3) Initial conditions sets initial conditions for hot big bang. Who sets up initial conditions for inflation?
Properties of initial conditions:
Scale invariant, no fluctuation in composition, gaussian, gravity waves (?!)
eps=Hdot/H^2 << 1
k^3 P(k) \ptopto k^{ns-1} ~ constant mostly for most models. The exact value depends upon the inflationary model.
Gaussian - but small departures can vary depending upon the parameters of the inflationary paradigm.
Comoving wavenumber:
Figure:
log(scale) vs log a
Comoving horizon ~1/(aH) grows as a function of a. For H constant, the comoving horizon grows as a goes down, and the observable scales are within the horizon.
Modes grow out of the horizon sequentially, largest first, and come in to the horizon smallest first...
There is an attractor solution to how the modes behave once they go out of the horizon. So long wavelength modes do not affect shorter ones.
ds^2=-dt^2 + e^{2rho(t)+2xi(t)} dx^2
Where rho = ln(a).
rhodot^2=8piG/3(1/2phidot^2+V(phi))
rhoddot=4 piG \phidot^2
phiddot+3rhodotphidot+Vdash(phi)=0
delta phi ~H.
xi = H\delta phi/phi. Time delay fluctuations
epsilon=-Hdot/H^2=Kin/(Kin+pot energy)
Amplitude of the gravity waves: H/Mpl.
Time translation symmetry same as scale invariance.
Deviations from deSitter (Hdot and Hddot) enter the departure from scale invariance.
Gaussian fluctuations:
If V(phi) nequal to 1/2m^2 phi^2, then the fluctuations are gaussian. Harmonic oscillator solutions.
V''' will give non-gaussianity. Gravitational interactions do not contribute much.
The Lagrangian in case of V''' is:
L3=V'''(delta phi)^3.
L2=0.5\del(delta phi)^2
del delta phi is of order H delta phi.
L3/L2=V''' del phi/H^2 ~ V''' phidot /H^3 Hdel phi/phidot ~ fNL zeta. These lead to very small non-gaussianities of the order fnl ~ eps^2. This is because V has to obey the slow roll conditions.
But the fluctuations have self gravity. And this causes terms of the following type:
(del_mu delta phi)^2 del phidot.
NG is of the order L3/L2 ~ eps zeta where zeta ~10^-5.
Fluctuations can be in composition or in initial conditions, A, ns, rather than in Omegab h^2 and Omegam h^2. The first ones cannot be erased, the second ones can be.
1) Source of non-Gaussianity are small in both the above cases. So fNL is large for multifield, when zeta differs from being a clock.
2) cs<<1, the fluctuation speed and how it differs from the speed of light.
1) 60 e-folds.
2) Density fluctuations
3) Initial conditions sets initial conditions for hot big bang. Who sets up initial conditions for inflation?
Properties of initial conditions:
Scale invariant, no fluctuation in composition, gaussian, gravity waves (?!)
eps=Hdot/H^2 << 1
k^3 P(k) \ptopto k^{ns-1} ~ constant mostly for most models. The exact value depends upon the inflationary model.
Gaussian - but small departures can vary depending upon the parameters of the inflationary paradigm.
Comoving wavenumber:
Figure:
log(scale) vs log a
Comoving horizon ~1/(aH) grows as a function of a. For H constant, the comoving horizon grows as a goes down, and the observable scales are within the horizon.
Modes grow out of the horizon sequentially, largest first, and come in to the horizon smallest first...
There is an attractor solution to how the modes behave once they go out of the horizon. So long wavelength modes do not affect shorter ones.
ds^2=-dt^2 + e^{2rho(t)+2xi(t)} dx^2
Where rho = ln(a).
rhodot^2=8piG/3(1/2phidot^2+V(phi))
rhoddot=4 piG \phidot^2
phiddot+3rhodotphidot+Vdash(phi)=0
delta phi ~H.
xi = H\delta phi/phi. Time delay fluctuations
epsilon=-Hdot/H^2=Kin/(Kin+pot energy)
Amplitude of the gravity waves: H/Mpl.
Time translation symmetry same as scale invariance.
Deviations from deSitter (Hdot and Hddot) enter the departure from scale invariance.
Gaussian fluctuations:
If V(phi) nequal to 1/2m^2 phi^2, then the fluctuations are gaussian. Harmonic oscillator solutions.
V''' will give non-gaussianity. Gravitational interactions do not contribute much.
The Lagrangian in case of V''' is:
L3=V'''(delta phi)^3.
L2=0.5\del(delta phi)^2
del delta phi is of order H delta phi.
L3/L2=V''' del phi/H^2 ~ V''' phidot /H^3 Hdel phi/phidot ~ fNL zeta. These lead to very small non-gaussianities of the order fnl ~ eps^2. This is because V has to obey the slow roll conditions.
But the fluctuations have self gravity. And this causes terms of the following type:
(del_mu delta phi)^2 del phidot.
NG is of the order L3/L2 ~ eps zeta where zeta ~10^-5.
Fluctuations can be in composition or in initial conditions, A, ns, rather than in Omegab h^2 and Omegam h^2. The first ones cannot be erased, the second ones can be.
1) Source of non-Gaussianity are small in both the above cases. So fNL is large for multifield, when zeta differs from being a clock.
2) cs<<1, the fluctuation speed and how it differs from the speed of light.