Physical Cosmology 19/5/2017

Physical Cosmology 19/5/2017

Alessandro Melchiorri

[email protected] slides can be found here:

oberon.roma1.infn.it/alessandro/cosmo2017

Cosmological «Circuit»

Generator of Perturbations (Inflation)

Amplifier (Gravity)

Low band pass filter.

Cosmological and

Astrophysical effects

Tend to erase small scale (large k) perturbations

Power Spectrum

Each Fourier component is a complex number, which can be written in the form

The mean square amplitude of the Fourier components defines the power spectrum :

where the average is taken over all possible orientations of the wavenumber. If δ(⃗r) is isotropic, then no information is lost,

statistically speaking, if we average the power spectrum over all angles and we get an isotropic power spectrum:

Correlation function

Let us consider the autocorrelation function of the density field (usually called the correlation function):

Where the brackets indicates an average over a volume V.

We can write:

and, performing the integral we have:

Correlation function

Since the correlation function is a real number, assuming an isotropic power spectrum we have:

If the density field is gaussian, we have that the value of δ at a randomly selected point is drawn from the Gaussian

probability distribution:

where the standard deviation σ can be computed from the power spectrum:

Summary

In practice, our theory cannot predict the exact value of

in a region of the sky. But if we assume that the initial perturbations are gaussian we can predict the correlation function, the variance of the fluctuations and its the power spectrum P(k).

These are things that we can measure using, for example, galaxy surveys and assuming that galaxies trace the

CDM distribution.

Cosmological «Circuit»

Generator of Perturbations (Inflation)

Amplifier (Gravity)

Low band pass filter.

Cosmological and

Astrophysical effects

Tend to erase small scale (large k) perturbations

Power spectrum for CDM

The analysis we have presented up to now is very qualitative and approximated (just to have an idea…)

The true power spectrum for CDM density fluctuations can be computed by integrating a system of differential equations.

We will see this in better detail in the next lectures.

In any case, we assume a power law as initial power spectrum as:

The motivation of using this type of primordial spectrum comes from inflation (we will discuss this also in a future lecture). We have two free parameters: the amplitude A and the spectral index n^s.

Power spectrum for CDM

The first numerical results on the CDM power spectrum appeared around 1980.

Bardeen, Bond, Kaiser and Szalay (BBKS) in 1986 proposed the following fitting function:

where: This is correct fit for a pure CDM model (no baryons or massive neutrinos).

Note the dependence on

that defines the epoch of equivalence.

P(k) for LCDM (from numerical computations).

Spectral index is assumed

n=0.96

Note these oscillations in the

CDM P(k).

Gravitational feedback from

baryons.

The position of the peak is related to size of the

horizon at equivalence, i.e.

to the matter density since radiation is fixed.

Primor dial r

egime

Damping (scales that enter

ed horizon befor

e quality)

If we plot the P(k) in function of h/Mpc, then the dependence is just on

Effect of the Cosmological parameters

Cold dark matter

Effect of the Cosmological parameters

Cold dark matter

Effect of the Cosmological parameters

Baryon density

Effect of the Cosmological parameters

Baryon density

Effect of the Cosmological parameters

Spectral index

Effect of the Cosmological parameters

Spectral index

CDM vs data (1996)

If we assume a flat universe made just of matter with

we get too much power on small scales

to match observations.

Already in 1996 the best fit was for

i.e. suggesting a low matter density universe.

CDM vs data (2003)

The 2dfGRS provided

the following constraints:

(assuming just CDM, no massive neutrinos)

2 sigma detection of baryons:

Measurements from SDSS

Massive Neutrinos

We saw that massive neutrinos contribute to the matter density when they are non-relativistic as:

However, until they are relativistic, neutrino have

w=1/3, i.e. perturbations in the neutrino component dissipates when they enter the horizon.

The wavelength of the horizon when they start to be non relativistic is given by:

Evolution of the density fields in the synchronous gauge (top panels) and the conformal Newtonian gauge (bottom panels) in the Ων = 0.2 CDM+HDM

model for 3 wavenumbers k= 0.01, 0.1 and 1.0 Mpc−1. In each figure, the five lines represent δc , δb , δγ , δν and δh for the CDM (solid), baryon (dash-

dotted), photon (long-dashed), massless neutrino (dotted), and massive neutrino (short-dashed) components, respectively.

Numerical evolution of density fluctuations

Model with CDM, neutrinos, massive neutrinos, photons, baryons

Focus just on the Top panel

(Syncronous Gauge)

These fluctuations enter the horizon

just after decoupling

CDM, Baryons and MN grow as a^2 and a

Photons and

massless neutrinos oscillates and damp

after entering the horizon

Focus just on the Top panel

(Syncronous Gauge)

CDM grow as a^2, ln aˆ2 and a

Until decoupling baryons oscillates and are damped, they quickly follow CDM

afterwards.

Massive neutrinos

(nearly 1.5 eV in this case)

are semi

relativistic at equality and damped.

They slowly catch CDM afterwards.

These fluctuations enter the horizon just after equality

Photons and massless neutrinos oscillates and damp

after entering the horizon

Focus just on the Top panel

(Syncronous Gauge)

Until decoupling baryons oscillates and are damped, they quickly follow CDM

afterwards.

Massive neutrinos are first

relativistic and strongly damped because of free streaming.

They try to catch CDM afterwards.

These fluctuations enter the horizon well before

equality

Photons and massless neutrinos oscillates and damp

after entering the horizon

CDM grow as a^2, ln aˆ2 and a

small feedback from b and mnu

Baryons vs Massive Neutrinos

If a perturbation in baryons enters the horizon before

decoupling oscillates with a decreasing amplitude because of diffusion damping (photons can go from hotter to

colder regions after several scattering).

After decoupling, baryons fall in the CDM potential well.

If a perturbation in massive neutrinos enter the horizon before the non relativistic regime is strongly damped

because of free streaming (neutrinos are collision-less).

Neutrino are very light and also afterwards they still

suffer from free streaming and practically don’t cluster.

Massive Neutrinos

The growth of the fluctuations is therefore suppressed on all scales below the horizon when the neutrinos become

nonrelativistic

the small scale suppression is given by:

Larger is the neutrino mass, large is the suppression.

Hu et al, arXiv:astro-ph/9712057

eff

Massive Neutrinos

Massive Neutrinos

m_ν = 0 eV m_ν = 1 eV

m_ν = 7 eV m_ν = 4 eV Ma ’96

...but we have degeneracies...

• Lowering the matter density suppresses the power spectrum


• This is virtually degenerate with non-zero neutrino mass

Inclusion of CMB data is important to break degeneracies

Conservative limit from CMB+P(k) measurements:

Mass fluctuations

Given a theoretical model a quantity can be often easily compared with observations is the

variance of fluctuations on a sphere of R Mpc:

where:

Usually one assumes R=8 Mpc hˆ-1, where the linear approximation is valid.

P(k) for LCDM (from numerical computations).

Spectral index is assumed

n=0.96

The

gives the P(k)

amplitude around these scales

Examples from CAMB

http://lambda.gsfc.nasa.gov/toolbox/tb_camb_form.cfm

Output from CAMB

Output from CAMB

Output from CAMB

Output from CAMB

Inflation I

Paradoxes of the FRW model:

Flatness:

( )

∝

−

= Ω

− _{2 2 2/3}

1 t

t a

H

t k Radiation dominated era Matter dominated era

( )

(

)

2

0 1 1

1− Ω = − Ω

a H

t H

In order to have today we need:1− Ω₀ ≤ 0.2 10 4

2

1− Ω_rm ≤ × ⁻ 10 14

3

1− Ω_nuc ≤ × ⁻ 10 60

1

1− Ω_Planck ≤ × ⁻

At matter-radiation equality At nucleosynthesis

At Planck epoch a_Planck ≈ 2×10⁻³² s t_Planck ≈ 5×10⁻⁴⁴

Inflation II

• Horizons problem. Regions that are not causally connected at recombination

show the same temperature. Why ?

( ) ( )

o ls

A

ls

Hor rad

Mpc Mpc z

d

t d

Hor 0.03 2

13 4 .

0 ≈ ≈

≈

= θ

We have about 20000 not causally connected regions. Why they have a similar temperature ?

Inflation III

• Monopole problem. GUT predict that the GUT phase transition creates point-like topological defects that act as magnetic monopoles. The rest energy of the magnetic monopole is

predicted to be

( )

3 82

3

36 10

2 ) 1

10

( ≈ ⁻ ≈ ≈ m⁻

s ct t

n

GUT GUT

M

3 94

2 10 Tev

) (

)

(t_GUT = n_M t_GUT m_M c ≈ m⁻ ρM

3 104

4 10 Tev

)

(t_GUT ≈ σT_GUT ≈ m⁻ ρ_γ

) 10

( )

10

(t ¹⁶ s t ¹⁶ s

M

−

− > =

= ρ_γ

ρ

The universe should be matter

dominated already in the early universe.

This is impossible because, for example, of BBN constraints !!

Let’s suppose one monopole par horizon.

Inflation IV

• The solution is to suppose a period of accelerated expansion called inflation in the early universe.

• Let’s model inflation as a cosmological constant acting from tⁱ to t^f

( )

( )

( ) ^{t t}

t t

t

t t

t t

e a

e a

t t

a t

a

f

f i

i

f t

t H i

t t H i

i i

f i

i i

<

≤

≤

<

⎪ ⎩

⎪ ⎨

⎧

=

−

−

2 / ) 1

(

) (

2 / 1

/ /

Costant

= Λ

= _i Hi

Inflation V

• We can define as number of e-foldings the number N from :

( )

( )

N i

f e

t a

t

a = ^{N = Hi}

(

)

(

)

s t

H N

i f

GUT i

34

1 36

1

10

10 100

−

−

−

≈

−

≈

≈

=

3 2 105

Tev/m 8 10

3 ≈

≈ G

H_i

i π

ρ 2 3

0 0.004Tev/m 8

7 3 .

0 ≈

Λ ≈

G H ρ π

Can’t be the cosmological constant we see today since:

Inflation VI

( )

i

e i

a H

t k_{2 2} ²

1− Ω = − ∝ ⁻

( )

^{( )}

^{( )}

^{( )}

Ω

− ⁻ 1 ⁻ 1 10⁻ 1

1 ^{2 200 87}

The flatness problem is solved since for an exponential inflation:

Inflation VII

• The horizon problem is also solved since for an exponential expansion:

( ) ( )

t

i i

i i

Hor ct

t t a c dt a

t d

i

∫

=

0

2 /

1 2

/

( )

(

)

[ ( )

]

⁽

⁾

⎞

⎜⎜

⎝

⎛

+ −

=

∫ ∫

t

t i i i

t

i i

N i f

Hor e c t H

t t

H a

dt t

t a c dt

e a t

d

f

i i

( )

( )

^{( )}

d

m t

d

ls Hor

f Hor

i Hor

43 16

28

10 8

. 0 10

2

10 6

=

⇒

≈

×

≈

×

≈ ⁻

Inflation VIII

( )

^{( )}

(

)

d _{p f} = _{f p 0} ≈ 10⁻³⁴ 1.4×10⁴ ≈ 3×10⁻²³ ≈ 0.9

( )

( )

d _{p i} = ^−N _{p f} ≈ 10⁻⁴⁴

( )

d_{Hor i} ≈ 6×10⁻²⁸

We can look at the problem in a different way, if

we go back in time the size of our universe just after inflation was:

This means that before inflation our entire universe was contained in a region of:

Well inside the horizon at that time:

Inflation IX

• Also the Monopole problem is solved provided that inflation takes place after GUT.

• …today:

( )

^{( )}

^{( )}

n_{M f} ^N _{M i M i}

( ) t

≈ 10

Mpc

n