Physical Cosmology 27/5/2016
Alessandro Melchiorri
alessandro.melchiorri@roma1.infn.it slides can be found here:
oberon.roma1.infn.it/alessandro/cosmo2016
The Cosmic Microwave Background
Discovered By Penzias and Wilson in 1965.
It is an image of the universe at the time of recombination (near
baryon-photons decoupling), when the universe was just a few thousand years old (z~1000).
The CMB frequency spectrum
is a perfect blackbody at T=2.73 K:
this is an outstanding confirmation of the hot big bang model.
Uniform...
Dipole...
Galaxy (z=0)
The Microwave Sky
COBE
Imprint left by primordial
tiny density inhomogeneities (z~1000)..
Fluctuation generator
Fluctuation amplifier
Dens Hot Sm oot e
h
Ra Cool
refi ed
Clum py
CMB map from COBE
1992
CMB map from WMAP
2003
CMB map from Planck
2015
In the past 20 years we measured the CMB sky with improved sensitivity, frequency range and angular resolution.
7 degree 0.3 degree 0.08 degree
Planck maps for 9 frequencies
Main foreground:
Synchrotron
Main
foreground:
Dust
In practice: we see only dust for channels above 353 Ghz
( )
Δ( )
=∑
+(
⋅)
Δ
2 1
2
1 (2 1)
2
1 γ γ
γ π
γ C P
T T T
T
The CMB Angular Power Spectrum
R.m.s. of has power per decade in l:ΔT / T l +
(
l 1)
Cl / 2π( ) ( ) ( )
∑ + ≈ ∫ +
=
Δ l l C d l
l C T
T l
l
rms l ln
2 1 4
1 / 2 2
π π
Doroshkevich, A. G.; Zel'Dovich, Ya. B.; Syunyaev, R. A.
Soviet Astronomy, Vol. 22, p.523, 1978
A Brief History of the CMB Anisotropies Angular Spectrum (Theoretical predictions)
Wilson, M. L.; Silk, J., Astrophysical Journal, Part 1, vol. 243, Jan. 1, 1981, p. 14-25.
1981
A Brief History of the CMB Anisotropies Angular Spectrum (Theoretical predictions)
Bond, J. R.; Efstathiou, G.; Royal Astronomical Society, Monthly Notices (ISSN 0035-8711), vol. 226, June 1, 1987, p. 655-687, 1987
A Brief History of the CMB Anisotropies Angular Spectrum (Theoretical predictions)
Chung-Pei Ma, Edmund Bertschinger, Astrophys.J. 455 (1995) 7-25
A Brief History of the CMB Anisotropies Angular Spectrum (Theoretical predictions)
Hu, Wayne; Scott, Douglas; Sugiyama, Naoshi; White, Martin.
Physical Review D, Volume 52, Issue 10, 15 November 1995, pp.5498-5515
A Brief History of the CMB Anisotropies Angular Spectrum (Theoretical predictions)
A Brief History of the CMB Anisotropies Angular Spectrum (Experimental Data)
In 1995 Big Bang Model was nearly dead…
Collection of CMB anisotropy data from C. Lineweaver et al., 1996
A Brief History of the CMB Anisotropies Angular Spectrum (Experimental Data)
Test flight: P. Mauskopf et al, ApJ letters 536, L59, (2000), astro-ph/9911444 A. Melchiorri et al, ApJ letters 536, L63, (2000),astro-ph/9911445 LDB-B00: P. de Bernardis et al, Nature 404 (2000) 955-969, asro-ph/0004404
A. E. Lange et al, PRD D63 (2001) 042001, astro-ph/0005004
Maxima: A. Jaffe et al, Phys. Rev. Lett, (2001) 86, 3475-3479, astro-ph/0007333 LDB-B01: B. Netterfield et al, ApJ in press, astro-ph/0104460
P. de Bernardis et al, ApJ in press, astro-ph/0105296.
BOOMERanG Experiment
Boomerang’s Track
(1 lap in 10.6 days...)
Blue and Red
are two separate
analyses of same data
Beam
uncertainty (correlated)
1.8% of the sky Boomerang Power Spectrum
20% calibration uncertainty
(correlated)
Error bars
10% correlated
11 days of observations 4 channels at 150GHz
CMB anisotropies pre-WMAP (January 2003)
Wilkinson Microwave Anisotropy Probe
Best fit model
cosmic variance
Temperature
Temperature- polarization
1 deg
85% of sky
Spergel et al, 2003
BOOM03 Flight
Launched: January 6, 2003
! Polarization sensitive receivers 145/245/345 GHz
! Flight January 2003
! 195 hours (11.7 days) of data fsky= 1.8%
! First results published in July 2005
– Masi et al. astro-ph/0507509
– Jones et al. astro-ph/0507494
– Piacentini et al. astro-ph/0507507
– Montroy et al. astro-ph/0507514
– MacTavish et al. astro-ph/0507503
Feet
[MacTavish et al. 2005]
A Brief History of the CMB Anisotropies Angular Spectrum (2012)
Planck 2015
CMB observations
@May 2016
The shape of the CMB angular spectrum depends on
cosmological parameters !
How to get a bound on a cosmological parameter
DATA
Fiducial cosmological model:
(Ωbh2 , Ωmh2 , h , ns , τ, Σmν )
PARAMETER ESTIMATES
Cosmic Microwave Background Anisotropies: Theory
http://space.mit.edu/home/tegmark/varenna.html
CMB Anisotropy: BASICS
• Friedmann Flat Universe with 5 components:
Baryons, Cold Dark Matter (w=0, always),
Photons, Massless Neutrinos, Cosmological Constant.
• Linear Perturbation theory. Conformal Newtonian Gauge. Scalar modes only.
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• Perturbation Variables:
CMB Anisotropy: BASICS
Key point: we work in Fourier space :
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CMB Anisotropy: BASICS
CDM:
Baryons:
Photons:
Neutrinos:
Their evolution is governed by a nasty set of coupled partial differential equations:
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Numerical Integration
- Early Codes (1995) integrate the full set of equations (about 2000 for each k mode, approx, 2 hours CPU time for obtaining one single spectrum).
COSMICS first public Boltzmann code http://arxiv.org/abs/astro-ph/9506070.
- Major breakthrough with line of sight integration method with CMBFAST
(Seljak&Zaldarriaga, 1996, http://arxiv.org/abs/astro-ph/9603033). (5 minutes of CPU time)
- Most supported and updated code at the moment CAMB (Challinor, Lasenby, Lewis), http://arxiv.org/abs/astro-ph/9911177 (Faster than CMBFAST).
Another recent code (CLASS) is available at http://class-code.net.
- Both on-line versions of CAMB and CMBFAST available on LAMBDA website.
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CMB Anisotropy: BASICS
CDM:
Baryons:
Photons:
Neutrinos:
Their evolution is governed by a nasty set of coupled partial differential equations:
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First Pilar of the standard model of structure formation:
Standard model: Evolution of perturbations is passive and coherent.
Active and decoherent models of structure formation
(i.e. topological defects see Albrecht et al, http://arxiv.org/
abs/astro-ph/9505030):
Linear differential operator
Perturbation Variables
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Oscillations supporting evidence for passive and coherent scheme.
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CMB anisotropies
The temperature fluctuation in a direction can be expressed as a line-of-sight integral:
We have 3 terms:
Gravity: photons coming out from CDM potential wells will suffer a redshift.
Intrinsic: where we have more baryons we have more photons (tight coupling) and a blue shift.
Doppler: baryons move and they leave a doppler effect when scattering off photons.
CMB anisotropies @LSS
We have that:
ne is the free electron fraction.
In the limit of infinitely thin LSS the function
approaches to a Dirac
delta-function centered at recombination.
CMB anisotropies @LSS
As a first approximation we have just to compute the value of the first three terms at the time of
recombination !
For simplicity we also neglect the last term.
This term is called Integrated Sachs-Wolfe effect and is zero when the universe is dominated by matter.
Equations for CDM
Continuity Euler
Poisson
Here we have the
total density perturbation.
We will assume to be dominated by CDM.
Equation for photons
The relativistic species (photons and neutrinos) are not
characterized by a simple velocity field, but by a distribution function f whose evolution is governed by the Boltzmann
equation,
Here p is the magnitude of the momentum, γ is a direction cosine of the momentum, and C is a collision term having to do with scattering.
Primary anisotropies
We will begin by assuming that, after the end of the radiation epoch, most of the mass in the Universe is in the form of cold dark matter:
Then the gravitational potential is completely determined by the CDM.
We also make another approximation: we assume tight coupling between photons and baryons. Specifically, we
assume that the mean free time τ between photon collisions is small compared to the other important time scales:
This is an excellent approximation right up until around the time of last scattering.
Primary anisotropies
In a tight coupling approximation, the photon and baryon densities are coupled adiabatically:
The behavior of the photon-baryon fluid is therefore
characterized by a single variable. We will find it convenient to take as our variable the fractional temperature fluctuation Θ,
given by:
Primary anisotropies
With these approximations, the dynamics of the photon-baryon fluid is described by the single equation:
Where:
Conformal time
Baryon/photon ratio
Gravitational driving
Primary anisotropies
It is possible to write:
With the approximations that we’re making, there are no
anisotropic stresses, so the two gravitational potentials are simply related to each other:
Primary anisotropies
The Poisson equation:
In Fourier space since k is co-moving we have:
So if the expansion is matter dominated we have that
δ≈a and by using the Poisson equation in Fourier space we see that Ψ is constant with time.
The gravitational driving term therefore simplifies to
Primary anisotropies
If we assume that R is also slowly varying with time, we have:
This is the equation for a simple harmonic oscillator, with solution:
where:
Primary anisotropies
In this approximation, then, each Fourier mode represents an acoustic plane wave propagating at sound speed cs.
There is a simple physical picture underlying this result. The baryon-photon fluid wants to fall into the potential wells, but it is supported by radiation pressure. The balance between
pressure and gravity sets up acoustic oscillations.
Let’s make things even simpler and set R = 0. Then
In many theories, the initial perturbation is adiabatic, meaning that the matter and radiation fluctuations are the same at any particular point. In this case we have the initial conditions:
Primary anisotropies
In this very simple case we therefore have:
Since we have to compute:
Ignoring the Doppler term for the moment, note that the other two terms give a pure cosine oscillation in k:
LS
with frequency given by the sound horizon at LS (≈10 Mpc).
Primary anisotropies
Let us consider now the Doppler term.Using the continuity equation and the relation δ = 3Θ, we find that
Since for R=0 we have and since we assume ψ time independent, we have:
Since the r.m.s. value of is , the r.m.s. Doppler contribution to equation is:
This has the same amplitude as the (Θ + Ψ) contribution, but is 90 degrees out of phase in both time (it goes like a sine
instead of a cosine) and space (it has an extra factor of i). This has the rather disastrous consequence of completely erasing the Doppler peaks: the total ∆T/T is the quadrature sum (we are considering the power spectrum) is:
Primary anisotropies
The k dependence, which led to the peaks, is gone.
Primary anisotropies
Let us remove the assumption R=0, but keep the
approximation that R is time-independent. Then the solution for Θ(η) changes in two ways.
1) The sound speed gets smaller by a factor:
2) The driving term F (η) gets bigger by a factor The adiabatic solution to Θ is now:
By allowing R to be nonzero, we have increased the amplitude of the cosine oscillations by a factor (1 + 3R). Furthermore,
there is now an offset in the combined Sachs-Wolfe and
adiabatic contributions to ∆T/T: in the limit R → 0, we found that Θ + Ψ oscillated symmetrically about zero; now it
oscillates about −RΨ.
Primary anisotropies
We also have a smaller amplitude of the Doppler contribution, since v is proportional to cs and cs has gotten smaller. Since the cosine oscillations are now larger in amplitude than the sine oscillations, we do indeed expect to see a series of peaks at
Why does including the dynamical effect of the baryons effect these changes in the solution? The essential reason is that
baryons contribute to the effective mass of the photon-baryon fluid, but not to the pressure. The effect of the baryons,
therefore, is to slow down the oscillations, and also to make the fluid fall deeper into the potential wells.
Primary anisotropies
Hu, Sugiyama, Silk, Nature 1997, astro-ph/9604166
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Baryonic abundance and CMB peaks
CMB spectrum is therefore strongly dependent from R, i.e. the baryon
abundance.
Increasing the baryon density
increases the odd peaks and erases the even peaks.
Projection
A mode with wavelength λ will show up on an angular scale θ ∼ λ/R, where R is the distance to the last-scattering surface, or in other words, a mode with wavenumber k shows up at multipoles l∼k.
The spherical Bessel function jl(x) peaks at x ∼ l, so a single Fourier mode k does indeed contribute
most of its power around multipole lk = kR, as expected. However, as the figure shows, jl does have significant power beyond the first peak,
meaning that the power contributed by a Fourier mode “bleeds” to l-values different from lk.
Moreover for an open universe (K is the curvature) :
l=30
l=60
l=90
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Projection
In Fourier space we have oscillations With frequency (or physical scale):
In Legendre space oscillations are smeared and have a frequency that Depends on the angular diameter distance at recombination.
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