Perturbations from Inflation
Heling Deng
- Perturbations
- Inflation
- Perturbations from inflation
- Perturbations
- Inflation
- Perturbations from inflation
f(t,x^{i})
f(t,xi)
g_{\mu\nu}=\bar{g}_{\mu\nu}+\delta g_{\mu\nu}
gμν=gˉμν+δgμν
T_{\mu\nu}=\bar{T}_{\mu\nu}+\delta T_{\mu\nu}
Tμν=Tˉμν+δTμν
\delta G_{\mu\nu}\propto\delta T_{\mu\nu}
δGμν∝δTμν
Metric
Matter
Einstein Equations
=\bar{f}(t)+\delta f(t,x^{i})
=fˉ(t)+δf(t,xi)
{tr}E=0
trE=0
ds^{2}=-dt^{2}+a(t)^{2}\delta_{ij}dx^{i}dx^{j}
ds2=−dt2+a(t)2δijdxidxj
Background metric
\delta g_{00}=-2\Phi
δg00=−2Φ
\delta g_{0i}=aB_{i}
δg0i=aBi
\delta g_{ij}=a^{2}\left(2\Psi\delta_{ij}+E_{ij}\right)
δgij=a2(2Ψδij+Eij)
Perturbations
\Phi,\Psi,B,E
Φ,Ψ,B,E
B_i^{V},\ E_i^{V}
BiV, EiV
E_{ij}^{T}
EijT
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[(1−2Ψ)δij+E,ij]dxidxj
Background fluid
\bar{T}_{\ \nu}^{\mu}=\left(\bar{\rho}+\bar{p}\right)\bar{u}^{\mu}\bar{u}_{\nu}+\delta_{\nu}^{\mu}\bar{p}
Tˉ νμ=(ρˉ+pˉ)uˉμuˉν+δνμpˉ
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=(ρˉ+δρ+pˉ+δp)(uˉμ+δuμ)(uˉν+δuν)+δνμ(pˉ+δp)
\delta T_{\ 0}^{0}=-\delta\rho
δT 00=−δρ
\delta T_{\ i}^{0}=a^{-2}(\bar{\rho}+\bar{p})(aB_{i}-\delta u_{i})
δT i0=a−2(ρˉ+pˉ)(aBi−δui)
\delta T_{\ 0}^{i}=(\bar{\rho}+\bar{p})\delta u_{i}
δT 0i=(ρˉ+pˉ)δui
\delta T_{\ j}^{i}=\delta p\delta_{ij}
δT ji=δpδij
Perturbations
Gauge
Gauge
t\to t+\alpha(t,\bold{x})
t→t+α(t,x)
\bold{x}\to \bold{x}+\bold{\nabla} \beta(t,\bold{x})
x→x+∇β(t,x)
\Phi\to\Phi-\dot{\alpha}
Φ→Φ−α˙
\Psi\to\Psi+H\alpha
Ψ→Ψ+Hα
B\to B+\frac{\alpha}{a}-a\dot{\beta}
B→B+aα−aβ˙
E\to E-\beta
E→E−β
\Phi_{B}\equiv\Phi-\frac{d}{dt}\left[a^{2}\left(\dot{E}-\frac{B}{a}\right)\right]
ΦB≡Φ−dtd[a2(E˙−aB)]
\Psi_{B}\equiv\Psi+a^{2}H\left(\dot{E}-\frac{B}{a}\right)
ΨB≡Ψ+a2H(E˙−aB)
Gauge
\delta\rho\to\delta\rho-\dot{\bar{\rho}}\alpha
δρ→δρ−ρˉ˙α
\delta p\to\delta p-\dot{\bar{p}}\alpha
δp→δp−pˉ˙α
\delta u\to\delta u+\alpha
δu→δu+α
\delta\rho_B\equiv\delta\rho-a^{2}\dot{\bar{\rho}}\left(\dot{E}-\frac{B}{a}\right)
δρB≡δρ−a2ρˉ˙(E˙−aB)
\delta u_B\equiv\delta u + a^{2}\left(\dot{E}-\frac{B}{a}\right)
δuB≡δu+a2(E˙−aB)
\delta p_B\equiv\delta p-a^{2}\dot{\bar{p}}\left(\dot{E}-\frac{B}{a}\right)
δpB≡δp−a2pˉ˙(E˙−aB)
t\to t+\alpha(t,\bold{x})
t→t+α(t,x)
\bold{x}\to \bold{x}+\bold{\nabla} \beta(t,\bold{x})
x→x+∇β(t,x)
Choosing a Gauge
Newtonian Gauge
B=E=0
B=E=0
ds^{2}=-(1+2\Phi)dt^{2}+a^{2}(1-2\Psi)d\bold{x}^2
ds2=−(1+2Φ)dt2+a2(1−2Ψ)dx2
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[(1−2Ψ)δij+E,ij]dxidxj
Synchronous
\Phi=B=0
Φ=B=0
Comoving
\delta u=0
δu=0
Uniform-density
\delta \rho=0
δρ=0
Spatially-flat
\Psi=E = 0
Ψ=E=0
- Perturbations
- Inflation
- Perturbations from inflation
Comoving Radius
\frac{1}{aH}
aH1
today
recombination
inflation
hot big bang
Time
Inflation:
\frac{d}{dt} \left( \frac{1}{aH} \right) < 0
dtd(aH1)<0
\Leftrightarrow
⇔
\ddot{a} > 0
a¨>0
\Leftrightarrow
⇔
\rho + 3p < 0
ρ+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=∂μϕ∂νϕ+δνμ[21(∂ϕ)2−V(ϕ)]
\rho = \frac{1}{2}\dot\phi^2 + V
ρ=21ϕ˙2+V
p = \frac{1}{2}\dot\phi^2 - V
p=21ϕ˙2−V
\ddot{\phi}+3H\dot{\phi}+V^\prime=0
ϕ¨+3Hϕ˙+V′=0
KG Equation
Slow-roll Approximation
\dot{\phi}^2 \ll V,
ϕ˙2≪V,
Slow-roll conditions
|\ddot{\phi}| \ll |V^\prime|
∣ϕ¨∣≪∣V′∣
\epsilon_V\equiv \frac{M_{pl}^2}{2} \left( \frac{V^\prime}{V} \right) ^2 \ll 1
ϵV≡2Mpl2(VV′)2≪1
\eta_V \equiv M_{pl}^2 \frac{|V^{\prime \prime}|}{V} \ll 1
ηV≡Mpl2V∣V′′∣≪1
Slow-roll parameters
Hubble hierarchy parameters
\epsilon\equiv-\frac{\dot{H}}{H^{2}} \ll 1
ϵ≡−H2H˙≪1
\eta\equiv\frac{\dot{\epsilon}}{H\epsilon} \ll 1
η≡Hϵϵ˙≪1
- Perturbations
- Inflation
- Perturbations from inflation
Comoving Radius
\frac{1}{aH}
aH1
today
reheating
recombination
inflaton
photons, neutrinos, electrons, DM...
Time
\phi=\bar{\phi}+\delta \phi
ϕ=ϕˉ+δϕ
Comoving Radius
(aH)^{-1}
(aH)−1
today
recombination
\dot{\mathcal{R}}_k\approx0
R˙k≈0
Time
"Conservation" Outside the Horizon
\mathcal{R}\equiv-\Psi+H\delta u
R≡−Ψ+Hδu
for
k \ll aH
k≪aH
\dot{\mathcal{R}}_k\approx0
R˙k≈0
"adiabatic solution"
\left(p=p(\rho)\right)
(p=p(ρ))
Comoving Radius
(aH)^{-1}
(aH)−1
today
recombination
\dot{\mathcal{R}}_k\approx0
R˙k≈0
Time
Mukhanov-Sasaki Equation
Comoving gauge
\delta u = 0 \ (\delta \phi = 0),\ E = 0
δu=0 (δϕ=0), E=0
v_{\bold{k}}^{\prime \prime} +\left( k^2 -\frac{z^{\prime \prime}}{z} \right)v_{\bold{k}} =0
vk′′+(k2−zz′′)vk=0
where
v_{\bold{k}}=z\mathcal{R}_{\bold{k}},
vk=zRk,
\mathcal{R}\equiv-\Psi+H\delta u
R≡−Ψ+Hδu
^{\prime} = \frac{d}{d\tau} = a\frac{d}{dt}
′=dτd=adtd
z = \frac{a\dot{\bar{\phi}}}{H},
z=Haϕˉ˙,
Naïve de Sitter Solutions
v_{\bold{k}}^{\prime \prime} +\left( k^2 -\frac{z^{\prime \prime}}{z} \right)v_{\bold{k}} =0
vk′′+(k2−zz′′)vk=0
- Subhorizon
(k \gg aH)
(k≫aH)
- Superhorizon
(k \ll aH)
(k≪aH)
v_{\bold{k}} \propto z
vk∝z
oscillation
"frozen"
v_{\bold{k}}\propto e^{\pm ik\tau}
vk∝e±ikτ
\to \ \mathcal{R}_{\bold{k}}=z^{-1} v_{\bold{k}}\propto const
→ Rk=z−1vk∝const
\frac{z^{\prime\prime}}{z}\sim(aH)^{2}
zz′′∼(aH)2
Slow-roll Solutions
v_{\bold{k}}^{\prime \prime} +\left( k^2 -\frac{z^{\prime \prime}}{z} \right)v_{\bold{k}} =0
vk′′+(k2−zz′′)vk=0
v_{\mathbf{k}}^{\prime\prime}+\left(k^{2}-\frac{\nu^{2}-\frac{1}{4}}{\tau^{2}}\right)v_{\mathbf{k}}=0
vk′′+(k2−τ2ν2−41)vk=0
where
\nu\equiv\frac{3}{2}+\epsilon+\frac{1}{2}\eta,
ν≡23+ϵ+21η,
\epsilon=-\frac{\dot{H}}{H^{2}},
ϵ=−H2H˙,
\eta=\frac{\dot{\epsilon}}{H\epsilon}
η=Hϵϵ˙
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)]
initial condition?
Quantization
v_{\bold{k}}^{\prime \prime} +\left( k^2 -\frac{z^{\prime \prime}}{z} \right)v_{\bold{k}} =0
vk′′+(k2−zz′′)vk=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)=∫(2π)3/2d3k[ak−vk(τ)eik⋅x+ak+vk∗(τ)e−ik⋅x]
\left[v(\tau,\mathbf{x}),v^\prime(\tau,\mathbf{y})\right]=i\delta(\mathbf{x}-\mathbf{y})
[v(τ,x),v′(τ,y)]=iδ(x−y)
\left[a_{\mathbf{k}}^{-},a_{\mathbf{k}^{\prime}}^{+}\right]=\delta(\mathbf{k}-\mathbf{k}^{\prime})
[ak−,ak′+]=δ(k−k′)
v_{k}v_{k}^{*\prime}-v_{k}^{*}v_{k}^{\prime}=i
vkvk∗′−vk∗vk′=i
v_{\mathbf{k}}=a_{\mathbf{k}}^{-}v_{k}+a_{-\mathbf{k}}^{+}v_{k}^{*}
vk=ak−vk+a−k+vk∗
Early Time
v_{\bold{k}}^{\prime \prime} +\left( k^2 -\frac{z^{\prime \prime}}{z} \right)v_{\bold{k}} =0
vk′′+(k2−zz′′)vk=0
v_{\bold{k}}^{\prime \prime} +k^2v_{\bold{k}} =0
vk′′+k2vk=0
v_{k}=Ae^{-ik\tau}
vk=Ae−ikτ
v_{k}(\tau _i)=\frac{1}{\sqrt{2k}} e^{-ik\tau}
vk(τi)=2k1e−ikτ
Let
v_{k}v_{k}^{*\prime}-v_{k}^{*}v_{k}^{\prime}=i
vkvk∗′−vk∗vk′=i
Zero-Point Fluctuation
\langle v_{\mathbf{k}}v_{\mathbf{k}^{\prime}}\rangle=\langle0|v_{\mathbf{k}}v_{\mathbf{k}^{\prime}}|0\rangle
⟨vkvk′⟩=⟨0∣vkvk′∣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
=⟨0∣(ak−vk+a−k+vk∗)(ak′−vk′+a−k′+vk′∗)∣0⟩
=|v_{k}|^{2}\delta(\mathbf{k}+\mathbf{k}^{\prime})
=∣vk∣2δ(k+k′)
P_{v}(k)\equiv{|v_{k}|^{2}}
Pv(k)≡∣vk∣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)⟩
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{\frac{2}{\pi}}\left(\frac{C_{1}}{\sqrt{k}}e^{-ik\tau}+\frac{C_{2}}{\sqrt{k}}e^{ik\tau}\right)
vk(τ)=π2(kC1e−ikτ+kC2eikτ)
C_{1}=\frac{\sqrt{\pi}}{2},\ C_{2}=0
C1=2π, C2=0
v_{k}(\tau)=\frac{\sqrt{\pi}}{2}(-\tau)^{1/2}H_{\nu}^{(1)}(-k\tau)
vk(τ)=2π(−τ)1/2Hν(1)(−kτ)
(k\gg aH)
(k≫aH)
Outside-the-Horizon Solution
v_{k}(\tau)=\frac{\sqrt{\pi}}{2} (-\tau)^{1/2} H_{\nu}^{(1)}(-k\tau)
vk(τ)=2π(−τ)1/2Hν(1)(−kτ)
v_{k}(\tau)=i\frac{(-\tau)^{1/2}\Gamma(\nu)}{2\sqrt{\pi}}\left(\frac{-k\tau}{2}\right)^{-\nu}
vk(τ)=i2π(−τ)1/2Γ(ν)(2−kτ)−ν
(k\ll aH)
(k≪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)=z2Pν(k)=z2∣vk∣2∝ϵhcHhc2k−3∝k−3−2ϵhc−ηhc
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)∝k31↔⟨f(x)f(y)⟩=⟨f(λx)f(λy)⟩
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}^{(1)}=E_{,ij}^{S}-\frac{1}{3}\delta_{ij}\nabla ^2E^{S}
Eij(1)=E,ijS−31δij∇2ES
E_{ij}^{(2)}=E_{i,j}^{V}+E_{j,i}^{V}
Eij(2)=Ei,jV+Ej,iV
\delta^{ij}E^{V}_{i,j}=0
δijEi,jV=0
\delta^{jk}E^{(3)}_{ij,k}=\delta^{jk}E^{T}_{ij,k}=0
δjkEij,k(3)=δjkEij,kT=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+BiV
\bold{\nabla}\times\mathbf{B}^{(1)}=0
∇×B(1)=0
\bold{\nabla} \cdot \bold{B}^{(2)} = 0
∇⋅B(2)=0
\to \ \bold{B}^{(1)}=-\bold{\nabla}B^S
→ B(1)=−∇BS
Perturbations from Inflation Heling Deng