Physical Cosmology 13/5/2016

Physical Cosmology 13/5/2016

Alessandro Melchiorri

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

oberon.roma1.infn.it/alessandro/cosmo2016

Cosmological «Circuit»

Generator of Perturbations (Inflation)

Amplifier (Gravity)

Low band pass filter.

Cosmological and

Astrophysical effects

Tend to erase small scale (large k) perturbations

n is the spectral index.

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

Note:

Increasing CDM shifts the peak to the right.

Equality happens earlier (larger z).

Horizon at equality is smaller.

k of equality is larger

Effect of the Cosmological parameters

Baryon density

Note:

converting CDM to baryons

simply destroys small scale

power.

Baryons alone cannot form structure.

Effect of the Cosmological parameters

Spectral index

Note:

increasing n shifts power from large to small scales.

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:

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