Perturbations from Inflation

Heling Deng

  • Perturbations
  • Inflation
  • Perturbations from inflation
  • Perturbations
  • Inflation
  • Perturbations from inflation
f(t,x^{i})
f(t,xi)f(t,x^{i})
g_{\mu\nu}=\bar{g}_{\mu\nu}+\delta g_{\mu\nu}
gμν=g¯μν+δgμνg_{\mu\nu}=\bar{g}_{\mu\nu}+\delta g_{\mu\nu}
T_{\mu\nu}=\bar{T}_{\mu\nu}+\delta T_{\mu\nu}
Tμν=T¯μν+δTμνT_{\mu\nu}=\bar{T}_{\mu\nu}+\delta T_{\mu\nu}
\delta G_{\mu\nu}\propto\delta T_{\mu\nu}
δGμνδTμν\delta G_{\mu\nu}\propto\delta T_{\mu\nu}

Metric

Matter

Einstein Equations

=\bar{f}(t)+\delta f(t,x^{i})
=f¯(t)+δf(t,xi)=\bar{f}(t)+\delta f(t,x^{i})
{tr}E=0
trE=0{tr}E=0
ds^{2}=-dt^{2}+a(t)^{2}\delta_{ij}dx^{i}dx^{j}
ds2=dt2+a(t)2δijdxidxjds^{2}=-dt^{2}+a(t)^{2}\delta_{ij}dx^{i}dx^{j}

Background metric

\delta g_{00}=-2\Phi
δg00=2Φ\delta g_{00}=-2\Phi
\delta g_{0i}=aB_{i}
δg0i=aBi\delta g_{0i}=aB_{i}
\delta g_{ij}=a^{2}\left(2\Psi\delta_{ij}+E_{ij}\right)
δgij=a2(2Ψδij+Eij)\delta g_{ij}=a^{2}\left(2\Psi\delta_{ij}+E_{ij}\right)

Perturbations

\Phi,\Psi,B,E
Φ,Ψ,B,E\Phi,\Psi,B,E
B_i^{V},\ E_i^{V}
BiV, EiVB_i^{V},\ E_i^{V}
E_{ij}^{T}
EijTE_{ij}^{T}

Scalar

Vector

Tensor

Scalar-vector-tensor Decomposition

ds^{2}=-(1+2\Phi)dt^{2}+2aB_{,i}dx^{i}dt+a^{2}\left[\left(1-2\Psi\right)\delta_{ij}+E_{,ij}\right]dx^{i}dx^{j}
ds2=(1+2Φ)dt2+2aB,idxidt+a2[(12Ψ)δij+E,ij]dxidxjds^{2}=-(1+2\Phi)dt^{2}+2aB_{,i}dx^{i}dt+a^{2}\left[\left(1-2\Psi\right)\delta_{ij}+E_{,ij}\right]dx^{i}dx^{j}

Background fluid

\bar{T}_{\ \nu}^{\mu}=\left(\bar{\rho}+\bar{p}\right)\bar{u}^{\mu}\bar{u}_{\nu}+\delta_{\nu}^{\mu}\bar{p}
T¯ νμ=(ρ¯+p¯)u¯μu¯ν+δνμp¯\bar{T}_{\ \nu}^{\mu}=\left(\bar{\rho}+\bar{p}\right)\bar{u}^{\mu}\bar{u}_{\nu}+\delta_{\nu}^{\mu}\bar{p}
T_{\ \nu}^{\mu}=\left(\bar{\rho}+\delta\rho+\bar{p}+\delta p\right)(\bar{u}^{\mu}+\delta u^{\mu})(\bar{u}_{\nu}+\delta u_{\nu})+\delta_{\nu}^{\mu}\left(\bar{p}+\delta p\right)
T νμ=(ρ¯+δρ+p¯+δp)(u¯μ+δuμ)(u¯ν+δuν)+δνμ(p¯+δp)T_{\ \nu}^{\mu}=\left(\bar{\rho}+\delta\rho+\bar{p}+\delta p\right)(\bar{u}^{\mu}+\delta u^{\mu})(\bar{u}_{\nu}+\delta u_{\nu})+\delta_{\nu}^{\mu}\left(\bar{p}+\delta p\right)
\delta T_{\ 0}^{0}=-\delta\rho
δT 00=δρ\delta T_{\ 0}^{0}=-\delta\rho
\delta T_{\ i}^{0}=a^{-2}(\bar{\rho}+\bar{p})(aB_{i}-\delta u_{i})
δT i0=a2(ρ¯+p¯)(aBiδui)\delta T_{\ i}^{0}=a^{-2}(\bar{\rho}+\bar{p})(aB_{i}-\delta u_{i})
\delta T_{\ 0}^{i}=(\bar{\rho}+\bar{p})\delta u_{i}
δT 0i=(ρ¯+p¯)δui\delta T_{\ 0}^{i}=(\bar{\rho}+\bar{p})\delta u_{i}
\delta T_{\ j}^{i}=\delta p\delta_{ij}
δT ji=δpδij\delta T_{\ j}^{i}=\delta p\delta_{ij}

Perturbations

Gauge

Gauge

t\to t+\alpha(t,\bold{x})
tt+α(t,x)t\to t+\alpha(t,\bold{x})
\bold{x}\to \bold{x}+\bold{\nabla} \beta(t,\bold{x})
xx+β(t,x)\bold{x}\to \bold{x}+\bold{\nabla} \beta(t,\bold{x})
\Phi\to\Phi-\dot{\alpha}
ΦΦα˙\Phi\to\Phi-\dot{\alpha}
\Psi\to\Psi+H\alpha
ΨΨ+Hα \Psi\to\Psi+H\alpha
B\to B+\frac{\alpha}{a}-a\dot{\beta}
BB+αaaβ˙ B\to B+\frac{\alpha}{a}-a\dot{\beta}
E\to E-\beta
EEβ E\to E-\beta
\Phi_{B}\equiv\Phi-\frac{d}{dt}\left[a^{2}\left(\dot{E}-\frac{B}{a}\right)\right]
ΦBΦddt[a2(E˙Ba)]\Phi_{B}\equiv\Phi-\frac{d}{dt}\left[a^{2}\left(\dot{E}-\frac{B}{a}\right)\right]
\Psi_{B}\equiv\Psi+a^{2}H\left(\dot{E}-\frac{B}{a}\right)
ΨBΨ+a2H(E˙Ba) \Psi_{B}\equiv\Psi+a^{2}H\left(\dot{E}-\frac{B}{a}\right)

Gauge

\delta\rho\to\delta\rho-\dot{\bar{\rho}}\alpha
δρδρρ¯˙α\delta\rho\to\delta\rho-\dot{\bar{\rho}}\alpha
\delta p\to\delta p-\dot{\bar{p}}\alpha
δpδpp¯˙α\delta p\to\delta p-\dot{\bar{p}}\alpha
\delta u\to\delta u+\alpha
δuδu+α\delta u\to\delta u+\alpha
\delta\rho_B\equiv\delta\rho-a^{2}\dot{\bar{\rho}}\left(\dot{E}-\frac{B}{a}\right)
δρBδρa2ρ¯˙(E˙Ba)\delta\rho_B\equiv\delta\rho-a^{2}\dot{\bar{\rho}}\left(\dot{E}-\frac{B}{a}\right)
\delta u_B\equiv\delta u + a^{2}\left(\dot{E}-\frac{B}{a}\right)
δuBδu+a2(E˙Ba)\delta u_B\equiv\delta u + a^{2}\left(\dot{E}-\frac{B}{a}\right)
\delta p_B\equiv\delta p-a^{2}\dot{\bar{p}}\left(\dot{E}-\frac{B}{a}\right)
δpBδpa2p¯˙(E˙Ba)\delta p_B\equiv\delta p-a^{2}\dot{\bar{p}}\left(\dot{E}-\frac{B}{a}\right)
t\to t+\alpha(t,\bold{x})
tt+α(t,x)t\to t+\alpha(t,\bold{x})
\bold{x}\to \bold{x}+\bold{\nabla} \beta(t,\bold{x})
xx+β(t,x)\bold{x}\to \bold{x}+\bold{\nabla} \beta(t,\bold{x})

Choosing a Gauge

Newtonian Gauge

B=E=0
B=E=0B=E=0
ds^{2}=-(1+2\Phi)dt^{2}+a^{2}(1-2\Psi)d\bold{x}^2
ds2=(1+2Φ)dt2+a2(12Ψ)dx2ds^{2}=-(1+2\Phi)dt^{2}+a^{2}(1-2\Psi)d\bold{x}^2
ds^{2}=-(1+2\Phi)dt^{2}+2aB_{,i}dx^{i}dt+a^{2}\left[\left(1-2\Psi\right)\delta_{ij}+E_{,ij}\right]dx^{i}dx^{j}
ds2=(1+2Φ)dt2+2aB,idxidt+a2[(12Ψ)δij+E,ij]dxidxjds^{2}=-(1+2\Phi)dt^{2}+2aB_{,i}dx^{i}dt+a^{2}\left[\left(1-2\Psi\right)\delta_{ij}+E_{,ij}\right]dx^{i}dx^{j}

Synchronous

\Phi=B=0
Φ=B=0\Phi=B=0

Comoving

\delta u=0
δu=0\delta u=0

Uniform-density

\delta \rho=0
δρ=0\delta \rho=0

Spatially-flat

\Psi=E = 0
Ψ=E=0\Psi=E = 0
  • Perturbations
  • Inflation
  • Perturbations from inflation

Comoving Radius

\frac{1}{aH}
1aH\frac{1}{aH}

today

recombination

inflation

hot big bang

Time

Inflation:

\frac{d}{dt} \left( \frac{1}{aH} \right) < 0
ddt(1aH)<0\frac{d}{dt} \left( \frac{1}{aH} \right) < 0
\Leftrightarrow
\Leftrightarrow
\ddot{a} > 0
a¨>0\ddot{a} > 0
\Leftrightarrow
\Leftrightarrow
\rho + 3p < 0
ρ+3p<0\rho + 3p < 0

A Single Scalar Field

T_{\ \nu}^{\mu}=\partial^{\mu}\phi\partial_{\nu}\phi+\delta_{\nu}^{\mu}\left[\frac{1}{2}(\partial \phi)^{2}-V(\phi)\right]
T νμ=μϕνϕ+δνμ[12(ϕ)2V(ϕ)]T_{\ \nu}^{\mu}=\partial^{\mu}\phi\partial_{\nu}\phi+\delta_{\nu}^{\mu}\left[\frac{1}{2}(\partial \phi)^{2}-V(\phi)\right]
\rho = \frac{1}{2}\dot\phi^2 + V
ρ=12ϕ˙2+V\rho = \frac{1}{2}\dot\phi^2 + V
p = \frac{1}{2}\dot\phi^2 - V
p=12ϕ˙2Vp = \frac{1}{2}\dot\phi^2 - V
\ddot{\phi}+3H\dot{\phi}+V^\prime=0
ϕ¨+3Hϕ˙+V=0\ddot{\phi}+3H\dot{\phi}+V^\prime=0

KG Equation

Slow-roll Approximation

\dot{\phi}^2 \ll V,
ϕ˙2V,\dot{\phi}^2 \ll V,

Slow-roll conditions

|\ddot{\phi}| \ll |V^\prime|
ϕ¨V|\ddot{\phi}| \ll |V^\prime|
\epsilon_V\equiv \frac{M_{pl}^2}{2} \left( \frac{V^\prime}{V} \right) ^2 \ll 1
ϵVMpl22(VV)21\epsilon_V\equiv \frac{M_{pl}^2}{2} \left( \frac{V^\prime}{V} \right) ^2 \ll 1
\eta_V \equiv M_{pl}^2 \frac{|V^{\prime \prime}|}{V} \ll 1
ηVMpl2VV1\eta_V \equiv M_{pl}^2 \frac{|V^{\prime \prime}|}{V} \ll 1

Slow-roll parameters

Hubble hierarchy parameters

\epsilon\equiv-\frac{\dot{H}}{H^{2}} \ll 1
ϵH˙H21\epsilon\equiv-\frac{\dot{H}}{H^{2}} \ll 1
\eta\equiv\frac{\dot{\epsilon}}{H\epsilon} \ll 1
ηϵ˙Hϵ1\eta\equiv\frac{\dot{\epsilon}}{H\epsilon} \ll 1
  • Perturbations
  • Inflation
  • Perturbations from inflation

Comoving Radius

\frac{1}{aH}
1aH\frac{1}{aH}

today

reheating

recombination

inflaton

photons, neutrinos, electrons, DM...

Time

\phi=\bar{\phi}+\delta \phi
ϕ=ϕ¯+δϕ\phi=\bar{\phi}+\delta \phi

Comoving Radius

(aH)^{-1}
(aH)1(aH)^{-1}

today

recombination

\dot{\mathcal{R}}_k\approx0
R˙k0\dot{\mathcal{R}}_k\approx0

Time

"Conservation" Outside the Horizon

\mathcal{R}\equiv-\Psi+H\delta u
RΨ+Hδu\mathcal{R}\equiv-\Psi+H\delta u

for

k \ll aH
kaHk \ll aH
\dot{\mathcal{R}}_k\approx0
R˙k0\dot{\mathcal{R}}_k\approx0

"adiabatic solution"

\left(p=p(\rho)\right)
(p=p(ρ))\left(p=p(\rho)\right)

Comoving Radius

(aH)^{-1}
(aH)1(aH)^{-1}

today

recombination

\dot{\mathcal{R}}_k\approx0
R˙k0\dot{\mathcal{R}}_k\approx0

Time

Mukhanov-Sasaki Equation

Comoving gauge

\delta u = 0 \ (\delta \phi = 0),\ E = 0
δu=0 (δϕ=0), E=0\delta u = 0 \ (\delta \phi = 0),\ E = 0
v_{\bold{k}}^{\prime \prime} +\left( k^2 -\frac{z^{\prime \prime}}{z} \right)v_{\bold{k}} =0
vk+(k2zz)vk=0v_{\bold{k}}^{\prime \prime} +\left( k^2 -\frac{z^{\prime \prime}}{z} \right)v_{\bold{k}} =0

where

v_{\bold{k}}=z\mathcal{R}_{\bold{k}},
vk=zRk,v_{\bold{k}}=z\mathcal{R}_{\bold{k}},
\mathcal{R}\equiv-\Psi+H\delta u
RΨ+Hδu\mathcal{R}\equiv-\Psi+H\delta u
^{\prime} = \frac{d}{d\tau} = a\frac{d}{dt}
=ddτ=addt^{\prime} = \frac{d}{d\tau} = a\frac{d}{dt}
z = \frac{a\dot{\bar{\phi}}}{H},
z=aϕ¯˙H, z = \frac{a\dot{\bar{\phi}}}{H},

Naïve de Sitter Solutions

v_{\bold{k}}^{\prime \prime} +\left( k^2 -\frac{z^{\prime \prime}}{z} \right)v_{\bold{k}} =0
vk+(k2zz)vk=0v_{\bold{k}}^{\prime \prime} +\left( k^2 -\frac{z^{\prime \prime}}{z} \right)v_{\bold{k}} =0
  • Subhorizon
(k \gg aH)
(kaH)(k \gg aH)
  • Superhorizon
(k \ll aH)
(kaH)(k \ll aH)
v_{\bold{k}} \propto z
vkzv_{\bold{k}} \propto z

oscillation

"frozen"

v_{\bold{k}}\propto e^{\pm ik\tau}
vke±ikτv_{\bold{k}}\propto e^{\pm ik\tau}
\to \ \mathcal{R}_{\bold{k}}=z^{-1} v_{\bold{k}}\propto const
 Rk=z1vkconst \to \ \mathcal{R}_{\bold{k}}=z^{-1} v_{\bold{k}}\propto const
\frac{z^{\prime\prime}}{z}\sim(aH)^{2}
zz(aH)2\frac{z^{\prime\prime}}{z}\sim(aH)^{2}

Slow-roll Solutions

v_{\bold{k}}^{\prime \prime} +\left( k^2 -\frac{z^{\prime \prime}}{z} \right)v_{\bold{k}} =0
vk+(k2zz)vk=0v_{\bold{k}}^{\prime \prime} +\left( k^2 -\frac{z^{\prime \prime}}{z} \right)v_{\bold{k}} =0
v_{\mathbf{k}}^{\prime\prime}+\left(k^{2}-\frac{\nu^{2}-\frac{1}{4}}{\tau^{2}}\right)v_{\mathbf{k}}=0
vk+(k2ν214τ2)vk=0v_{\mathbf{k}}^{\prime\prime}+\left(k^{2}-\frac{\nu^{2}-\frac{1}{4}}{\tau^{2}}\right)v_{\mathbf{k}}=0

where

\nu\equiv\frac{3}{2}+\epsilon+\frac{1}{2}\eta,
ν32+ϵ+12η,\nu\equiv\frac{3}{2}+\epsilon+\frac{1}{2}\eta,
\epsilon=-\frac{\dot{H}}{H^{2}},
ϵ=H˙H2,\epsilon=-\frac{\dot{H}}{H^{2}},
\eta=\frac{\dot{\epsilon}}{H\epsilon}
η=ϵ˙Hϵ\eta=\frac{\dot{\epsilon}}{H\epsilon}
v_{k}(\tau)=\sqrt{-\tau}\left[C_{1}H_{\nu}^{(1)}(-k\tau)+C_{2}H_{\nu}^{(2)}(-k\tau)\right]
vk(τ)=τ[C1Hν(1)(kτ)+C2Hν(2)(kτ)]v_{k}(\tau)=\sqrt{-\tau}\left[C_{1}H_{\nu}^{(1)}(-k\tau)+C_{2}H_{\nu}^{(2)}(-k\tau)\right]

initial condition?

Quantization

v_{\bold{k}}^{\prime \prime} +\left( k^2 -\frac{z^{\prime \prime}}{z} \right)v_{\bold{k}} =0
vk+(k2zz)vk=0v_{\bold{k}}^{\prime \prime} +\left( k^2 -\frac{z^{\prime \prime}}{z} \right)v_{\bold{k}} =0

Commutation

v(\tau,\mathbf{x})=\int\frac{d^{3}\mathbf{k}}{(2\pi)^{3/2}}\left[a_{\mathbf{k}}^{-}v_{k}(\tau)e^{i\mathbf{k}\cdot\mathbf{x}}+a_{\mathbf{k}}^{+}v_{k}^{*}(\tau)e^{-i\mathbf{k}\cdot\mathbf{x}}\right]
v(τ,x)=d3k(2π)3/2[akvk(τ)eikx+ak+vk(τ)eikx]v(\tau,\mathbf{x})=\int\frac{d^{3}\mathbf{k}}{(2\pi)^{3/2}}\left[a_{\mathbf{k}}^{-}v_{k}(\tau)e^{i\mathbf{k}\cdot\mathbf{x}}+a_{\mathbf{k}}^{+}v_{k}^{*}(\tau)e^{-i\mathbf{k}\cdot\mathbf{x}}\right]
\left[v(\tau,\mathbf{x}),v^\prime(\tau,\mathbf{y})\right]=i\delta(\mathbf{x}-\mathbf{y})
[v(τ,x),v(τ,y)]=iδ(xy)\left[v(\tau,\mathbf{x}),v^\prime(\tau,\mathbf{y})\right]=i\delta(\mathbf{x}-\mathbf{y})
\left[a_{\mathbf{k}}^{-},a_{\mathbf{k}^{\prime}}^{+}\right]=\delta(\mathbf{k}-\mathbf{k}^{\prime})
[ak,ak+]=δ(kk)\left[a_{\mathbf{k}}^{-},a_{\mathbf{k}^{\prime}}^{+}\right]=\delta(\mathbf{k}-\mathbf{k}^{\prime})
v_{k}v_{k}^{*\prime}-v_{k}^{*}v_{k}^{\prime}=i
vkvkvkvk=iv_{k}v_{k}^{*\prime}-v_{k}^{*}v_{k}^{\prime}=i
v_{\mathbf{k}}=a_{\mathbf{k}}^{-}v_{k}+a_{-\mathbf{k}}^{+}v_{k}^{*}
vk=akvk+ak+vkv_{\mathbf{k}}=a_{\mathbf{k}}^{-}v_{k}+a_{-\mathbf{k}}^{+}v_{k}^{*}

Early Time

v_{\bold{k}}^{\prime \prime} +\left( k^2 -\frac{z^{\prime \prime}}{z} \right)v_{\bold{k}} =0
vk+(k2zz)vk=0v_{\bold{k}}^{\prime \prime} +\left( k^2 -\frac{z^{\prime \prime}}{z} \right)v_{\bold{k}} =0
v_{\bold{k}}^{\prime \prime} +k^2v_{\bold{k}} =0
vk+k2vk=0v_{\bold{k}}^{\prime \prime} +k^2v_{\bold{k}} =0
v_{k}=Ae^{-ik\tau}
vk=Aeikτv_{k}=Ae^{-ik\tau}
v_{k}(\tau _i)=\frac{1}{\sqrt{2k}} e^{-ik\tau}
vk(τi)=12keikτv_{k}(\tau _i)=\frac{1}{\sqrt{2k}} e^{-ik\tau}

Let

v_{k}v_{k}^{*\prime}-v_{k}^{*}v_{k}^{\prime}=i
vkvkvkvk=iv_{k}v_{k}^{*\prime}-v_{k}^{*}v_{k}^{\prime}=i

Zero-Point Fluctuation

\langle v_{\mathbf{k}}v_{\mathbf{k}^{\prime}}\rangle=\langle0|v_{\mathbf{k}}v_{\mathbf{k}^{\prime}}|0\rangle
vkvk=0vkvk0\langle v_{\mathbf{k}}v_{\mathbf{k}^{\prime}}\rangle=\langle0|v_{\mathbf{k}}v_{\mathbf{k}^{\prime}}|0\rangle
=\langle0|(a_{\mathbf{k}}^{-}v_{k}+a_{-\mathbf{k}}^{+}v_{k}^{*})(a_{\mathbf{k}^{\prime}}^{-}v_{k^{\prime}}+a_{-\mathbf{k}^{\prime}}^{+}v_{k^{\prime}}^{*})|0\rangle
=0(akvk+ak+vk)(akvk+ak+vk)0=\langle0|(a_{\mathbf{k}}^{-}v_{k}+a_{-\mathbf{k}}^{+}v_{k}^{*})(a_{\mathbf{k}^{\prime}}^{-}v_{k^{\prime}}+a_{-\mathbf{k}^{\prime}}^{+}v_{k^{\prime}}^{*})|0\rangle
=|v_{k}|^{2}\delta(\mathbf{k}+\mathbf{k}^{\prime})
=vk2δ(k+k)=|v_{k}|^{2}\delta(\mathbf{k}+\mathbf{k}^{\prime})
P_{v}(k)\equiv{|v_{k}|^{2}}
Pv(k)vk2P_{v}(k)\equiv{|v_{k}|^{2}}

Power Spectrum

P_{v}(k)\leftrightarrow\langle v(\tau,\mathbf{x})v(\tau,\mathbf{y})\rangle
Pv(k)v(τ,x)v(τ,y)P_{v}(k)\leftrightarrow\langle v(\tau,\mathbf{x})v(\tau,\mathbf{y})\rangle

Gaussianity

Back to Slow-roll Solutions

v_{k}(\tau)=\sqrt{-\tau}\left[C_{1}H_{\nu}^{(1)}(-k\tau)+C_{2}H_{\nu}^{(2)}(-k\tau)\right]
vk(τ)=τ[C1Hν(1)(kτ)+C2Hν(2)(kτ)]v_{k}(\tau)=\sqrt{-\tau}\left[C_{1}H_{\nu}^{(1)}(-k\tau)+C_{2}H_{\nu}^{(2)}(-k\tau)\right]
v_{k}(\tau)=\sqrt{\frac{2}{\pi}}\left(\frac{C_{1}}{\sqrt{k}}e^{-ik\tau}+\frac{C_{2}}{\sqrt{k}}e^{ik\tau}\right)
vk(τ)=2π(C1keikτ+C2keikτ)v_{k}(\tau)=\sqrt{\frac{2}{\pi}}\left(\frac{C_{1}}{\sqrt{k}}e^{-ik\tau}+\frac{C_{2}}{\sqrt{k}}e^{ik\tau}\right)
C_{1}=\frac{\sqrt{\pi}}{2},\ C_{2}=0
C1=π2, C2=0C_{1}=\frac{\sqrt{\pi}}{2},\ C_{2}=0
v_{k}(\tau)=\frac{\sqrt{\pi}}{2}(-\tau)^{1/2}H_{\nu}^{(1)}(-k\tau)
vk(τ)=π2(τ)1/2Hν(1)(kτ)v_{k}(\tau)=\frac{\sqrt{\pi}}{2}(-\tau)^{1/2}H_{\nu}^{(1)}(-k\tau)
(k\gg aH)
(kaH)(k\gg aH)

Outside-the-Horizon Solution

v_{k}(\tau)=\frac{\sqrt{\pi}}{2} (-\tau)^{1/2} H_{\nu}^{(1)}(-k\tau)
vk(τ)=π2(τ)1/2Hν(1)(kτ)v_{k}(\tau)=\frac{\sqrt{\pi}}{2} (-\tau)^{1/2} H_{\nu}^{(1)}(-k\tau)
v_{k}(\tau)=i\frac{(-\tau)^{1/2}\Gamma(\nu)}{2\sqrt{\pi}}\left(\frac{-k\tau}{2}\right)^{-\nu}
vk(τ)=i(τ)1/2Γ(ν)2π(kτ2)νv_{k}(\tau)=i\frac{(-\tau)^{1/2}\Gamma(\nu)}{2\sqrt{\pi}}\left(\frac{-k\tau}{2}\right)^{-\nu}
(k\ll aH)
(kaH)(k\ll aH)
P_{\mathcal{R}}(k)=\frac{P_{\nu}(k)}{z^{2}}=\frac{|v_{k}|^{2}}{z^{2}}\propto\frac{H_{hc}^{2}}{\epsilon_{hc}}k^{-3}\propto k^{-3-2\epsilon _{hc}-\eta _{hc}}
PR(k)=Pν(k)z2=vk2z2Hhc2ϵhck3k32ϵhcηhcP_{\mathcal{R}}(k)=\frac{P_{\nu}(k)}{z^{2}}=\frac{|v_{k}|^{2}}{z^{2}}\propto\frac{H_{hc}^{2}}{\epsilon_{hc}}k^{-3}\propto k^{-3-2\epsilon _{hc}-\eta _{hc}}

nearly scale-invariant

P_{f}(k)\propto\frac{1}{k^{3}}\leftrightarrow\langle f(\mathbf{x})f(\mathbf{y})\rangle=\langle f(\lambda\mathbf{x})f(\lambda\mathbf{y})\rangle
Pf(k)1k3f(x)f(y)=f(λx)f(λy)P_{f}(k)\propto\frac{1}{k^{3}}\leftrightarrow\langle f(\mathbf{x})f(\mathbf{y})\rangle=\langle f(\lambda\mathbf{x})f(\lambda\mathbf{y})\rangle

At Hubble crossing

Scalar-vector-tensor Decomposition

E_{ij}=E_{ij}^{(1)}+E_{ij}^{(2)}+E_{ij}^{(3)}
Eij=Eij(1)+Eij(2)+Eij(3)E_{ij}=E_{ij}^{(1)}+E_{ij}^{(2)}+E_{ij}^{(3)}
E_{ij}^{(1)}=E_{,ij}^{S}-\frac{1}{3}\delta_{ij}\nabla ^2E^{S}
Eij(1)=E,ijS13δij2ESE_{ij}^{(1)}=E_{,ij}^{S}-\frac{1}{3}\delta_{ij}\nabla ^2E^{S}
E_{ij}^{(2)}=E_{i,j}^{V}+E_{j,i}^{V}
Eij(2)=Ei,jV+Ej,iVE_{ij}^{(2)}=E_{i,j}^{V}+E_{j,i}^{V}
\delta^{ij}E^{V}_{i,j}=0
δijEi,jV=0\delta^{ij}E^{V}_{i,j}=0
\delta^{jk}E^{(3)}_{ij,k}=\delta^{jk}E^{T}_{ij,k}=0
δjkEij,k(3)=δjkEij,kT=0\delta^{jk}E^{(3)}_{ij,k}=\delta^{jk}E^{T}_{ij,k}=0

Scalar-vector-tensor Decomposition

B_{i}=B_{i}^{(1)}+B_{i}^{(2)}=-B^S_{,i}+B_{i}^{V}
Bi=Bi(1)+Bi(2)=B,iS+BiVB_{i}=B_{i}^{(1)}+B_{i}^{(2)}=-B^S_{,i}+B_{i}^{V}
\bold{\nabla}\times\mathbf{B}^{(1)}=0
×B(1)=0\bold{\nabla}\times\mathbf{B}^{(1)}=0
\bold{\nabla} \cdot \bold{B}^{(2)} = 0
B(2)=0\bold{\nabla} \cdot \bold{B}^{(2)} = 0
\to \ \bold{B}^{(1)}=-\bold{\nabla}B^S
 B(1)=BS \to \ \bold{B}^{(1)}=-\bold{\nabla}B^S
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