Linear Perturbation Evolution in minimal model
Equation of motions (DraftXXX) ¶ The equations of motion for the matter perturbations come from conservation of the stress tensor
∇ μ T μ ν = 0 \boxed{
\nabla^\mu T_{\mu \nu} = 0} ∇ μ T μν = 0 Developing the covariant derivative
∇ μ T μ ν = ∂ μ T ν μ + Γ μ α μ T ν α − Γ μ ν α T α μ = 0 \nabla^\mu T_{\mu \nu} = \partial_\mu T^\mu_\nu + \Gamma^\mu_{\mu \alpha} T^{\alpha}_\nu - \Gamma^\alpha_{\mu \nu} T^\mu_\alpha = 0 ∇ μ T μν = ∂ μ T ν μ + Γ μα μ T ν α − Γ μν α T α μ = 0 For ν = 0 \nu = 0 ν = 0 conitinuity equation (equation of density perturbation evolution):
∂ η δ ρ = − 3 H ( δ ρ + δ P ) + 3 ψ ˙ ( ρ ˉ + P ˉ ) − ∂ i q i \boxed{
\partial_\eta \delta \rho = -3 \mathcal{H} (\delta \rho + \delta P) + 3 \dot \psi (\bar \rho + \bar P) - \partial_i q^i} ∂ η δ ρ = − 3 H ( δ ρ + δ P ) + 3 ψ ˙ ( ρ ˉ + P ˉ ) − ∂ i q i For ν = i \nu = i ν = i Euler equation (equation of momentum perturbation evolution):
∂ η ( ( ρ ˉ + P ˉ ) v i ) = − 4 H q i − ( ρ ˉ + P ˉ ) ∂ i ϕ − ∂ i δ P − ∂ j Π i j \boxed{
\partial_\eta ((\bar{\rho} + \bar{P})v^i) = -4 \mathcal{H} q^i - (\bar \rho + \bar P ) \partial^i \phi - \partial^i \delta P - \partial_j \Pi^{ij}
} ∂ η (( ρ ˉ + P ˉ ) v i ) = − 4 H q i − ( ρ ˉ + P ˉ ) ∂ i ϕ − ∂ i δ P − ∂ j Π ij Proof 1 (Equation of motion)
We evaluate equations (3) (4)
Matter : non-relativistic fluid, P m = 0 P_m = 0 P m = 0 Π m i j = 0 \Pi^{ij}_m = 0 Π m ij = 0
δ ˙ m = − ∇ ⋅ v m + 3 ψ ˙ v ˙ m = − H v m − ∇ ϕ \begin{align*}
\dot \delta_m &= - \nabla \cdot \mathbf{v}_m + 3 \dot \psi \\
\dot{\mathbf{v}}_m &= - \mathcal{H} \mathbf{v}_m - \nabla \phi
\end{align*} δ ˙ m v ˙ m = − ∇ ⋅ v m + 3 ψ ˙ = − H v m − ∇ ϕ Radiation : relativistic fluid, P r = 1 3 ρ r P_r = \dfrac{1}{3} \rho_r P r = 3 1 ρ r Π r i j = 0 \Pi^{ij}_r = 0 Π r ij = 0
δ ˙ r = − 4 3 ∇ ⋅ v r + 4 ψ ˙ v ˙ r = − 1 4 ∇ δ r − ∇ ϕ \begin{align*}
\dot \delta_r &= - \dfrac{4}{3} \nabla \cdot \mathbf{v}_r + 4 \dot \psi \\
\dot{\mathbf{v}}_r &= - \dfrac{1}{4} \nabla \delta_r - \nabla \phi
\end{align*} δ ˙ r v ˙ r = − 3 4 ∇ ⋅ v r + 4 ψ ˙ = − 4 1 ∇ δ r − ∇ ϕ Initial conditions ¶ Adiabatic fluctuations ¶ Single–field inflation predicts adiabatic initial fluctuations .
This means all perturbations come from the same local shift in time of the background Universe.
At each point ( τ , x ) (\tau, \mathbf{x}) ( τ , x ) τ + δ τ ( x ) \tau + \delta \tau(\mathbf{x}) τ + δ τ ( x )
The local density of species I I I
δ ρ I ( τ , x ) ≡ ρ ˉ I ( τ + δ τ ( x ) ) − ρ ˉ I ( τ ) , . \delta \rho_I(\tau, \mathbf{x}) \equiv \bar{\rho}_I(\tau + \delta \tau(\mathbf{x})) - \bar{\rho}_I(\tau) ,. δ ρ I ( τ , x ) ≡ ρ ˉ I ( τ + δ τ ( x )) − ρ ˉ I ( τ ) , . For small δ τ \delta \tau δ τ
δ ρ I = ρ ˉ I ′ , δ τ ( x ) , . \delta \rho_I = \bar{\rho}_I' , \delta \tau(\mathbf{x}) ,. δ ρ I = ρ ˉ I ′ , δ τ ( x ) , . The key point is that the same δ τ \delta \tau δ τ :
δ τ = δ ρ I ρ ˉ I ′ = δ ρ J ρ ˉ J ′ for all I , J \delta \tau = \frac{\delta \rho_I}{\bar{\rho}_I'} = \frac{\delta \rho_J}{\bar{\rho}_J'} \quad \text{for all } I,J δ τ = ρ ˉ I ′ δ ρ I = ρ ˉ J ′ δ ρ J for all I , J This means all components fluctuate together, with no relative perturbation between species , which defines an adiabatic mode .
Using
ρ I ˉ ′ ∝ ( 1 + ω I ) ρ I \bar{\rho_I}' \propto (1 + \omega_I) \rho_I ρ I ˉ ′ ∝ ( 1 + ω I ) ρ I and define
δ I ≡ δ ρ I ρ I ˉ \delta_I \equiv \dfrac{\delta \rho_I}{\bar{\rho_I}} δ I ≡ ρ I ˉ δ ρ I We have
δ I 1 + ω I = δ J 1 + ω J \dfrac{\delta_I}{1 + \omega_I} = \dfrac{\delta_J}{1 + \omega_J} 1 + ω I δ I = 1 + ω J δ J For matter and radiation components
δ r = 4 3 δ m \delta_r = \dfrac{4}{3} \delta_m δ r = 3 4 δ m Total density perturbation is dominated by background species since δ I \delta_I δ I
δ ρ t o t = ρ ˉ t o t δ t o t = ∑ I ρ I ˉ δ I \rm \delta \rho_{tot} = \bar{\rho}_{tot} \delta_{tot} = \sum_I \bar{\rho_{I}} \delta_I δ ρ tot = ρ ˉ tot δ tot = I ∑ ρ I ˉ δ I At early times, the universe is radiation dominated , so it natural to set the initial conditions for all superhorizon Fourier modes.
Equation (67)
ϕ = const \boxed{\phi = \text{const}} ϕ = const
on superhorizon scales.
From equation (60)
δ ≈ δ r = − 2 ϕ \boxed{
\delta \approx \delta_r = -2 \phi
} δ ≈ δ r = − 2 ϕ
on superhorizon scales.
The velocity divergence θ \theta θ ϕ \phi ϕ 0i Einstein equation
ϕ ˙ + H ϕ ∝ ( ρ ˉ + P ˉ ) θ \dot \phi + \mathcal{H} \phi \propto (\bar \rho + \bar P) \theta ϕ ˙ + H ϕ ∝ ( ρ ˉ + P ˉ ) θ
so θ \theta θ negligible (decaying mode) on superhorizon scales (with ϕ = const \phi = \text{const} ϕ = const
The pressure perturbation δ P \delta P δ P
δ P = c s 2 δ ρ = c 2 s ( − 2 ρ ˉ ϕ ) \delta P = c^2_s \delta \rho = c^2 s(-2 \bar \rho \phi) δ P = c s 2 δ ρ = c 2 s ( − 2 ρ ˉ ϕ ) Thus, all matter perturbation variables (δ \delta δ δ P \delta P δ P θ \theta θ σ \sigma σ single constant ϕ \phi ϕ
Proof 2 (Initial conditions)
From equation (60) k ≪ H k \ll \mathcal{H} k ≪ H ∇ 2 ϕ \nabla^2 \phi ∇ 2 ϕ
− 3 H ( ϕ ˙ + H ϕ ) = 4 π G a 2 δ ρ -3 \mathcal{H}(\dot \phi + \mathcal{H} \phi) = 4 \pi G a^2 \delta \rho − 3 H ( ϕ ˙ + H ϕ ) = 4 π G a 2 δ ρ Since ϕ = const \phi = \text{const} ϕ = const ϕ ˙ = 0 \dot \phi = 0 ϕ ˙ = 0
− 3 H 2 ϕ = 4 π G a 2 δ ρ -3 \mathcal{H}^2 \phi = 4 \pi G a^2 \delta \rho − 3 H 2 ϕ = 4 π G a 2 δ ρ Using Friedmann equation
3 H 2 = 8 π G a 2 ρ ˉ 3 \mathcal{H}^2 = 8 \pi G a^2 \bar \rho 3 H 2 = 8 π G a 2 ρ ˉ we have
− 8 π G a 2 ρ ˉ ϕ = 4 π G a 2 δ ρ \begin{align*}
- 8 \pi G a^2 \bar \rho \phi &= 4 \pi G a^2 \delta \rho \\
\end{align*} − 8 π G a 2 ρ ˉ ϕ = 4 π G a 2 δ ρ Therefore,
δ = − 2 ϕ = const \begin{align*}
\delta &= -2 \phi = \text{const}
\end{align*} δ = − 2 ϕ = const Comoving curvature perturbation ¶ The gravitational potential ϕ \phi ϕ
R = − ϕ + H ρ ˉ + P ˉ δ q \mathcal{R} = - \phi + \dfrac{\mathcal{H}}{\bar \rho + \bar P} \delta q R = − ϕ + ρ ˉ + P ˉ H δ q