Lecture 3 - Temperature anisotropy from sound waves (continued) - - - PowerPoint PPT Presentation
Lecture 3 - Temperature anisotropy from sound waves (continued) - Cosmological parameter dependence of the temperature power spectrum Stone: Fluctuations entering the horizon This is a tricky concept, but it is important Suppose
(continued)
temperature power spectrum
including the ones that exceed the Hubble length (which we loosely call our “horizon”)
fluctuations”, but just assume their existence
longer and longer wavelengths
10 Gpc/h today 1 Gpc/h today 100 Mpc/h today 10 Mpc/h today 1 Mpc/h today “enter the horizon” Radiation Era Last scattering Matter Era
radiation equality?
Entered the horizon during the radiation era
by [ ]2 with q -> l/rL
adiabatic or not?
[We will show it later.]
by [ ]2 with q -> l/rL
Going back to the original tight-coupling equation..
Then the equation simplifies to
where
Then the equation simplifies to
where
The solution is
Then the equation simplifies to
where
The solution is
where
and the scalar curvature perturbation, ψ.
and the scalar curvature perturbation, ψ.
and the scalar curvature perturbation, ψ.
Will come back to this later. For now, let’s ignore any viscosity.
and the scalar curvature perturbation, ψ.
Will come back to this later. For now, let’s ignore any viscosity.
and the scalar curvature perturbation, ψ.
“non-adiabatic” pressure
and the scalar curvature perturbation, ψ.
“non-adiabatic” pressure We shall ignore this
where
damp
Kodama & Sasaki (1986, 1987)
where
damp
Poisson Equation & oscillation solution for radiation
where
damp
(super-horizon scales, q << aH) regardless of the contents of the Universe
radiation-dominated, matter-dominated, or whatever
Bardeen, Steinhardt & Turner (1983); Weinberg (2003); Lyth, Malik & Sasaki (2005)
Integrate
, then
integration constant Bardeen, Steinhardt & Turner (1983); Weinberg (2003); Lyth, Malik & Sasaki (2005)
, then
integration constant
Bardeen, Steinhardt & Turner (1983); Weinberg (2003); Lyth, Malik & Sasaki (2005)
The solution is
where Kodama & Sasaki (1986, 1987); Baumann, Green, Meyers & Wallisch (2016)
The solution is
where i.e.,
ADI ADI
Kodama & Sasaki (1986, 1987); Baumann, Green, Meyers & Wallisch (2016)
The adiabatic solution is with
Therefore, the solution is a pure cosine
Kodama & Sasaki (1986, 1987); Baumann, Green, Meyers & Wallisch (2016)
Silk (1968) “Silk damping”
The solution is
where
The solution is
where Chluba & Grin (2013) non-zero value!
The solution is
where Hu & Sugiyama (1996) Bashinsky & Seljak (2004) Phase shift!
The solution is
where Hu & Sugiyama (1996) Bashinsky & Seljak (2004) Phase shift!
damps exponentially
to a higher order in the tight-coupling approximation
between photons and baryons is small. We write
[d is an arbitrary dimensionless variable]
*In this limit, viscosity πγ is exponentially suppressed. This result comes from the Boltzmann equation but we do not derive it here. It makes sense physically.
between photons and baryons is small. We write
[d2 is an arbitrary dimensionless variables]
where
the scale factor, we obtain
equation in the 2nd-order tight coupling. We do not derive it
Kaiser (1983)
the scale factor, we obtain
equation in the 2nd-order tight coupling. We do not derive it
Kaiser (1983) given by the velocity potential
dynamics
we obtain, for q >> aH,
New term, giving damping! where
we obtain, for q >> aH,
New term, giving damping! where Important for high frequencies (large multipoles)
we obtain, for q >> aH,
New term, giving damping! Exponential dampling! SOLUTION:
we obtain, for q >> aH,
New term, giving damping! Exponential dampling! SOLUTION: Silk Silk “diffusion length” = length traveled by photon’s random walks
Planck Collaboration (2016)
Landau
is thus exp(–2q2/qsilk2)
rL/√2 = 1370. Seems too large, compared to the exact calculation
scattering surface, σ~250 K
Planck Collaboration (2016)
Planck Collaboration (2016)
high frequencies during the radiation era:
last-scattering surface (when R is no longer small)
high frequencies during the radiation era:
last-scattering surface (when R is no longer small)
Slightly improved solution, with a weak time dependence of R using the WKB method [Peebles & Yu (1970)]
given by setting R=0 because sound waves are not important at large scales
Weinberg “Cosmology”, Eq. (6.5.7) where T(q), S(q), θ(q) are “transfer functions” that smoothly interpolate two limits as
q q q
with qEQ = aEQHEQ ~ 0.01 Mpc–1, giving lEQ=qEQrL ~ 140
“EQ” for “matter-radiation Equality epoch”
given by setting R=0 because sound waves are not important at large scales
Weinberg “Cosmology”, Eq. (6.5.7) where T(q), S(q), θ(q) are “transfer functions” that smoothly interpolate two limits as
q q q
with qEQ = aEQHEQ ~ 0.01 Mpc–1, giving lEQ=qEQrL ~ 140
“EQ” for “matter-radiation Equality epoch”
given by setting R=0 because sound waves are not important at large scales
Weinberg “Cosmology”, Eq. (6.5.7) where T(q), S(q), θ(q) are “transfer functions” that smoothly interpolate two limits as
q q q
with qEQ = aEQHEQ ~ 0.01 Mpc–1, giving lEQ=qEQrL ~ 140
“EQ” for “matter-radiation Equality epoch”
given by setting R=0 because sound waves are not important at large scales
Weinberg “Cosmology”, Eq. (6.5.7)
q q q
q -> 0(*)
This should agree with the Sachs-Wolfe result: Φ/3; thus,
given by setting R=0 because sound waves are not important at large scales
Weinberg “Cosmology”, Eq. (6.5.7)
q q q
Shift the zero-point of
Reduce the amplitude of
` ≈ 302 × qrs/⇡
` ≈ 302 × qrs/⇡
B
t d u e t
e c a y i n g p
e n t i a l d u r i n g t h e r a d i a t i
e r a
` ≈ 302 × qrs/⇡
Silk damping
` ≈ 302 × qrs/⇡
` ≈ 302 × qrs/⇡
Zero-point shift of the
` ≈ 302 × qrs/⇡
WKB factor (1+R)-1/4 and Silk damping compensate the zero- point shift
Weinberg “Cosmology”, Eq. (6.5.7) where T(q), S(q), θ(q) are “transfer functions” that smoothly interpolate two limits as
q q q
“EQ” for “matter-radiation Equality epoch”
` ≈ 302 × qrs/⇡
Smaller matter density
power at high multipoles by a factor of 5 compared to
the Sachs-Wolfe plateau
the odd peaks relative to the even peaks
boosting of the 3rd and 5th peaks not so