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The Cosmic Microwave Background


Academic year: 2021

Condividi "The Cosmic Microwave Background"


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Physical Cosmology 27/5/2016

Alessandro Melchiorri

alessandro.melchiorri@roma1.infn.it slides can be found here:



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.




Galaxy (z=0)

The Microwave Sky


Imprint left by primordial

tiny density inhomogeneities (z~1000)..


Fluctuation generator

Fluctuation amplifier

Dens Hot Sm oot e


Ra Cool

refi ed

Clum py


CMB map from COBE



CMB map from WMAP



CMB map from Planck



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:





In practice: we see only dust for channels above 353 Ghz


( )


( )






2 1


1 (2 1)


1 γ γ

γ π

γ C P



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


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.


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



1.8% of the sky Boomerang Power Spectrum

20% calibration uncertainty


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- 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




[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


Fiducial cosmological model:

bh2 , Ωmh2 , h , ns , τ, Σmν )



Cosmic Microwave Background Anisotropies: Theory



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.



• Perturbation Variables:

CMB Anisotropy: BASICS

Key point: we work in Fourier space :



CMB Anisotropy: BASICS





Their evolution is governed by a nasty set of coupled partial differential equations:



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.



CMB Anisotropy: BASICS





Their evolution is governed by a nasty set of coupled partial differential equations:



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/


Linear differential operator

Perturbation Variables



Oscillations supporting evidence for passive and coherent scheme.



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


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


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:


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:



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:


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




Baryonic abundance and CMB peaks

CMB spectrum is therefore strongly dependent from R, i.e. the baryon


Increasing the baryon density

increases the odd peaks and erases the even peaks.



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) :







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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