑃̄)∂_𝑖𝛹 - ∂_𝑖𝛿𝑃 - ∂^𝑗𝛱_𝑖𝑗.  ▮

•Matter  Consider a non-relativistic fluid (i.e. matter), with 𝑃_𝑚 = 0 and 𝛱^𝑖𝑗_𝑚.
   The continuity and Euler equations then simplify considerably
    (6.85-6)    𝛿ʹ_𝑚 = -𝛁 ⋅ 𝐯_𝑚 + 3𝛷ʹ,     𝐯ʹ_𝑚 = -𝓗𝐯_𝑚 - 𝛁𝛹.
  Combining the time derivative of (6.85) with the divergence of (6.86), we find  
    (6.87)    𝛿ʺ_𝑚 + 𝓗𝛿ʹ_{𝑚← friction} = ∇²𝛹 _{← gravity} + 3(𝛷ʺ + 𝓗𝛷ʹ).
Except for the terms arising from the time dependence of potential, this is the same as the Newtonian equation for the matter perturbations. We will apply this to the clustering of dark matter perturbation.

•Radiation  For a relativistic perfect fluid (i.e. radiation with no viscosity), we have 𝑃_𝑟 = 1/3 𝜌_𝑟 and 𝛱^𝑖𝑗_𝑟 = 0. The continuity and Euler equations then become
    (6.88-9)    𝛿ʹ_𝑟 = -4/3 𝛁 ⋅ 𝐯_𝑟 + 4𝛷ʹ,      𝐯ʹ_𝑟 = -1/4 𝛁𝛿_𝑟 - 𝛁𝛹.
   Combining the time derivative of (6.88) with the divergence of (6.89), we get
     (6.90)    𝛿ʺ_𝑟 - 1/3 ∇²𝛿_𝑟 = 4/3 ∇²𝛹 + 4𝛷ʺ,  
   where ∇²𝛿_𝑟 is about pressure and ∇²𝛹 is about gravity. We see that radiation perturbations don't experience the Hubble friction, but fee an additional pressure-induced force. Below how this equation allows for sound waves in the primordial plasma will be shown.

          Sound waves and Summary 
For future reference we collect here the linearized continuity and Euler equations:
    (6.91)  𝛿ʹ = -(1 + 𝑃̄/𝜌̄)(∂_𝑖𝑣^𝑖 - 3𝛷ʹ) - 3𝓗(𝛿𝑃/𝛿𝜌 - 𝑃̄/𝜌̄)𝛿,
    (6.92)  𝑣_𝑖ʹ = -[𝓗 + 𝑃̄ʹ/(𝜌̄ + 𝑃̄)]𝑣_𝑖 - 1/(𝜌̄ + 𝑃̄) (∂_𝑖𝛿𝑃 + ∂^𝑗𝛱_𝑖𝑗) - ∂_𝑖𝛹.
We typically work with these equations in Fourier space. For scalar fluctuations, we obtain the equations for the individual Fourier modes by the following replacement:     (6.93)  ∂_𝑖 → 𝑖𝑘_𝑖,     𝑣_𝑖 →  𝑖𝑘̂_𝑖𝑣,     𝛱_𝑖𝑗 →  -(𝜌̄ + 𝑃̄)𝑘̂_⟨𝑖𝑘̂_𝑗⟩𝛱, where 𝑘̂_⟨𝑖𝑘̂_𝑗⟩ ≡ 𝑘̂_𝑖𝑘̂_𝑗 - 1/3 𝛿_𝑖𝑗. Note that we have absorbed factors of 𝑘 ≡ ∣𝐤∣ in 𝑣 and 𝛱-as well as rescaled the anisotropic stress by a factor of (𝜌̄ + 𝑃̄)-so that 𝑣(𝜂, 𝐤) and 𝛱(𝜂, 𝐤) are not quite the Fourier transforms of 𝑣(𝜂, 𝐱) and 𝛱(𝜂, 𝐱). Equations (6.91) and (6.92) then lead to
    (6.94)    𝛿ʹ = -(1 + 𝑃̄/𝜌̄)(𝑘𝑣^𝑖 - 3𝛷ʹ) - 3𝓗(𝛿𝑃/𝛿𝜌 - 𝑃̄/𝜌̄)𝛿,
    (6.95)    𝑣_𝑖ʹ = -[𝓗 + 𝑃̄ʹ/(𝜌̄ + 𝑃̄)]𝑣 - 1/(𝜌̄ + 𝑃̄) 𝑘𝛿𝑃 + 2/3 𝑘𝛱 - 𝑘𝛹.
   These equations apply separately for all components that aren't directly coupled to each other. For example, dark matter doesn't interact with other matter (except through gravity), so it satisfies its own conservation equations. In general, we will have separation equations for each species with perturbation (𝛿_𝑎, 𝑣_𝑎, 𝛿𝑃_𝑎, 𝛱_𝑎). The gravitational coupling between the different species is captured by the expansion rate 𝓗 and the metric potentials 𝛷 and 𝛹. We must make further simplifying assumptions and additional equations to describe evolution of the four perturbations (𝛿, 𝑣, 𝛿𝑃, 𝛱).

•  Sometime we will assume a perfect fluid. Such a fluid is characterized by strong interactions which keep the pressure isotropic, 𝛱 = 0. For a barotropic fluid, with 𝑃 = 𝑃(𝜌), we can write as 𝛿𝑃 = 𝑐_𝑠², where the sound speed is     
    (6.96)    𝑐_𝑠² ≡ 𝑑𝑃/𝑑𝜌 = 𝑃̄ʹ/𝜌̄ʹ.
   For general fluid, the pressure may not just depend on density and the sound speed defined as
    (6.97)    𝑐_𝑠² ≡ (∂𝑃/∂𝜌)_𝑆 = 𝑃̄ʹ/𝜌̄ʹ,
   where the subscript indicate that the derivative is taken at fixed entropy and the second equality assumes that the expansion of the back ground is adiabatic (i.e. there is no entropy production). The perturbations are called adiabatic if
    (6.98)    𝛿𝑃 = 𝑐_𝑠²𝛿𝜌 = 𝑃̄ʹ/𝜌̄ʹ 𝛿𝜌.
   Note that the perturbations in a barotropic fluid are necessarily adiabatic, but this is not the case. If the perturbations are adiabatic, then they are described by only independent variables, 𝛿 and 𝑣 in the continuity and Euler equations.

•  Decoupled or weakly interacting species (e.g. neutrinos) cannot be described by the above method, so we have to solve the Boltzmann equation for the evolution of the perturbed distribution function 𝑓 (see Appendix B).

•  Decoupled dark matter is collisionless and has a negligible velocity dispersion. It behave like a pressureless perfect fluid although it has no interactions and therefore is not the traditional one [6].

          6.1.4 Einstein Equations 
Wee see from (91) and (6.92) that the evolution of each matter species is influenced by the metric potential 𝛷 and 𝛹. To close the system of equations, we need equations for the evolution of the metric perturbations. These follows from the Einstein equation. Deriving the linearized Einstein equation in Newtonian gauge is conceptually straightforward, but a bit tedious. So we do the computation once by hand and we had better use Mathematica or alternative to perform algebra.
   We require the perturbation to the Einstein tensor, 𝐺_𝜇𝜈 ≡ 𝑅_𝜇𝜈 - 1/2 𝑅𝑔_𝜇𝜈, so we first need to calculate the perturbed Ricci tensor 𝑅_𝜇𝜈 and Ricci scalar 𝑅.

• Ricci tensor  Recall that the Ricci tensor can be expressed in terms of the connection coefficients as [RE (2.120)]
    (6.99)    𝑅_𝜇𝜈 =  ∂_𝜆𝛤^𝜆_𝜇𝜈 - ∂_𝜈𝛤^𝜆_𝜇𝜆 + 𝛤^𝜆_𝜆𝜌𝛤^𝜌_𝜇𝜈 - 𝛤^𝜌_𝜇𝜆𝛤^𝜆_𝜈𝜌.
   Substituting the perturbed connection coefficient (6.64)-(6.69),
    (6.100)   𝑅₀₀ =  -3𝓗ʹ + ∇²𝛹 + 3𝓗(𝛷ʹ + 𝛹ʹ) + 3𝛷ʺ,
    (6.101)   𝑅_0𝑖 = 2∂𝑖(𝛷ʹ + 𝓗𝛹),
    (6.102)   𝑅_𝑖𝑗 = [𝓗ʹ + 2𝓗² - 𝛷ʺ + ∇²𝛷 -2(𝓗ʹ + 2𝓗²)(𝛷 + 𝛹) - 𝓗𝛹' -5𝓗𝛷ʹ]𝛿_𝑖𝑗 + ∂_𝑖∂_𝑗(𝛷 - 𝛹).

Example   The temporal component of Ricci tensor is
    (6.103)   𝑅₀₀ =  ∂_𝜌𝛤^𝜌₀₀ - ∂₀𝛤^𝜌_0𝜌 + 𝛤^𝛼₀₀𝛤^𝜌_𝛼𝜌 - 𝛤^𝛼_0𝜌𝛤^𝜌_0𝛼.
   When 𝜌 = 0, 𝑅₀₀ = 0, so we get
    (6.104)   𝑅₀₀ =  ∂_𝑖𝛤^𝑖₀₀ - ∂₀𝛤^𝑖_0𝑖 + 𝛤^𝑖_𝑖𝜌𝛤^𝜌₀₀ - 𝛤^𝜌_0𝑖𝛤^𝑖_0𝜌 =  ∂_𝑖𝛤^𝑖₀₀ - ∂₀𝛤^𝑖_0𝑖 + 𝛤^𝑖_𝑖0𝛤⁰₀₀ + ^𝑂(2)𝛤^𝑖_𝑖𝑗𝛤^𝑗₀₀ - ^𝑂(2)𝛤⁰_0𝑖𝛤^𝑖₀₀ -𝛤^𝑗_0𝑖𝛤^𝑖_0𝑗.
                 = ∇²𝛹 - 3∂₀(𝓗 - 𝛷ʹ) + 3(𝓗 + 𝛹ʹ)(𝓗 - 𝛷ʹ) - (𝓗 - 𝛷ʹ)²𝛿^𝑗_𝑖𝛿^𝑖_𝑗 = -3𝓗ʹ + ∇²𝛹 + 3𝓗(𝛷ʹ + 𝛹ʹ) + 3𝛷ʺ.  ▮

Exercise 6.7   Derive (6.101) and (6.102). Verify the results using the Mathematica notebook provide on the book's website.

• Ricci scalar  We can compute according to 𝑅 = 𝑅^𝜇_𝜇 = 𝑔^𝜇𝜈𝑅_𝜇𝜈
    (6.105)   𝑅 = 𝑔⁰⁰𝑅₀₀ + ^𝑂(2)2𝑔^0𝑖𝑅_0𝑖 + 𝑔^𝑖𝑗𝑅_𝑖𝑗.
   Since, according to (6.71), 𝑔^𝑖𝑗 = diag[-𝑎^-2(1 - 2𝛹), 𝑎^-2(1 + 2𝛷)𝛿^𝑖𝑗]
    (6.106)   𝑎²𝑅 = -(1 - 2𝛹)𝑅₀₀ + (1 + 2𝛷)𝛿^𝑖𝑗𝑅_𝑖𝑗 = (1 - 2𝛹)[3𝓗ʹ - ∇²𝛹 - 3𝓗(𝛷ʹ + 𝛹ʹ) - 3𝛷ʺ] + 3(1 + 2𝛷)[𝓗ʹ + 2𝓗∇² - 𝛷ʺ + ∇²𝛷 - 2(𝓗ʹ + 2𝓗²)(𝛷 + 𝛹) - 𝓗𝛹ʹ - 5𝓗𝛷ʹ] + (1 + 2𝛷)∇²(𝛷 - 𝛹),     dropping nonlinear terms              𝑎²𝑅 = 6(𝓗ʹ + 𝓗²) - 2∇²𝛷 -12(𝓗ʹ + 𝓗²)𝛹 - 6𝛷ʺ - 6𝓗(𝛹ʹ + 3𝛷ʹ),
   where the first term is the homogeneous part discussed in Chapter 2 and the other part is the linear correction.

• Einstein equation  We are ready to compute the Einstein equation
    (6.107)   𝐺^𝜇_𝜈 = 8π𝐺𝛵^𝜇_𝜈.
   We chose to work with one index raised since it simplify the form of the energy-momentum tensor. When 𝜇 = 𝜈 = 0,
    (6.109)   𝐺⁰₀ = 𝑔^0𝜇𝐺_𝜇0 = 𝑔⁰⁰[𝑅₀₀ - 1/2 𝑔₀₀𝑅 = -𝑎^-2(1 - 2𝛹)𝑅₀₀ - 1/2 𝑅,
   where we used that 𝑔^0𝑖 = 0 in Newtonian gauge. Substituting  (6.100) and (6.107),
    (6.110)   𝛿𝐺⁰₀ = -𝑎^-2[∇²𝛷 - 3𝓗(𝛷ʹ + 𝓗𝛹)],
   Since 𝛵⁰₀ ≡ -(𝜌̄ + 𝛿𝜌) in (6.33),
    (6.111)   ∇²𝛷 - 3𝓗(𝛷ʹ + 𝓗𝛹) = 4π𝐺𝑎²𝛿𝜌,
   where 𝛿𝜌 ≡ ∑_𝑎𝛿𝜌_𝑎 is the total density perturbation. Equation (6.111) is the relativistic generalization of the Poisson equation. Inside the Hubble radius-i.e. for Fourier modes with 𝑘 ≫ 𝓗-we have ∣∇²𝛷∣ ≫ 3𝓗∣(𝛷ʹ + 𝓗𝛹)∣, so that (6.111) reduces to ∇²𝛷 ≈ 4π𝐺𝑎²𝛿𝜌, which is the Poisson equation in the Newtonian limit.The GR corrections in (6.111) start to become important on scales comparable to the Hubble radius.
   Next, we consider the purely spatial part of the Einstein equation,
     (6.112)    𝐺^𝑖_𝑗 = 𝑔^𝑖𝑘𝐺_𝑘𝑗 = 𝑔^𝑖𝑘[𝑅_𝑘𝑗 - 1/2 𝑔_𝑘𝑗𝑅] = 𝑎^-2(1 + 2𝛷)𝛿^𝑖𝑘𝑅_𝑘𝑗 - 1/2 𝛿^𝑖_𝑗𝑅,
   We will first focus on the tracefree part of 𝐺^𝑖_𝑗 which is sourced by the anisotropic stress 𝛱^𝑖_𝑗.  Using (6.102) this gives [RE (6.5)(6.9)(6.33)]
     (6.113)     ∂_⟨𝑖∂_𝑗⟩(𝛷 - 𝛹) = 8π𝐺𝑎²𝛱_𝑖𝑗.   ^{[verification needed]}
   With the replacement ∂_⟨𝑖∂_𝑗⟩ → -𝑘_𝑖𝑘_𝑗 and 𝛱_𝑖𝑗 = -(𝜌̄ + 𝑃̄)𝑘̂_𝑖𝑘̂_𝑗, [RE (6.39)] the corresponding equation in Fourier space reads
     (6.114)     𝑘²(𝛷 - 𝛹) = 8π𝐺𝑎²(𝜌̄ + 𝑃̄)𝛱,
where 𝛱 ≡ ∑_𝑎𝛱_𝑎. On large scales, dark matter and baryons can be described as perfect fulids with negligible anisotropic stress. Photons only start to develope an anisotropic stress component during the matter-dominated era when their energy density is subdominant. The only relevant source in (6.113) are free-streaming nutrinos. To describe the neutrino-induced anisotropic stress requires beyond the fluid approximation; see Appendix B. The effect is relatively small, so to the level of accuracy here it can be ignored. Then (6.113) implies 𝛷 ≈ 𝛹.
   The time-space part of the Einstein equation equation is obtained in the same way.
     (6.115)    𝐺⁰_𝑖 = 𝑔^0𝜇𝐺_𝜇𝑖 = 𝑔⁰⁰𝑅_0𝑖 = -𝑎^-2𝑅_0𝑖 = -2𝑎^-2∂_𝑖(𝛷ʹ + 𝓗𝛹).
   Comparing this to 𝛵⁰_𝑖 = ∂_𝑖𝑞 in (6.33),
     (6.116)    𝛷ʹ + 𝓗𝛹 = -4π𝐺𝑎²𝑞.
   With this, the Poisson equation (6.111) can be written as
     (6.117)    ∇²𝛷 = 4π𝐺𝑎²𝜌̄∆,  
   where the comoving density contrast ∆ was defined in (6.52). We see that in terms of the comoving density contrast the relativistic generalization of the Poisson equation takes the same form as in /Newtonian treatment, but is now valid on all scales. This will allow us to relate solutions for the gravitational potential directly to the solutions for the dominant source of matter perturbations.
   Finally, we look at the trace of space-space Einstein equation as follows
     (6.118)    𝛷ʺ + 1/3 ∇²(𝛹 - 𝛷) + (2𝓗ʹ + 𝓗²)𝛹 + 𝓗𝛹ʹ + 2𝓗𝛷ʹ =  4π𝐺𝑎²𝛿𝑃,
   where 𝛿𝑃 ≡ ∑_𝑎𝛿𝑃_𝑎. Assuming 𝛹 ≈ 𝛷, this reduces to
     (6.119)    𝛷ʺ + 3𝓗𝛷ʹ + (2𝓗ʹ + 𝓗²)𝛷 = 4π𝐺𝑎²𝛿𝑃,
   which becomes a closed equation for the evolution of 𝛷 if we write 𝛿𝑃 = 𝑐_𝑠²𝛿𝜌 and use the Poisson equation to relate 𝛿𝜌 to 𝛷.

Exercise 6.8   Derive (6.118).

          Summary
   The linearized Einstein equations are
     (6.120)    ∇²𝛷 - 3𝓗(𝛷ʹ + 𝓗𝛹) = 4π𝐺𝑎²𝛿𝜌,
     (6.121)    -(𝛷ʹ + 𝓗𝛹) = 4π𝐺𝑎²𝑞,
     (6.122)    ∂_⟨𝑖∂_𝑗⟩(𝛷 - 𝛹) = 8π𝐺𝑎²𝛱_𝑖𝑗,
     (6.123)    𝛷ʺ + 𝓗𝛹ʹ + 2𝓗𝛷ʹ + 1/3 ∇²(𝛹 - 𝛷) + (2𝓗ʹ + 𝓗²)𝛹 = 4π𝐺𝑎²𝛿𝑃.
In the absence of anisotropic stress, 𝛱 ≈ 0, (6.122) implies that the two metric potentials are equal, 𝛷 ≈ 𝛹, and (6.123) reduces to
     (6.124)    𝛷ʺ + 3𝓗𝛷ʹ + (2𝓗ʹ + 𝓗²)𝛷 = 4π𝐺𝑎²𝛿𝑃.
Equation (6.120) and (6.121) can be combined into ∇²𝛷 = 4π𝐺𝑎²𝜌̄∆, which has same form as Newtonian Poisson equation, but is now valid on all scale.
   Combing the Einstein equations with the conservation equations (6.91) and (6.92) leads to a closed system of equation (after specifying the equation of state and the sound speed of each fluid component). We will study the solutions of these equations for various situation.

          6.2 Initial Conditions 
At sufficiently early times, all scales of interest to current observations were outside of Hubble radius. On such superhorizon scales, the evolution of the perturbations becomes very simple, especially for adiabatic initial conditions.

          6.2.1 Superhorizon Limit 
Consider the superhorizon limit of the continuity equations (6.85) and (6.88) for matter and radiation:
     (6.125)    𝛿ʹ_𝑚 = 3𝛷ʹ,
     (6.126)    𝛿ʹ_𝑟 = 4𝛷ʹ,
where the fact that the velocity terms can be dropped follows from the superhorizon limit of the Euler equations (6.83) and (6.86). Integrating these equations for photons, neutrinos, baryons and cold dark matter gives
     (6.127)    𝛿_𝑟 = 4𝛷 + 𝐶_𝑟     →     𝛿_𝑟 = 4𝛷 + 𝐶_𝑟
                𝛿_𝜈 = 4𝛷 + 𝐶_𝜈     →     𝛿_𝜈 = 𝛿_𝑟 + 𝑆_𝜈
                𝛿_𝑐 = 3𝛷 + 𝐶_𝑐     →     𝛿_𝑐 = 3/4 𝛿_𝑟 + 𝑆_𝑐
                𝛿_𝑏 = 3𝛷 + 𝐶_𝑏     →     𝛿_𝑏 = 3/4 𝛿_𝑟 + 𝑆_𝑏
where 𝐶_𝑎's are integration constants. Note that these constants are function of the wavevecter 𝐤 in general. 𝑆_𝑎's (for 𝑎 = 𝜈, 𝑐, 𝑏) are called isocurvature modes.
   A preferred set of initial conditions are given by the adiabatic mode with 𝑆_𝜈 = 𝑆_𝑐 = 𝑆_𝑏 = 0 and 𝐶_𝑟 ≠ 0. In that case, the initial values of all perturbations are determined by a single degree of freedom. We will assume adiabatic initial conditions throughout the the rest of this book and these also seems to be preferred by observations (see Section 8.4.1).
   Next, we look at the Einstein equation (6.120):
     (6.128)    ∇²𝛷 - 3𝓗(𝛷ʹ + 𝓗𝛹) = 4π𝐺𝑎²(𝜌̄_𝛾𝛿_𝛾 + 𝜌̄_𝜈𝛿_𝜈 + 𝜌̄_𝑐𝛿_𝑐 + 𝜌̄_𝑏𝛿_𝑏),
where we ignored the anisotropic stress of the neutrinos, so that 𝛷 ≈ 𝛹. On large scale, we can drop ∇²𝛷 and at the early times the matter contributions are negligible,
     (6.129)    3𝓗(𝛷ʹ + 𝓗𝛹) = -4π𝐺𝑎²(𝜌̄_𝛾𝛿_𝛾 + 𝜌̄_𝜈𝛿_𝜈).
Specializing to adiabatic initial conditions, we have 𝛿_𝛾 = 𝛿_𝜈 and 4π𝐺𝑎²(𝜌̄_𝛾𝛿_𝛾 + 𝜌̄_𝜈𝛿_𝜈) = 3𝓗²/2. [RE (2.516)] Substituting 𝓗 = 1/𝜂, [In radiation-domonated era, 𝑎 ∝ 𝜂 according to (2.149), so  𝓗 = 𝑎ʹ/𝑎 = 𝜂ʹ/𝜂 = 1/𝜂.]
     (6.130)    𝜂𝛷ʹ + 𝛷 = -1/2 𝛿_𝛾.
Taking a time derivative, and using (6. 126) to replace to 𝛿ʹ_𝑟, this becomes
     (6.131)    𝜂𝛷ʺ + 4𝛷ʹ = 0.    [(𝜂𝛷ʹ)ʹ + 𝛷ʹ = -1/2 𝛿ʹ_𝛾  ⇒  𝛷ʹ + 𝜂𝛷ʺ + 𝛷ʹ = -1/2 4𝛷ʹ  ⇒ 𝜂𝛷ʺ + 4𝛷ʹ = 0.]
The two solutions to this equation are the growing mode 𝛷 = const and decaying mode 𝛷 ∝ 𝜂^-3. [𝛷 = const or 𝑐₁/𝜂³ + 𝑐₂.] Growing mode 𝛷 = 𝛷_𝑖 in (6.130) implies
     (6.132)    𝛿_𝛾(𝜂_𝑖, 𝐤) = -2𝛷_𝑖(𝐤).
We see then that the integration constant in (6.127) is fixed by the primordial value 𝐶_𝑟 = -6𝛷_𝑖. For adiabatic fluctuations in all components are then related as
     (6.133)    𝛿_𝑟 = 𝛿_𝜈 = 4/3 𝛿_𝑐 = 4/3 𝛿_𝑏 = - 2𝛷_𝑖   (superhorizon).

Gauge dependence   One important subtlety is the gauge dependence of the superhorizon dynamics. Considr the Poisson equation written in terms of the comoving density contrast
     (6.134)    ∆ = - 2/3 𝑘²/𝓗² 𝛷.
First, the comoving density contrast ∆ is much smaller than 𝛿 on superhorizon scales. Second, it evolves as ∆ ∝𝓗^-2 ∝ 𝜂². So on superhorizon scale, we don't have to worry about the qualitative difference in the two different gauges and we cannot measure the density contrast! On subhorizon scale we instead have for our observation
     (6.135)   (∆ - 𝛿)/∆ = - 3/4 𝓗(𝛷ʹ + 𝓗𝛷)/𝑘²𝛷 →(𝑘 ≫ 𝓗)→ 0,
so that there is no difference between ∆ and 𝛿 on small scales. This is a general feature: the gauge problem disappear for all measureable quantities on subhorizon scales.

   Had we taken the anisotropic stress due to the neutrinos into account, we would instead have found (see Problem 6.2)
     (6.136)   𝛿_𝛾(𝜂_𝑖, 𝐤) = -2𝛹_𝑖(𝐤) = -2𝛷_𝑖(𝐤)(1 + 2/5 𝑓_𝜈)^-1, where we have introduced the fractional neutrino density 𝑓_𝜈 ≡ 𝜌_𝜈/(𝜌_𝜈 + 𝜌_𝛾).

Exercise 6.9   Using the result from Chapter 3, show that
     (6.137)   𝑓_𝜈 ≈ 0.41,
   so that 𝛹_𝑖 ≈ 0.86 𝛷_𝑖.
[Solution]   Since 𝜌_𝜈 = 7/8 𝑁_eff (4/11)^4/3𝜌_𝛾 [RE (3.66)]
    (a)   𝑓_𝜈 ≡ 𝜌_𝜈/(𝜌_𝜈 + 𝜌_𝛾) = 7/8 𝑁_eff (4/11)^4/3/(7/8 𝑁_eff (4/11)^4/3 + 1) ≈ 0.41.
where 𝑁_eff = 3.046. [RE p.88] From (6.136)
    (b)   𝛹_𝑖 = 𝛷_𝑖(1 + 2/5)^-1 ≈ 0.86.  ▮

          6.2.2 Adiabatic Perturbations 
   Adiabatic perturbations can be created by starting with a homogeneous universe and performing a common, local shift in time of all background quantities, 𝜂 → 𝜂 + π(𝑡, 𝐱). After this time shift, some parts of the universe are "ahead" and others are "behind" in the evolution. As we will see in Chapter 8, quantum fluctuations during inflation produce precisely such a time shift in the background quantities. The induced pressure and density perturbations are
     (6.138)    { 𝛿𝑃_𝑎(𝑡, 𝐱) = 𝑃̄_𝑎ʹπ(𝑡, 𝐱),     𝛿𝜌_𝑎(𝑡, 𝐱) = 𝜌̄_𝑎ʹπ(𝑡, 𝐱)     ⇒     { 𝛿𝑃_𝑎/𝛿𝜌_𝑎 = 𝑃̄_𝑎ʹ/𝜌_𝑎ʹ,     𝛿𝜌_𝑎/𝛿𝜌_𝑏 = 𝜌̄_𝑎ʹ/𝜌̄_𝑏ʹ
where we used that π(𝑡, 𝐱) is the same for all species 𝑎. The first relation on the right implies
     (6.139)    𝛿𝑃_𝑎 = 𝑐²_𝑠,𝑎𝛿𝜌_𝑎,
so that the perturbations are indeed adiabatic in the sense defined  in (6.98). Referring to similar equation (2.106), using 𝜌̄_𝑎ʹ = -3𝓗(1 + 𝜔_𝑎)𝜌̄_𝑎, the the second relation on the right can be written as
     (6.140)    𝛿_𝑎/(1 + 𝜔_𝑎) = 𝛿_𝑏/(1 + 𝜔_𝑏)     for all species 𝑎 and 𝑏.
Perturbations in matter (𝜔_𝑚 = 0) and radiation (𝜔_𝑟 = 1/3) obey
     (6.141)    𝛿_𝑚 = 3/4 𝛿_𝑟,
which is the same relation as for adiabatic mode in (6.127). An important corollary is that 𝛿𝜌 ≡ ∑_𝑎𝜌̄_𝑎𝛿_𝑎 is dominated by the species that carries the dominant energy density, 𝜌̄_𝑎 since all of the 𝛿_𝑎's are comparable.
   Finally, we note that the time shift ʹπ(𝑡, 𝐱) has the interpretation of the Goldstone boson associated with spontaneous breaking of time translation symmetry. This symmetry breaking arises because of the time dependence of the cosmological background. The language of spontaneous symmetry breaking has recently become very popular for describing cosmological perturbations and an effective field theory was constructed in terms of the Goldstone mode [7].

Exercise 6.10   Show that the thermodynamic relation 𝛵𝑑𝑆/𝑉 = 𝑑𝑈 + 𝑃𝑑𝑉 implies
     (6.142)    𝛵𝑑𝑆/𝑉 = [𝜌_𝜈 - 3/4 𝑟_𝜈(𝜌 + 𝑃)](𝛿_𝜈 - 𝛿_𝛾) + ∑_{𝑎=𝑐,𝑏}[𝜌_𝑎 - 𝑟_𝑎(𝜌 + 𝑃)](𝛿_𝜈 - 3/4 𝛿_𝛾).
where 𝑟_𝑎 ≡ 𝑛_𝑎/𝑛 are the fractional number densities of the species. For adiabatic perturbations, this means that 𝑑𝑆 = 0.

          6.2.3 Isocurvature Perturbations^*
The complement of adiabatic perturbation are isocurvature perturbations. For two component 𝑎 and 𝑏, the (density) isocurvature perturbation is
     (6.143)   𝑆_𝑎𝑏 ≡ 𝛿_𝑎/(1 + 𝜔_𝑎) -  𝛿_𝑏/(1 + 𝜔_𝑏),
We see from (6.140) that a nonzero  value of this quantity measures the deviation from the adiabatic mode. Special cases of this isocurvature mode are given in (6.127). The "isocurvature" was introduced because these perturbations can be chosen in such a way that the curvature perturbation 𝜁 vanishes. In that case an overdensity in one species compensate for an underdensity in another, resulting in no net curvature perturbation.
   Let us illustrate the special case of the neutrino density isocurvature mode. From the definition of 𝜁 in (6.54) we have
     (6.144)   𝜁 = 𝛷 - 𝛿𝜌/3(𝜌 + 𝑃) = 𝛷 - (𝑓_𝛾𝛿_𝛾 + 𝑓_𝜈𝛿_𝜈 + 𝑓_𝑏𝛿_𝑏 + 𝑓_𝑐𝛿_𝑐)/(3 + 𝑓_𝛾 + 𝑓_𝜈),
where we used the continuity equation for 𝜌̄ʹ^{[citation needed]} and introduced 𝑓_𝑎 ≡ 𝜌_𝑎/𝜌. At early times, we can ignore byrons and dark matter, 𝑓_𝑏,𝑐 ≈ 0, so that 𝑓_𝛾 ≈ 1 - 𝑓_𝜈 and hence
     (6.145)   𝜁 = 𝛷 - 𝛿_𝛾/4 + 𝑓_𝜈(𝛿_𝛾 - 𝛿_𝜈)/4 = - 1/4 (𝐶_𝛾 + 𝑓_𝜈𝑆_𝜈),
where we used (6.127). The curvature perturbation vanishes for 𝑆_𝜈 = -𝐶_𝛾/𝑓_𝜈 and the relation between the density contrast is
     (6.146)   𝛿_𝛾 = 4𝛷 - 𝑓_𝜈𝑆_𝜈 = 𝛿_𝜈 - 𝑆_𝜈 = 4/3 𝛿_𝑐 = 4/3 𝛿_𝑏.
In Problem 6.2 we will show that
     (6.147)   𝑆_𝜈 = (15 + 4𝑓_𝜈)/(1 - 𝑓_𝜈) 𝛷_𝑖 ≈ 68.8 𝛷_𝑖, where the final equality is for 𝑓_𝜈 ≈ 0.41.
   There are many different types of isocurvature perturbation. The general pheonomenology of isocurvature perturvations is described in [8]. Because isocurvature perturbations have not been seen in CMB data, adiabatic initial conditions are preferred. They are also attractive on theoretical grounds, because adiabatic perturbations arise naturally in the simplest inflationary models (see Chapter 8) and are frozen on superhorizon scales (see Section 6.2.4). The latter property is important for the predictive  power of inflation as it allows us to be agnostic about the uncertain physics of reheating. But primordial isocurvature perturbations can be washed out by thermal equilibrium and their amplitude is strongly model-dependent and sensitive to the post-inflationary evolution. If all particle species are in thermal equilibrium after inflation and their local densities are determined by the temperature (with vanishing chemical potential) the the primordial perturbations are adiabatic. For instance, the neutrino density isocurvature mode discussed above could be due to spatial fluctuations in the chemical potential of neutrinos.
   Given the complexity of models with isocurvature fluctuations and the fact that observations don't seem to require them, we will assume adiabatic initial conditions for the rest of this book.

          6.2.4 Curvature Perturbations
Fig. 6.3 shows the superhorizon evolution of gravitational in the transition from radiation domination to matter domination. We see that the gravitational potential is a constant during the radiation and matter eras, but varies inbetween.

Exercise 6.11   Consider a universe dominated by a fluid with constant equation of state 𝜔 ≠ -1. Assuming adiabatic perturbations and vanishing anisotropic stress, show that the evolution of the gravitational potential obeys
     (6.148)  𝛷ʺ + 3(1 + 𝜔)𝓗𝛷ʹ - 𝜔∇²𝛷 = 0. 
[Solution]   Ignoring anisotropic stress, the linearlized Einstein equation is
   (a)   𝛷ʺ + 3𝓗𝛷ʹ + (2𝓗ʹ + 𝓗²)𝛷 = 4π𝐺𝑎²𝛿𝑃.    (b)   4π𝐺𝑎²𝛿𝑃 = 4π𝐺𝑎² 𝜔𝛿𝜌 = 𝜔[∇²𝛷 - 3𝓗(𝛷ʹ + 𝓗𝛷)]
   (c)   2𝓗ʹ + 𝓗² = -2[4π𝐺/3 (𝜌 +3𝜔𝜌)]𝑎² + (8π𝐺/3 𝜌)𝑎² = -3𝜔(8π𝐺/3 𝑎²) = -3𝜔𝓗²
   (d)    𝛷ʺ + 3(1 + 𝜔)𝓗𝛷ʹ - 𝜔∇²𝛷 = 0.  ▮ 
 
   This shows that the growing mode is a constant on superhorizon scales.
   It would be convenient to identify an alternative perturbation variable that stays constant on large scales even the equation of state changes. This is particularly important for the transition from inflation to the radiation-dominated universe. Since the equation of state during reheating is unknown. it is crucial that we can track a quantity whose evolution doesn't depend on these details. This will allow us to match the predictions made by inflation to the fluctuations in the primordial plasma after inflation. For adiabatic initial conditions, the conserved perturbations are the crucial that we can track a quantity whose devolution doesn't depend on details. This will allow us to match the predictions made by inflation to the fluctuations in the primordial plasma after inflation. For adiabatic initial conditions, the conserved perturbations are the curvature perturbations introduced in (6.54) and (6.55) in Newtonian gauge
     (6.149-50)  𝜁 = 𝛷 - 𝛿𝜌/3(𝜌̄ + 𝑃̄),     𝓡 = 𝛷 - 𝓗𝑣,  
where we use the continuity equation to replace 𝜌̄ʹ. To prove that 𝜁 is indeed conserved on super-Hubble scale we take the time derivative of (6.149):
     (6.151)  𝜁ʹ = 𝛷ʹ - 𝛿𝜌ʹ/3(𝜌̄ + 𝑃̄) + (𝜌̄ʹ + 𝑃̄ʹ)/3(𝜌̄ʹ + 𝑃̄ʹ)² 𝛿𝜌.
Using continuity equations   [RE (2.106)]
     (6.152)  𝜌̄ʹ = -3𝓗(𝜌̄ + 𝑃̄),    𝛿𝜌ʹ = -3𝓗(𝛿𝜌 + 𝛿𝑃) - ∂_𝑖𝑞^𝑖 + 3𝛷ʹ(𝜌̄ + 𝑃̄),     we get
     (6.153)  (𝜌̄ + 𝑃̄)𝜁ʹ/𝓗 = (𝛿𝑃 - 𝑃̄ʹ/𝜌̄ʹ 𝛿𝜌) + 1/3 ∂_𝑖𝑞^𝑖/𝓗.
For adiabatic perturbations, the term in brackets on the right-hand side vanishes, while ∂_𝑖𝑞^𝑖 = ∇²𝑞 is suppressed by a factor of 𝑘²/𝓗² on large scales, so that 𝜁ʹ/𝓗 →(𝑘 ≪ 𝓗)→ 𝑂(𝑘²/𝓗²) ≈ 0. We have therefore established the conservation of the curvature perturbation on superhorizon scales.

Exercise 6.12   Show that the comoving curvature perturbation satisfies
     (6.154)  (𝜌̄ + 𝑃̄)𝓡ʹ/𝓗 = (𝛿𝑃 - 𝑃̄ʹ/𝜌̄ʹ 𝛿𝜌) + 𝑃̄ʹ/𝜌̄ʹ ∇²𝛷/4π𝐺𝑎²,
   and is therefore also conserved on superhorizon scales. Of course, this had to be true since we showed in (6.59) that 𝓡 is equal to 𝜁 on large scales.

Exercise 6.13   Consider a universe dominated by a fluid with constant equation of state. Show that the superhorizon limit of the curvature perturbation is
     (6.155)    𝓡 →(𝑘 ≪ 𝓗)→ (5 + 3𝜔)/(3 + 3𝜔) 𝛷.
   Use this to show that the amplitude of super-Hubble modes of 𝛷 drops by a factor of 9/10 in the radiation-to-mater transition (as seen in Fig. 6.3).
[Solution]   Consider the curvature perturbation equation 𝜁 and a constant equation of state 𝜔 induce
   (a)   𝜁 = 𝛷 - 𝛿𝜌/3(𝜌 + 𝑃) →(𝑘 ≪ 𝓗)→ 𝜁 = 𝛷 - 𝛿/3(1 + 𝜔)
On superhorizon scales we have 𝛿 = -2𝛷 and hence
   (b)   𝓡 ≈ 𝜁 = 𝛷 + 2𝛷/3(1 + 𝜔) = (5 + 3𝜔)/(3 + 3𝜔) 𝛷.  
In the radiation and matter eras, 𝜔 = 1/3 and 0, respectively
   (c)   𝓡 = { 3/2 𝛷_RD,     5/3 𝛷_MD
   (d)   since 𝓡 = const, 3/2 𝛷_RD = 5/3 𝛷_MD     ⇒      𝛷_MD = 9/10 𝛷_RD.  ▮

          6.2.5 Primordial Power Spectrum
Using the result of Exercise 6.13, the superhorizon value of the gravitational potential during the radiation era is
     (6.156)    𝛷(𝜂, 𝐤) = 2/3 𝓡_𝑖(𝐤)     (superhorizon),
where 𝓡_𝑖 ≡ 𝓡(0, 𝐤) is the initial value of the conserved curvature perturbation. Ignoring the effect of neutrinos, so that 𝛹 ≈ 𝛷, and assuming adiabatic initial conditions, we then also have
     (6.156)    𝛿_𝑟 = 4/3 𝛿_𝑚 = -2𝛷 = -4/3 𝓡_𝑖     (superhorizon),
The initial amplitudes of all fluctuations are determined by 𝓡_𝑖. We take the power spectrum of the primordial curvature perturbation to be
     (6.157)    𝒫_𝓡(𝑘, 𝑡_𝑖) = 𝛢𝑘^𝑛,
where 𝛢 and 𝑛 are constants. In Chapter 8, we will show that inflation naturally predicts a nearly scale-invariant spectrum, with 𝑛 ≈ 1.

          6.3 Growth of Matter Perturbations
An important scale in structure formation is the horizon at matter-radian equality. A mode entering the horizon at the time of euality has wavenumber 𝑘_eq = 𝓗_eq. Mode with 𝑘 > 𝑘_eq enter horizon during the radiation era, while modes with 𝑘 < 𝑘_eq enter during the matter era. This leads to a scale -dependent evolution of matter perturbation. We will noe derive this evolution again connecting it more rigorously to the superhorizon limit of fluctuations.

          6.3.1 Evolution of the Potential
Let us begin with the evolution of the gravitational potential. We will neglect anisotropic stress and assume adiabatic perturbations. Analytic solutions for 𝛷 can be found when the modes are on superhorizon scales, or when either radiation or matter dominate the background (see Fig. 6.4).
   When the universe is dominated by a component with a constant equation of state, evolution equation is (see Exercise 6.11)
    (6.159)   𝛷ʺ + 3(1 + 𝜔)𝓗𝛷ʹ - 𝜔∇²𝛷 = 0. 
During the matter domination, we have 𝜔 ≈ 0  and (6.159) takes the same form on subhorizon scales as well as superhorizon scales. The solutions for the growing and decaying modes are
    (6.160)   𝛷(𝑎, 𝐤) = 𝐶₁(𝐤) + 𝐶₂(𝐤)𝑎^-5/2     (matter era),  
[Derivation   (2.149) 𝑎 ∝𝜂².   ⇒   𝓗 = 𝑎ʹ/𝑎 = (𝜂²)ʹ/𝜂² = 2/𝜂.   ⇒   𝛷ʺ + 6/𝜂 𝛷ʹ = 0.   ⇒    𝛷 = 𝐶₁ + 𝐶₂𝜂^-5 = 𝐶₁ + 𝐶₂𝑎^-5/2.  ▮]
where we used 𝑎 ∝ 𝜂² to write the decaying mode in terms of the scale factor. The amplitude 𝐶₁(𝐤) and 𝐶₁(𝐤) are determined by the initial conditions and the evolution during the radiation era. We see that growing modes of the gravitational potential is a constant on all scales during the matter era.
   During the radiation era, we have 𝜔 ≈ 1/3, we find Fourier mode now satisfies
    (6.161)   𝛷ʺ + 4/𝜂 𝛷ʹ + 1/3 𝑘²𝛷 = 0.   [RE (5.14) ∇² = -𝑘² in Fourier mode]
writing 𝛷 = 𝑢(𝜑)/𝜑, with 𝜑 = 𝑘𝜂/√3, we have
    (6.162)   𝑑²𝑢/𝑑𝜑² + 2/𝜑 𝑑𝑢/𝑑𝜑 + (1 - 2/𝜑²)𝑢 = 0.  
The solutions of this equation are the spherical Bessel functions (see Appendix D)
    (6.163)   𝑗₁(𝜑) = + sin(𝜑)/𝜑² - cos(𝜑)/𝜑 = + 𝜑/3 + O(𝜑³),
    (6.164)   𝑛₁(𝜑) = + cos(𝜑)/𝜑² - sin(𝜑)/𝜑 = - 𝜑/𝜑² + O(𝜑⁰),  
Since 𝑛₁(𝜑) blows up for small 𝜑 (early times), we reject that solution on the basis of initial conditions. The solution for the gravitational potential during the radiation era then is
    (6.165)   𝛷(𝜂, 𝐤) = 2𝓡_𝑖 [sin(𝜑) - 𝜑 cos(𝜑)]/𝜑³     (radiation era),
where the overall normalization was determined  by 𝛷(0, 𝐤) = 2/3 𝓡(0, 𝐤) ≡ 2/3 𝓡_𝑖(𝐤) [RE Exercise 6.13]. Notice that (6.165) is valid on all scales. Taking the limits 𝜑 ≪ 1 and 𝜑 ≫ 1 on superhorizon and subhorizon scales respectively. These are
    (6.166)   𝛷(𝜂, 𝐤) ≈ { 2/3 𝓡_𝑖   (superhorizon); -6𝓡_𝑖 cos(1/√3 𝑘𝜂)/(𝑘𝜂)²   (subhorizon).  
During the radiation era, subhorizon modes of 𝛷 therefore oscillate with a frequency 1/√3 𝑘 and have an amplitude that decays as 𝜂^-2 ∝ 𝑎^-2.
   Fig. 6.5 shows  numerical solutions for the evolution of the gravitational potential for 3 representative wavelengths. As predicted, the potential is constant when the modes are outside horizon. Two modes enter the horizon during radiation era and the amplitudes decrease as 𝑎^-2. The resulting amplitudes in the matter era are strongly suppressed. During the matter era the potential is a constant on all scales. The long wavelength mode is only suppressed by a factor of 9/10 coming from the radiation-to-matter transition.
   In Problem 6.1 we will be asked to derive an analytic solution for 𝛷 superhorizon evolution:
    (6.167)   𝛷(𝑎, 𝐤) = 2𝓡_𝑖/30𝑦³ [16√(1 + 𝑦) + 9𝑦³ + 2𝑦² - 8𝑦 - 16]     (superhorizon),
where 𝑦 ≡ 𝑎/𝑎_eq. It gives the same result: this solution reduces to a constant at both early times (𝑦 ≪ 1) and late times (𝑦 ≫ 1) and accounts for the 9/10 factor in the transition described above.

Exercise 6.14   Consider the evolution equation (6.124) for the gravitational potential. Show that during the dark energy-dominated era, this equation can be written as
    (6.168)   𝑑2𝛷/𝑑𝑎2 + 5/𝑎 𝑑𝛷/𝑑𝑎 + 3/𝑎2𝛷 = 0,
   where 𝑎(𝜂) = [1 + 𝐻𝛬(𝜂0 - 𝜂)]-1, with 𝐻𝛬 =const. Note that this equation is valid on all scales. Show that the growing mode solution decays as 𝛷 ∝ 𝑎-1.
[Solution]   From (6.124)
   (a)   𝛷ʺ + 3𝓗𝛷ʹ + (2𝓗ʹ + 𝓗2)𝛷 = 4π𝐺𝑎2𝛿𝑃.
In the dark energy-dominated era we have 𝛿𝑃 = 0 and 𝐻𝛬 = const. From the corollary above (2.164)
   (b)   𝓗 ≡ 𝑎ʹ/𝑎 = 𝐻𝑎 = 𝑎̇/𝑎2 =  𝑎𝐻𝛬,   𝑑𝜂 = 𝑑𝑡/𝑎(𝑡),
        (c)   ∫ 𝑡0𝑡 (𝑑𝑎/𝑑𝑡)/𝑎2 𝑑𝑡 = ∫ 1𝑎 𝑑𝑎/𝑎2 = ∫ 𝜂0𝜂 𝐻𝛬 𝑑𝜂    ⇒    [- 1/𝑎]1𝑎 = 𝐻𝛬(𝜂0 - 𝜂),    ⇒    𝑎(𝜂) = 1/[1 + 𝐻𝛬(𝜂0 - 𝜂)].  
   (d)   𝛷ʹ = 𝑑𝛷/𝑑𝜂 = 𝑑𝑎/𝑑𝜂 𝑑𝛷/𝑑𝑎 = 𝑎𝓗 𝑑𝛷/𝑑𝑎 = 𝑎2𝐻𝛬 𝑑𝛷/𝑑𝑎
   (e)   𝛷ʺ = 𝑎2𝐻𝛬 𝑑/𝑑𝑎 (𝑎2𝐻𝛬 𝑑𝛷/𝑑𝑎) = 𝑎4𝐻𝛬2 𝑑2𝛷/𝑑𝑎2 + 2𝑎3𝐻𝛬2 𝑑𝛷/𝑑𝑎.
   (f)   2𝓗ʹ + 𝓗2 = 2(𝑎𝐻𝛬)ʹ + (𝑎𝐻𝛬)2 = 2𝑎ʹ𝐻𝛬 + 𝑎2𝐻𝛬2 = 2𝑎2𝐻𝛬2 + 𝑎2𝐻𝛬2 = 3𝑎2𝐻𝛬2.     So we have,
   (g)   (𝑎4𝐻𝛬2 𝑑2𝛷/𝑑𝑎2 + 2𝑎3𝐻𝛬2 𝑑𝛷/𝑑𝑎) + 3𝑎𝐻𝛬 𝑎2𝐻𝛬 𝑑𝛷/𝑑𝑎 + 3𝑎2𝐻𝛬2 = 0.    ⇒    𝑑2𝛷/𝑑𝑎2 + 5/𝑎 𝑑𝛷/𝑑𝑎 + 3/𝑎2𝛷 = 0.   Considering the ansaz 𝛷 = 𝑎𝑥, we get
   (h)   𝑥(𝑥 - 1)𝑎𝑥 - 2 - 5/𝑎 𝑥𝑎𝑥 - 1 + 3/𝑎2 𝑎𝑥 = 0,    ⇒    𝑥2 + 4𝑥 + 3 = (𝑥 + 1)(𝑥 + 3) = 0.    ⇒    𝑥 = {-1, -3}
Hence the growing mode solution decays as 𝛷 ∝ 𝑎-1.  ▮

          6.3.2 Clustering of Dark Matter
Next, the growth of dark matter perturbation will be described. In the relativistic framework we will reproduce and extend the evolution of subhorizon perturbations of a pressureless fluid treated before.

          Matter era In matter era since matter is the dominant component, the Poisson equation reads
    (6.169)   ∇²𝛷 ≈ 4π𝐺𝑎²𝜌̄_𝑚∆_𝑚.
The solution for the comoving density contrast ∆_𝑚 can be obtained directly from our previous result for 𝛷. Using (6.160) and 2𝜌̄_𝑚 ∝𝑎^-1,     (6.170)   ∆_𝑚(𝑎, 𝐤) = 𝑘²𝛷/4π𝐺𝑎²𝜌̄_𝑚 = 𝐶˜₁(𝐤) + 𝐶˜₂(𝐤)𝑎^-3/2.     (matter era),   [RE (5.14) ∇² = -𝑘² in Fourier mode]
just as in the Newtonian treatment, bu now valid on all scales. The growing mode of ∆_𝑚 evolves as 𝑎 outside the horizon, while density contrast 𝛿_𝑚 stays constant. Inside the horizon, 𝛿_𝑚 ≈ ∆_𝑚 and 𝛿_𝑚 in both gauges evolve as the scale factor 𝑎.

          Dark energy era   
  Since dark energy has no density fluctuations, the Poisson equation is still of form (6.169). since 𝑎²𝜌̄_𝑚 ∝ 𝑎^-1, we have 𝛷 ∝ ∆_𝑚/𝑎. The Einstein equation (6.124) then implies
    (6.171)   ∂_𝜂²(∆_𝑚/𝑎) + 3𝓗∂_𝜂(∆_𝑚/𝑎) + (2𝓗ʹ + 𝓗²(∆_𝑚/𝑎) = 0,     ⇒
    (6.172)   ∆_𝑚ʺ + 𝓗∆_𝑚ʹ + (𝓗ʹ - 𝓗²)∆_𝑚 = 0
Combing the Friedmann equations (2.156) and (2.157) for  a universe with matter and dark energy gives
    (6.173)   2(𝓗ʹ - 𝓗²) = (𝓗ʹ + 𝓗²) - 3𝓗² = -8π𝐺𝑎²𝑃̄ - 8π𝐺𝑎²𝜌̄ = +8π𝐺𝑎²𝜌̄_𝛬 - 8π𝐺𝑎²(𝜌̄_𝑚 + 𝜌̄_𝛬) = -8π𝐺𝑎²𝜌̄_𝑚,  
(6.172) becomes
    (6.174)   ∆_𝑚ʺ + 𝓗∆_𝑚ʹ - 4π𝐺𝑎²𝜌̄_𝑚∆_𝑚 = 0.
This is similar to the Newtonian equation (5.62), but is now valid on all scales. In the dark energy-dominated regime, we 𝓗² ≫ 4π𝐺𝑎²𝜌̄_𝑚 and we can drop the last term in (6.174) to get
    (6.175)   ∆_𝑚ʺ - 1/𝜂 ∆_𝑚ʹ ≈ 0,  
which has the final equation
    (6.176)   ∆_𝑚(𝑎, 𝐤) = 𝑘²𝛷/4π𝐺𝑎²𝜌̄_𝑚 = 𝐶˜₁(𝐤) + 𝐶˜₂(𝐤)𝑎^-2.     (dark energy era),   [Derivation   (2.149) 𝑎 ∝ -1/𝜂.   ⇒   𝓗 = 𝑎ʹ/𝑎 = (- 1/𝜂)ʹ/(- 1/𝜂) = - 1/𝜂.   ⇒   ∆_𝑚ʺ - 1/𝜂 ∆_𝑚ʹ = 0.   ⇒   ∆_𝑚 = 𝐶₁
+ 𝐶₂𝜂² = 𝐶₁ + 𝐶₂𝑎^-2.  ▮]
where we used -𝜂 ∝𝑎^-1. [correction: 𝜂 → -𝜂]. This recovers the suppression od the growth of structure that we found in the previous chapter, but it now hold on all scales.

Exercise 6.15   Show that (6.176) also follows directly from the solution to (6.168).
[Solution]   In Exercise 6.14, we showed that 𝛷 ∝ 𝑎^-1, 𝑎^-3. Since 𝛷 ∝ ∆_𝑚/𝑎, (u)∆_𝑚 ∝ 𝑎⁰ = Const, 𝑎^-2(/u).  ▮

          Radiation era
During the radiation era matter is a subdominant and we must work with continuity and Euler equations. In Section 6.1.3 we showed that this implies   [RE (6.87)]
    (6.177)   𝛿ʺ_𝑚 + 𝓗𝛿ʹ_𝑚 = ∇²𝛷 + 3(𝛷ʺ + 𝓗𝛷ʹ),
where 𝛷 = 𝛷_𝑟 + 𝛷_m is sourced by both radiation and matter. The contribution from 𝛷_𝑟 is rapidly oscillating on subhorizon scales, while 𝛷_m is a constant. The solution 𝛿_𝑚 inherits a "fast mode" by 𝛷_𝑟 and "slow mode" by 𝛷_m. It turns out that the fast mode is suppressed by a factor of (𝓗/𝑘)² relative to the slow mode [Weinberg]. This reflects the fact that the matter can't react to the fast change in the gravitational potential and effectively only evolves in response to the time-averaged potential. As a result 𝛿_𝑚 is sourced by even deep in the radiation era. Using ∇²𝛷 ≈ ∇²𝛷_𝑚 = 4π𝐺𝑎²𝜌̄_𝑚𝛿_𝑚 and 𝛷ʺ = 𝛷ʹ ≈ 0, we get
    (6.178)   𝛿ʺ_𝑚 + 𝓗𝛿ʹ_𝑚 - 4π𝐺𝑎²𝜌̄_𝑚𝛿_𝑚 ≈ 0,
where 4π𝐺𝑎²𝜌̄_𝑚 = 3/2 𝛺_m𝓗².^{[verification needed]} The conformal Hubble parameter for a universe with matter and radiation is
    (6.179)   𝓗/𝓗₀ = 𝛺_m/√𝛺_𝑟 √(1 + 𝑦)/𝑦,     𝑦 ≡ 𝑎/𝑎_eq,
where 𝑎_eq = 𝛺_𝑟/𝛺_m is the scale factor at matter-radiation equality. Using 𝑦 as the time variable, (6.178) becomes so-called Mészáros equation
    (6.180)   𝑑²𝛿_𝑚/𝑑𝑦² + (2 + 3𝑦)/2𝑦(1 + 𝑦) 𝑑𝛿_𝑚/𝑑𝑦 - 3/2𝑦(1 + 𝑦) 𝛿_𝑚 = 0,     whose solutions are
    (6.181)   𝛿_𝑚 ∝ { 1 + 3/2 𝑦;     (1 + 3/2 𝑦) ln[{√(1 + 𝑦) + 1}/{√(1 + 𝑦) - 1}] - 3√(1 + 𝑦).
In the limit 𝑦 ≪ 1 (RD) the growing mode solution is 𝛿_𝑚 ∝ ln 𝑦 ∝ ln 𝑎 i.e. grow logarithmically. In the limit 𝑦 ≫ 1 (MD) the solution is 𝛿_𝑚 ∝ 𝑦 ∝ 𝑎 significantly.
   Fig. 6.6 shows numerical solutions for the dark matter density contrast evolution for 3 Fourier modes. Notice that the 𝑘 > 𝑘_eq receives a small boost at horizon crossing before setting into the slow logarithmic growth.

          6.3.3 Matter Power Spectrum
We explain the shape of power spectrum for clustering of matter fluctuation. At a time 𝜂_𝑖 when all modes were still outside the Hubble radius, we take the initial power spectrum to be scale invariant, so that 𝑘³𝒫_𝑚(𝜂_𝑖, 𝑘) = const. We would like to see how the scale dependence of the spectrum evolves with time. (See also Section 5.2.3)
   Consider the time 𝜂_eq. Modes with 𝑘 < 𝑘_eq are outside the horizon and the spectrum must have the same initial spectrum (see Fig. 6.7). Modes with 𝑘 > 𝑘_eq evolved as ln 𝑎 inside the horizon in the radiation era. Their amplitude is enhanced by a factor of ln(𝑎_eq/𝑎_𝑘), where 𝑎_𝑘 is the moment of horizon crossing of mode 𝑘. Since 𝑘 = (𝑎𝐻)_𝑘 ∝ 𝑎_𝑘^-1 during the radiation era, this gives the (ln 𝑘)² scaling of the spectrum for 𝑘 > 𝑘_eq.
   Next, we take a time 𝜂₀ of today. All subhorizon modes grows as 𝑎. For 𝑘 < 𝑘_eq we have 𝑘³𝒫_𝑚 ∝ (𝑎_eq/𝑎_𝑘)² ∝ 𝑘⁴, where we used 𝑘 = (𝑎𝐻)_𝑘 ∝ 𝑎_𝑘^-1/2 in matter era. For 𝑘 < 𝑘_eq we instead get 𝑘³𝒫_𝑚 ∝ (𝑎₀/𝑎_eq)²ln(𝑎_eq/𝑎_𝑘)² ∝(ln 𝑘)² as before. Combining these results, we have
    (6.182)   𝒫_𝑚(𝜂₀, 𝑘) = { 𝑘^-3   𝑘 < 𝑘₀;     𝑘   𝑘₀ < 𝑘 < 𝑘_eq;     𝑘^-3(ln 𝑘)²   𝑘 > 𝑘_eq.
By definition the regime 𝑘 < 𝑘₀, with 𝑘₀ = 𝑎₀𝐻₀ ≈ 3 × 10^-4 𝘩 Mpc^-1, corresponds to superhorizon modes today. The scaling of the spectrum 𝑘 > 𝑘₀ on subhorizon mode is the same as that in Section 5.2.3.

          6.4 Evolution of Photons and Baryons
Fig. 6.8 shows the evolution of perturbations in dark matter, baryons and photons for a representative Fourier mode. Initially, the fluctuations in all components are of equal size (up to a factor of 4/3). Before decoupling , the baryons are tightly coupled to a photons and oscillate after horizon crossing. The dark matter experiences a slow logarithmic growth during the radiation era  a more rapid power law growth during the matter era. This explain why 𝛿_𝑐 ≫ 𝛿_𝑏 at decoupling. After decoupling, the baryons lose the pressure support of photons and fall into the gravitational potential wells  created by the dark matter. The density contrasts of dark matter and baryons eventually become equal again (see Exercise 5.2). It is crucial that the clustering of dark matter had a head start and then aided the growth of the baryon perturbations. Without the dark matter-assisted growth, the baryon wouldn't be able to cluster fast enough to explain the formation of galaxies.
   We will study the evolution of photons and baryons prior to decoupling. During that time, the photons and baryons acted as a single fluid with pressure provided by photons and extra mass density by the baryons. To build up intuition we will start by ignoring the baryons and study the photon perturbations alone.⁹ After that, we will how this dynamics is modified by the presence of baryons.

          6.4.1 Radiation Fluctuations
Consider the evolution of perturbation in the relativistic perfect fluid during the radiation era. Because the radiation component dominates we use the same trick and consider the Poisson equation, which now reads
    (6.183)  ∇²𝛷 = 3/2 𝓗²∆_𝑟,
In Fourier space, we have
    (6.184)  ∆_𝑟(𝜂, 𝐤) = -2/3 (𝑘𝜂)²𝛷(𝜂, 𝐤),
and using (6.165) we find
    (6.185)  ∆_𝑟(𝜂, 𝐤) = -4𝓡_𝑖 [sin(𝜑) - 𝜑 cos(𝜑)]/𝜑     (radiation era),
where 𝜑 ≡ 𝑘𝜂/√3. This solution is again valid on all scales. Taking the limits 𝜑 ≪ 1 and 𝜑 ≫ 1 gives the solutions as follows
    (6.186)  ∆_𝑟(𝜂, 𝐤) ≈ { -4/9 𝓡_𝑖(𝑘𝜂)²     (superhorizon);     4𝓡_𝑖 cos(1/√3 𝑘𝜂)     (subhorizon). We see that the comoving density contrast grows as ∆_𝑟 ∝ 𝜂² ∝ 𝑎² on superhorizon scales, while it oscillates with a constant amplitude on subhorizon scales.

Exercise 6.16 Using
    (6.187)  𝛿_𝑟 = -2/3 (𝑘𝜂)²𝛷 - 2𝜂𝛷ʹ - 2𝛷,
   determine the solution for 𝛿_𝑟 from (6.165). What are the superhorizon and subhorizon limits of the solution?
[Solution] Defining 𝜑 ≡ 𝑘𝜂/√3
   (a)   𝛿_𝑟 = -2𝜑²𝛷 - 2𝜑 𝑑𝛷/𝑑𝜑 - 2𝛷,
we get
   (b)   𝛿_𝑟 = -4𝓡_𝑖 [(𝜑² - 2)𝜑 cos(𝜑) + 2(𝜑² -1) sin(𝜑) ]/𝜑³
   (c)   𝛿_𝑟   →   { 4𝓡_𝑖 cos(𝜑)    𝜑 ≫ 1   (subhorizon);      4/3 𝓡_𝑖     𝜑 ≪ 1   (superhorizon).  ▮

   Next, we consider the evolution in the matter era. Since the radiation fluctuations are subdominant, we must use the continuity and Euler equations. In Section 6.1.3 we showed that the radiation density contrast satisfies
    (6.188)  𝛿_𝑟ʺ - 1/3 ∇²𝛿_𝑟 = 4/3 ∇²𝛷 + 4𝛷ʺ     ⇐    𝛿_𝑟ʹ = -4/3 𝛁 ⋅ 𝐯_𝑟 + 4𝛷ʹ,      𝐯_𝑟ʹ = -1/4 𝛁𝛿_𝑟 - 𝛁𝛷,
Recall that during matter domination, the gravitational potential is a constant on all scales, so that
    (6.189)  𝛿_𝑟ʺ - 1/3 ∇²𝛿_𝑟 = 4/3 ∇²𝛷 = const.
This is the equation of a harmonic oscillator with a constant driving force. The subhorizon fluctuations in the radiation density oscillate with a constant amplitude around a shifted equilibrium point, -4𝛷₀(𝐤), where 𝛷₀(𝐤) is the 𝑘-dependent amplitude of the gravitational potential in the matter era. This 𝑘-dependence arises from the initial conditions and nontrivial transfer function of the dark matter fluctuations. The solution for the density contrast is
    (6.190)  𝛿_𝑟(𝜂, 𝐤) = 𝐶(𝐤) cos(𝜑) + 𝐷(𝐤) sin(𝜑) - 4𝛷₀(𝐤),
where 𝜑 ≡ 𝑘𝜂/√3.

          6.4.2 Photon-Baryon Fluid
So far we have ignored the effects of baryons on the radiation fluid (except for their role in suppressing the photon aniso tropic stress). Let us fix this. We will work in the so-called tight-coupling approximation, where the coupling  between photons and baryons is so strong that they can be treated as a single photon-baryon fluid with velocity 𝐯_𝑏 = 𝐯_𝛾. Only the combined momentum density of the photons baryons is now conserved:
    (6.191)  𝐪 = (𝜌̄_𝛾 + 𝑃̄_𝛾)𝐯_𝛾 + (𝜌̄_𝑏 + 𝑃̄_𝑏)𝐯_𝑏 = 4/3 (1 + 𝑅)𝜌̄_𝛾𝐯_𝛾,     𝑅 ≡ 3/4 𝜌̄_𝑏/𝜌̄_𝛾 = 0.6 (𝛺_𝑏𝘩²/0.02)(𝑎/10^-3).
The fractional contribution from the baryons is characterized by the dimensionless parameter 𝑅. This parameter is small at early times, but grows linearly with 𝑎(𝑡) and becomes of order one around the time of recombination.
   Recall the Euler equation for the evolution of the momentum density
    (6.192)  𝐪ʹ + 4𝓗𝐪 = -(𝜌̄ + 𝑃̄)𝛁𝛹 - 𝛁𝛿𝑃,
where 𝛹 ≈ 𝛷 in the absence of anisotropic stress. Substituting (6.191) the left-hand side can be written as
    (6.193)  𝐪ʹ + 4𝓗𝐪 = ∂_𝜂(𝑎⁴𝐪)/𝑎⁴ = 4/3 𝜌̄_𝛾[(1 + 𝑅)𝐯_𝛾]ʹ,
where we have used that 𝑎⁴𝜌̄_𝛾 is a constant. Inserting 𝜌̄ + 𝑃̄ = 4/3 (1 + 𝑅)𝜌̄_𝛾 and 𝛿𝑃 = 1/3 𝜌̄_𝛾𝛿_𝛾 on the right-hand side, we get
    (6.194)  -(𝜌̄ + 𝑃̄)𝛁𝛹 - 𝛁𝛿𝑃 = -4/3 𝜌̄_𝛾[(1 + 𝑅)𝛁𝛹 + 1/4 𝛁𝛿_𝛾].
Combining the two equations then gives
    (6.195)  [(1 + 𝑅)𝐯_𝛾]ʹ _{← inertial mass} = -1/4 𝛁𝛿_𝛾 - (1 + 𝑅)𝛁𝛹 _{← gravitational mass}.
For 𝑅 = 0 this reduces to the Euler equation for the radiation fluid in (6.188). The correction ti this equation are: The coupling to the baryons adds extra "weight" to the photon-baryon fluid. This increases both the momentum density and the gravitational force term by a factor of (1 + 𝑅). The baryons, on the other hand, don't contribute to the pressure, so the pressure force does not receive this factor.
   Since the scattering of photons and baryons doesn't exchange energy, the continuity equation in (6.188) does not get modified and still reads
    (6.196)  𝛿_𝑟ʹ = -4/3 𝛁 ⋅ 𝐯_𝛾 + 4𝛷ʹ.
Combining the continuity and Euler equations we find
    (6.197)  𝛿_𝑟ʺ + 𝑅ʹ/(1 + 𝑅) 𝛿_𝑟ʹ - 1/3(1 + 𝑅) ∇²𝛿_{𝑟 ← pressure} = 4/3 ∇²𝛹 _{← gravity} + 4𝛷ʺ + 4𝑅ʹ/(1 + 𝑅) 𝛷ʹ.
which is the fundamental equation for the evolution of density perturbation in the photon-baryon fluid.
   From the pressure term in (6.197) we can read off the sound speed of the photon-baryon fluid
    (6.197)  𝑐_𝑠² ≡ 1/3(1 + 𝑅).
This is consistent with the definition of the definition of the adiabatic sound speed 𝑐_𝑠² = 𝛿𝑃/𝛿𝜌. To see this, we use 𝛿𝑃 = 𝛿𝑃_𝛾 = 1/3 𝛿𝜌_𝛾 and 𝛿𝜌 = 𝛿𝜌_𝛾 + 𝛿𝜌_𝑏, together with 𝛿𝜌_𝑏 = 3/4 (𝜌̄_𝑏/𝜌̄_𝛾)𝛿𝜌_𝛾 (for adiabatic perturbations).

          6.4.3 Cosmic Sound Waves
The dynamics associated with (6.197) is rather complex and will be discussed in detail in Chapter 7.

          Acoustic oscillations
For simplicity let us ignore the effects of gravity and consider solutions to the homogeneous equation:
    (6.199)  𝛿_𝑟ʺ + 𝑅ʹ/(1 + 𝑅) 𝛿_𝑟ʹ - 1/3(1 + 𝑅) ∇²𝛿_𝑟 = 0. There re two timescale: the expansion time and the oscillation time. On small scale, 𝑘 ≫ 𝓗/𝑐_𝑠, the latter is much shorter than the former. A WKB approximation can be used to the damped harmonic oscillator equation:
    (6.200)  𝛿_𝑟(𝜂, 𝐤) = 𝐶(𝐤) cos(𝑘𝑟_𝑠)/(1 + 𝑅)^1/4 + 𝐷(𝐤) sin(𝑘𝑟_𝑠)/(1 + 𝑅)^1/4,
where 𝑟_𝑠(𝜂) ≡ ∫^𝜂₀ 𝑐_𝑠(𝜂ʹ) 𝑑𝜂ʹ is the sound horizon at the time 𝜂. At the photon decoupling, 𝜂_*, the sound horizon 𝑟_𝑠(𝜂_*) ≈ 145 Mpc (or 500 million light-years). The functions 𝐶(𝐤) and 𝐷(𝐤) are fixed by superhorizon initial conditions. For the adiabatic initial conditions, the superhorizon modes are time independent, so the sine solution is not excited, 𝐷(𝐤) = 0. The amplitude of cosine solution 𝐶(𝐤) is determined by the value of the perturbation at horizon crossing. An important feature of solution is that all Fourier modes of a given wavelength reach the extrema at the same time in any direction. This phase coherence allows for the constructive interference of many waves, which leaves an imprint in cosmological observables.
   A numerical solution for the acoustic oscillations in the photon-baryon fluid is shown in Fig. 6.9. An important feature that is not captured by our simplified analysis is the damping of the fluctuations, which arises because the photons diffuse out of the regions high density into those of low density, washing out the density contrast. The so-called diffusion damping is especially relevant for small-scale fluctuations (high-𝑘 modes) and to describe it requires going beyond the tight-coupling approximation. A more accurate treatment must also include the time-dependent gravitational driving force that we ignored. In Chapter 7 we will treat the more subtle physics of the sound waves in the primordial plasma.

          CMB anisotropies
Looking at the CMB, we are seeing a snapshot of the primordial sound waves at the moment of sound decoupling. From (6.200) that the oscillation frequency depends on wavenumber 𝑘, meaning that different modes decouple at different phase and therefore different amplitudes at last-scattering (see Fig. 6.10). Evaluating the solution at decoupling, 𝜂 = 𝜂_*, gives the 𝑘-dependent amplitude of the fluctuations at last scattering. The amplitude has extreme at
    (6.201)  𝑘_𝑛 = 𝑛π/𝑟_𝑠(𝜂_*),       with 𝑛 = 1, 2, ∙ ∙ ∙ .
After projecting the Fourier modes onto the spherical surface of last-scattering, these becomes the peaks in the angular power spectrum of the CMB anisotropies. Note that the Power spectrum measures ∣𝛿_𝑟∣² so the maxima and minima in 𝛿_𝑟 both become peaks in the power spectrum, The positions of the peaks are determined by a combination of the sound horizon at decoupling and the angular diameter distance to the surface of last-scattering. The heights of the peaks carry information about the amount of dark matter and baryons in the universe. The size of the damping on small scales depends on the density of neutrinos. This is how the measurements of the CMB anisotropies by the WMAP, Planck satellites and other ground-based experiments allowed precise measurements of the cosmological parameters and revolutionized the field of cosmology. It will be described in much more detail in Chapter 7.

          Baryon acoustic oscillations^*
The same oscillations in the CMB anisotropies also leave an imprint in the clustering of galaxies. This arises because photons and baryons oscillates together before decoupling. The different Fourier mode of the baryons decouple at different phases in the evolution. Te initial conditions for gravitational growth of the barons include th oscillation of the photon-baryon fluid. This oscillatory feature gets transferred to the gravitational potential and the matter fluctuations, but because baryons are only about 15% of the total matter density, the effect is smaller than in the CMB. Since correlations in the galaxy distributions are inherited from the correlations in the matter density, these baryon acoustic oscillations (BAO) are imprinted in the galaxy power spectrum (see Fig. 6.11). In the following, a simple analytic derivation of the BAO signal will be given [12].
   The time when the baryons are released from the drag of the photons is known as the drag epoch. In fact the baryons decouple slightly later than the photons at 𝑧_𝑏 ≈ 1060 (compared to 𝑧_* ≈ 1090). After the drag epoch, the baryons behave as pressureless matter and we can write the total matter density contrast as
    (6.202)  𝛿_𝑚 = 𝑓_𝑏𝛿_𝑏 + (1 - 𝑓_𝑏)𝛿_𝑐,
where 𝑓_𝑏 ≡ 𝜌_𝑏/𝜌_𝑚 is the fractional baryon density. Near decopupling the solution for the totalmatter density is given by (6.181):
    (6.203)  𝛿_𝑚(𝜂, 𝐤) = 𝐶₁(𝐤) 𝐷₁(𝑦) + 𝐶₂(𝐤) 𝐷₂(𝑦),     𝑦(𝜂) ≡ 𝑎(𝜂)/𝑎_eq.
    (6.204)  𝐷₁(𝑦) = 2/3 + 𝑦 ≈ 𝑦,
                 𝐷₂(𝑦) = 𝐷₁(𝑦) n[{√(1 + 𝑦) + 1}/{√(1 + 𝑦) - 1}] - 2√(1 + 𝑦) ≈ 8/45 𝑦^-3/2.
The growing mode solution at late time is 𝛿_𝑚 ≈ 𝐶₁(𝐤) 𝑦. We would like to write this solution in terms of its initial conditions at baryon decoupling. To do this we match 𝛿_𝑚 and 𝛿_𝑚ʹ at the end of the drag epoch. The matching conditions are, using (6.203) and (6.202)
    (6.205)  𝐶₁(𝐤) 𝐷₁(𝜂_𝑑) + 𝐶₂(𝐤) 𝐷₂(𝜂_𝑑) = 𝑓_𝑏𝛿_𝑏(𝜂_𝑑) + (1 - 𝑓_𝑏)𝛿_𝑐(𝜂_𝑑),
            𝐶₁(𝐤) 𝐷₁ʹ(𝜂_𝑑) + 𝐶₂(𝐤) 𝐷₂ʹ(𝜂_𝑑) = 𝑓_𝑏𝛿_𝑏ʹ(𝜂_𝑑) + (1 - 𝑓_𝑏)𝛿_𝑐ʹ(𝜂_𝑑).
Solving this for 𝐶₁(𝐤) and only only keeping baryonic contribution which carries the oscillatory feature, we find
    (6.206)  𝐶₁(𝐤) = 𝑓_𝑏 𝐶₂ʹ/(𝐷₁𝐷₂ʹ - 𝐷₁ʹ𝐷₂) [𝛿_𝑏(𝜂_𝑑, 𝐤) - 𝐷₂/𝐷₂ʹ𝛿_𝑏ʹ(𝜂_𝑑, 𝐤)],  
where all functions of time are evaluated at 𝜂_𝑑. Note that ∂_𝜂𝑓 = 𝓗𝑦 ∂_𝑦𝑓. Using 𝑦_𝑑 ≫ 1, replacing all growth functions 𝐷_𝑖 by their asymptotic expressions, we get
    (6.207)  𝛿_𝑚(𝜂, 𝐤) = 𝐶₁(𝐤) ≈ 3𝑓_𝑏/5 [𝛿_𝑏(𝜂_𝑑, 𝐤) + 2/3 (𝑘𝜂_𝑑) 𝑣_𝑏(𝜂_𝑑, 𝐤)] 𝑎/𝑎_𝑑,
where we used 𝛿_𝑏ʹ = 𝑘𝑣_𝑏. On smaller scales than the horizon at the drag epoch, we have 𝑘𝜂_𝑑 ≫ 1 and the second term, proportional to the velocity 𝑣_𝑏 dominates. This is called the velocity overshoot. When the baryons are released at the drag epoch, they move according to their velocity and generate a new density perturbation. This is dominant effect on small scales.
   As long as the tight-coupling approximation holds, we have 𝑣_𝑏 = 𝑣_𝛾 = 3/4 𝛿_𝛾ʹ/𝑘. The cosine contribution in the solution (6.200) for 𝛿_𝛾 then leads to a dominant sine contribution in the solution (6.207) for 𝛿_𝑚. A recent measurement of the BAO signal is shown in 6.11. It is rather amazing how this signal arising from primordial sound waves in the early universe is so clearly visible to the clustering of the galaxies billions of years later. In position space space, the BAO feature maps to a peak around 110𝘩^-1Mpc. The position of this peak is an important "standard ruler." It depends on the sound horizon at the drag epoch and the angular diameter distance  to the the observed objects. Measurements of the BAO help to break degeneracies in the CMB data and played an important role in the measurements of the cosmological parameters (see Chapter 7).

          6.5 Gravitational Waves
On September 14, 2015, when the LIGO(Laser Interferometer Gravitational-Wave Oberservatory) detected the first gravitational waves (GWs) from the merger of two black holes, a new era of science was initiated! This detection marked the beginning of multi-messenger astronomy and in the future GWs may provide a new window into the early universe. Some mathematical background on cosmological GWs useful in Chapter 7 and 8 will be collected here. Further details can be found in [15].     GWs are tensor perturbations to the spatial metric,
    (6.208)  𝑑𝑠² = 𝑎²(𝜂) [-𝑑𝜂² + (𝛿_𝑖𝑗 + 𝘩_𝑖𝑗)𝑑𝑥^𝑖𝑑𝑥^𝑗].
Since the perturbation 𝘩_𝑖𝑗 is symmetric (𝘩_𝑖𝑗 = 𝘩_𝑗𝑖), transvers (∂_𝑖𝘩_𝑗𝑖 = 0), and traceless (𝘩_𝑗𝑖 = 0), it contains 6 - 3 -1 = 2 independent modes (corresponding to the 2 polarizations of GW). We write the Fourier modes of 𝘩_𝑖𝑗 as
    (6.209)  𝘩_𝑖𝑗(𝜂, 𝐤) = ∑_𝜆=+,× 𝘩_𝜆(𝜂, 𝐤) 𝜖^𝜆_𝑖𝑗(𝐤̂),
where 𝜖^𝜆_𝑖𝑗(𝐤̂) are two independent polarization tensors and 𝘩_𝜆(𝜂, 𝐤) are the corresponding mode functions. The polarization tensors can be taken to be real and satisfy 𝜖^𝜆_𝑖𝑗(𝐤̂) = 𝜖^𝜆_𝑖𝑗(-𝐤̂), so that 𝘩_𝜆(𝜂, 𝐱) is real if 𝘩^*_𝜆(𝜂, 𝐤) = 𝘩_𝜆(𝜂, -𝐤). The polarization tensors are symmetric (𝜖^𝜆_𝑖𝑗 = 𝜖^𝜆_𝑗𝑖), transverse (𝑘̂_𝑖𝜖^𝜆_𝑖𝑗 = 0) and traceless (𝜖^𝜆_𝑖𝑖 = 0). Our normalization for polarization basis can be
    (6.210)  ∑_𝑖,𝑗 𝜖^𝜆_𝑖𝑗𝜖^𝜆ʹ_𝑖𝑗 = 2𝛿^𝜆𝜆ʹ.  
Explicitly, the polarization tensors can be written as
    (6.211)  𝜖⁺_𝑖𝑗(𝐤̂) = 𝑚̂_𝑖𝑚̂_𝑗 - 𝑛̂_𝑖𝑛̂_𝑗,     𝜖^×_𝑖𝑗(𝐤̂) = 𝑚̂_𝑖𝑚̂_𝑗 + 𝑛̂_𝑖𝑛̂_𝑗
where 𝐦̂ and 𝐧̂ are two unit vectors orthogonal to 𝐤̂ and to each other. For a GW with a wavevector pointing in the 𝑧-direction direction, 𝐤̂ = (0, 0, 1), we can choose 𝐦̂ = 𝐱̂ and 𝐧̂ = 𝐲̂, so that
                               ⌈ 1    0    0 ⌉           ⌈ 0    1    0 ⌉      ⌈ 𝘩₊   𝘩_×   0 ⌉
    (6.212)  𝘩_𝑖𝑗 = 𝘩₊┃ 0   -1   0 ┃ + 𝘩_×┃1    0    0 ┃ = ┃𝘩_×  -𝘩₊   0┃
                               ⌊ 0    0    0 ⌋           ⌊ 0    0    0 ⌋      ⌊ 0     0     0 ⌋
Sometimes, it's useful to work in the so-called helicity basis, where
    (6.213)  𝜖^±2_𝑖𝑗 ≡ (𝜖⁺_𝑖𝑗 ± 𝑖𝜖^×_𝑖𝑗)/2,      𝘩_±2 ≡ 𝘩₊ ∓ 𝑖𝘩_×.
Under a rotation by an angle 𝜓 around the wavenumber 𝐤, we have 𝘩_±2 ↦ 𝑒^±2𝑖𝜓𝘩_±2, which shows that the mode 𝘩_±2 describe states of helicity +2 and -2 (also called "right-handed" and "left-handed" polarization respectively).
   Problem 6.4 shows that the linearlized Einstein equation implies following evolution equation for tensor
    (6.214)  𝘩_𝑖𝑗ʺ + 2𝓗𝘩_𝑖𝑗ʹ - ∇²𝘩_𝑖𝑗 = 16π𝐺𝑎²𝛱̂_𝑖𝑗,
where 𝛱̂_𝑖𝑗 is the transvers, traceless part of the anisotropic stress. In 𝛱̂_𝑖𝑗 = 0, this equation describes the free propagation of GWs in an expanding universe. By defining 𝑓_𝑖𝑗 ≡ 𝑎(𝜂)𝘩_𝑖𝑗, we can remove the Hubble friction term. Then each polarization mode satisfies
    (6.215)  𝑓_𝜆ʺ + (𝑘² - 𝑎ʺ/𝑎)𝑓_𝜆 = 0.  
Let us solve this for a generic scale factor 𝑎(𝜂) ∝ 𝜂^𝛽, which includes radiation domination (𝛽 = 1), matter domination (𝛽 = 2) and inflation (𝛽 ≈ -1) as special cases. The solution is  
    (6.216)  𝘩_𝜆(𝜂, 𝐤) = 𝐶_𝜆(𝐤)/𝑎(𝜂) 𝜂𝑗_𝛽-1(𝑘𝑛) + 𝐷_𝜆(𝐤)/𝑎(𝜂) 𝜂𝑦_𝛽-1(𝑘𝑛),
where 𝑗_𝛽 and 𝑦_𝛽 are spherical Bessel functions (see Appendix D) and the constants 𝐶_𝜆(𝐤) and 𝐷_𝜆(𝐤) must be fixed by the initial conditions.
   Note that for a power-law scale factor, we have 𝑎ʺ/𝑎 ∝ 𝓗². On sub-Hubble scales, 𝑘 ≫ 𝓗, we can drop the 𝑎ʺ/𝑎 term  in (6.215) and the solution is
    (6.217)  𝘩_𝜆(𝜂, 𝐤) = 𝐶_𝜆(𝐤)/𝑎(𝜂) 𝑒^𝑖𝑘𝑛 + 𝐷_𝜆(𝐤)/𝑎(𝜂) 𝑒^𝑖𝑘𝑛 𝑒^{-𝑖𝑘𝑛}     (for 𝑘 ≫ 𝓗),
which we can check also follows from the 𝑘𝑛 ≫ 1 limit of (6.216). On sub-Hubble scales, we get the expected plane wave solutions with an amplitude that decays as 𝑎^-1. On super-Hubble scales, 𝑘 ≪ 𝓗, we instead drop 𝑘² term and the solution becomes
    (6.218)  𝘩_𝜆(𝜂, 𝐤) = 𝐶_𝜆(𝐤)/𝑎(𝜂) + 𝐷_𝜆(𝐤) ∫ 𝑑𝜂/𝑎²(𝜂)     (for 𝑘 ≪ 𝓗).
The second term decays with the expansion of the universe, so that the growing mode of the GW is a constant on super-Hubble scales.¹⁰
   In cosmology we are interested in power spectrum of a stochastic background of GWs, 𝒫_𝘩(𝑘) ≡ (2π²/𝑘³)∆²_𝘩(𝑘). We define this power spectrum, so that the variance is
    (6.219)  ∑_𝑖,𝑗⟨𝘩²_𝑖𝑗(𝐱)⟩ = ∫ 𝑑 ln ∆²_𝘩(𝑘),
where we suppressed the time dependence. The power spectrum for the individual polarization modes then is
    (6.220)  ⟨𝘩_𝜆(𝐤) 𝘩_𝜆ʹ(𝐤ʹ)⟩ = 2π²/𝑘³ ∆²_𝘩(𝑘)/4 × (2π)³𝛿_𝐷(𝐤 + 𝐤ʹ) 𝛿_𝜆𝜆ʹ,   ^{[verification needed]}
where the factor of 4 follows from our normalization of the polarization basis in (6.210). In Chapter 8 we willshow how quantum fluctuations during inflation produce a scale-invariant spectrum of primordial GWs, while in Chapter 7, we will explain how these GWs affect the anisotropies in the CMB.

⁹ For simplicity, we will ignore the photon anisotropic stress. In reality this is only a good approximation because of the coupling to baryons.
¹⁰ In a radiation-dominated universe 𝑎ʺ/𝑎 vanishes identically, so that the solution (6.218) doesn't apply. In that case, the the 𝑘𝑛 ≪ 1 limit of (6.217), which again has a decaying mode and a constant mode.

          6.6 Summary
In this chapter, we developed the foundation of cosmological perturbation theory. By linearizing the Einstein Equations we derive the evolution equation for small fluctuations and solved them for various cases.
   Focusing on scalar perturbations, the most general perturbation of spacetime metric is
    (6.221)  𝑔_𝜇𝜈 = 𝑎²⌈-(1+ 2𝛢)                        ∂_𝑖𝛣            ⌉     
                               ⌊   ∂_𝑖𝛣            (1 + 2𝐶)𝛿_𝑖𝑗 + 2∂_⟨𝑖∂_𝑗⟩𝐸⌋,
where ∂_⟨𝑖∂_𝑗⟩ = ∂_𝑖∂_𝑗 - 1/3 ∇². The perturbed energy-momentum tensor is
    (6.222)  𝛵⁰₀ ≡ -(𝜌̄ + 𝛿𝜌),     𝛵^𝑖₀ ≡ (𝜌̄ + 𝑃̄)∂^𝑖𝑣,     𝛵⁰_𝑖 ≡ (𝜌̄ + 𝑃̄)∂_𝑖(𝑣 + 𝛣),     𝛵^𝑖_𝑗 ≡ (𝑃̄ + 𝛿𝑃)𝛿^𝑖_𝑗 + ∂^⟨𝑖∂_𝑗⟩𝛱,
and we often work in terms of the momentum density 𝑞 ≡ (𝜌̄ + 𝑃̄)𝑣 and the density contrast 𝛿 ≡ 𝛿𝜌/𝜌. Coordination transformation can change the values of these variables. The problem of unphyscial gauge mode can be addressed by working with combinations of perturbations that are invariant under coordinate transformation. Important gauge-invariant variables are the Bardeen potential and the comoving density contrast
    (6.223-5)  𝛹 ≡ 𝛢 + 𝓗(𝛣 - 𝐸ʹ) + (𝛣 - 𝐸ʹ)ʹ,     𝛷 ≡ -𝐶 + 1/3 ∇²𝐸 - 𝓗(𝛣 - 𝐸ʹ),      ∆ = 𝛿 + 𝜌̄ʹ/𝜌̄ (𝑣 + 𝛣).
as well as the two curvature perturbations
    (6.226-7)   𝛇 = -𝐶 + 1/3 ∇²𝐸 + 𝓗 𝛿𝜌/𝜌̄ʹ,     𝓡 = -𝐶 + 1/3 ∇²𝐸 - 𝓗(𝑣 + 𝛣).  
On superhorizon scales, 𝛇 and 𝓡 are equal and time independent if the matter perturbations are adiabatic. Perturbations are called adiabatic if
    (6.228)    𝛿𝑃 = 𝑐_𝑠²𝛿𝜌 = 𝑃̄ʹ/𝜌̄ʹ 𝛿𝜌,
and fluctuations in the densities of matter and radiation obey
    (6.229)    𝛿_𝑚 = 3/4 𝛿_𝑟.
The initial conditions of our universe are well described by adiabatic perturbations.
   An alternative to the gauge-invariant formalism is to work in a fixed gauge, track the evolution of all fluctuations and compute observables. Unphysical gauge modes must cancel in physical answers. Popular gauge are
   • Newtonian: 𝛣 = 𝐸 = 0     • Spatially flat: 𝐶 = 𝐸 = 0     • Synchronous: 𝛢 = 𝛣 = 0
   • Constant density: 𝛿𝜌 = 𝛣 = 0   • Comoving: 𝑣 = 𝛣 = 0
We derived the equations of motion for the perturbation in Newtonian gauge. The linearized Einstein equations are
     (6.230)    ∇²𝛷 - 3𝓗(𝛷ʹ + 𝓗𝛹) = 4π𝐺𝑎²𝛿𝜌,
     (6.231)    -(𝛷ʹ + 𝓗𝛹) = 4π𝐺𝑎²𝑞,
     (6.232)    ∂_⟨𝑖∂_𝑗⟩(𝛷 - 𝛹) = 8π𝐺𝑎²𝛱_𝑖𝑗,
     (6.233)    𝛷ʺ + 𝓗𝛹ʹ + 2𝓗𝛷ʹ + 1/3 ∇²(𝛹 - 𝛷) + (2𝓗ʹ + 𝓗²)𝛹 = 4π𝐺𝑎²𝛿𝑃.
where the sources on the right-hand side include a sum over all components. In the absence of anisotropic stress, (6.232) implies that 𝛹 ≈ 𝛷, and (6.233) reduces to
     (6.234)    𝛷ʺ + 3𝓗𝛷ʹ + (2𝓗ʹ + 𝓗²)𝛷 = 4π𝐺𝑎²𝛿𝑃.
(6.230) and (6.231) can be combined into
     (6.235)    ∇²𝛷 = 4π𝐺𝑎²𝜌̄∆,
which is of the same form as the Newtonian Poisson equation, but is now valid on all scales.
   The conservation of the energy-momentum tensor gives the continuity and Euler equations
     (6.236)    𝛿ʹ = -(1 + 𝜔)(𝛁 ⋅ 𝐯 - 3𝛷ʹ) - 3𝓗(𝑐_𝑠² - 𝜔)𝛿
     (6.237)    𝑣_𝑖ʹ = -[𝓗 - 3𝜔]𝐯 - 𝑐_𝑠²/(1 + 𝜌) 𝛁𝛿 - 𝛁𝛷,
where 𝑐_𝑠² = 𝛿𝑃/𝛿𝜌 is the sound speed of the fluid and 𝜌 = 𝑃̄/𝜌̄ is its equation of state. These equations apply separately for every non-interacting fluid.
   we discussed the solutions to the above equations in various special limits. Table 6.1 summarizes the results for 𝛷, ∆_𝑟 and ∆_𝑚. The density contrast 𝛿 is equal to ∆ on subhorizon scales, but differs on superhorizon scales, where it is a constant.
   Before decoupling, photons and baryons can be treated as a single tightly-coupled fluid. The density contrast of photon-baryon fluid evolves as
     (6.238)    𝛿_𝑟ʺ + 𝑅ʹ/(1 + 𝑅) 𝛿_𝑟ʹ - 1/3(1 + 𝑅) ∇²𝛿_𝑟 = 4/3 ∇²𝛹 + 4𝛷ʺ + 4𝑅ʹ/(1 + 𝑅) 𝛷ʹ,
where 𝑅 ≡ 3/4 𝜌̄_𝑏/𝜌̄_𝛾 ∝ 𝑎(𝑡). This is the equation of a harmonic oscillator with a gravitational driving force, which plays an important role in he physics of the CMB anisotropies (see Chapter 7).