A brief thermal history of the Universe

Martin Kunz Université de Genève

Overview

•  Equilibrium description –  Distribution functions –  Neutrino decoupling and the neutrino

background –  Photon decoupling, recombination and the CMB –  BBN and light element abundances –  Comparison to observations

•  Basics of Boltzmann equation –  hot and cold relics

•  Summary

Brief history of the Universe

Equilibrium distributions Short-range interactions maintaining

thermodynamic equilibrium: , T temperature, µ chem. pot.

f(k, t)d3k =

g

(2�)

3

�exp[(E � µ)/T ]± 1

��1d3k

E =p

k2 + m2

n =Z

f(k)d3k

� =Z

E(k)f(k)d3k

p =Z |k|2

3E(k)f(k)d3k

number density

energy density

pressure

Relativistic species, m << T

Crank handle, using m = µ = 0 -> E ~ k (use x=E/T as integration variable)

nB = T 3 g�(3)⇥2

nF =34nB

⇥B = T 4 g

30�2 ⇥F =

78⇥B

p =�

3

with ργ ~ a-4 => Tγ ~ 1/a -> expanding universe cools down

-> wrad = prad/ρrad = 1/3

-> Stefan-Boltzmann law ργ ~ T4

Massive species, m >> T

Expand and neglect +/-1 wrt exp(m/T)

E =p

k2 + m2 = mp

1 + k2/m2 ⇡ m + k2/(2m)

n = g

✓mT

2�

◆3/2

e�(m�µ)/T

� = mn +32nT

p = nT ⌧ �

Ekin =32kBT-> Ekin / particle:

Massive  par*cles  are  suppressed  by  Boltzmann  factor                    exp(-­‐m/T),  so  they  will  quickly  drop  out  of  thermal  equilibrium  when  T  <  m  -­‐>  ‘freeze  out’  -­‐>  effec*ve  μ  

Multiple relativistic species If we have several species at different temperatures:

⇥R =T 4

�

30�2g⇤ g⇤ =

X

i2B

gi

✓Ti

T�

◆4

+78

X

j2F

gj

✓Tj

T�

◆4

Entropy density: s =� + p

T/ T 3 d(sa3)/dt = 0

(use f and ) � + 3H(� + p) = 0

s =2�2

45g⇤ST 3

� g⇤S =X

i2B

gi

✓Ti

T�

◆3

+78

X

j2F

gj

✓Tj

T�

◆3

•  T� / g�1/3⇤S a�1

Now  we  are  ready  to  study  par.cle  evolu.on  in  the  early  universe!  

Neutrino decoupling

Interaction rate: Γ species in equil.: Γ >> H Expansion rate: H species decoupled: Γ << H

�(T ) = n(T )h�viT �F ' G2F E2 ' G2

F T 2 �F ⇠ G2F T 5

H(T ) =r

8�G

3p

⇥R '5.44mP

T 2 g⇤ = 2 +78(3⇥ 2 + 2⇥ 2)

) �F

H(T )' 0.24T 3G2

F mP '✓

T

1MeV

◆3

Neutrinos decouple when temperature drops below ~ 1 MeV because their interactions become too weak.

Temperature of ν background

Shortly after the neutrinos decouple, we reach T=0.5MeV=me and the entropy in electron-positron pairs is transferred to photons but not to the neutrinos. Photon + electron entropy g*S(Ta)3 is separately conserved:

g⇤(T�dec > T > me) = 2 +78⇥ 4 =

112

, g⇤(T < me) = 2

How much are the photons heated by the electron-positron annihilation?

Temperature of ν background

Shortly after the neutrinos decouple, we reach T=0.5MeV=me and the entropy in electron-positron pairs is transferred to photons but not to the neutrinos. Photon + electron entropy g*S(Ta)3 is separately conserved:

g⇤(T�dec > T > me) = 2 +78⇥ 4 =

112

, g⇤(T < me) = 2

(aT�)3after

(aT�)3before

=(g⇤)before

(g⇤)after=

114

Since (aTν) = (aTγ)before we now have Tγ = (11/4)1/3 Tν -> for T<0.5me : g* ~ 3.36 and g*S ~ 3.91 for radiation (γ+ν)

Photon decoupling

e+/e- annihilation stopped by baryon asymmetry, remaining electrons and photons stay in thermal contact (Compton scattering) until electrons and protons form neutral hydrogen (recombination). As number of free electrons ne drops, photons decouple when Γγ ~ H.

�� = ne⇤T , ⇤T =8⇥�2

EM

3m2e

nj = gj

✓mjT

2�

◆3/2

e(µj�mj)/T , j = e, p,H

µp + µe = µH

mp + me = mH + B

gp = ge = 2, gH = 4

due to interactions (number conservation) -> use this to remove µ’s

energy conservation, binding energy: B=13.6eV

nB = np + nH = ne + nH (np = ne)Baryon number density:

Photon decoupling II We can write nH

nenp⇡ gH

gpge

✓meT

2�

◆�3/2

eB/T

introduce Xe=ne/nB (fractional ionisation) and notice that (in equilibrium)

1�Xe

X2e

=nHnB

nenp=

4p

2�(3)p⇥

nB

n�

✓T

me

◆3/2

eB/T Saha eqn.

Assume recombination ~ Xe=0.1 and with η ~ 10-10 [why?] => Trec ~ 0.31 eV, zrec ~ 1300 (why Trec << B?) Go back to Γγ and compare to matter dominated expansion H=H0Ωm(1+z)3/2

⇒  Tdec ~ 0.26 eV, zdec ~ 1100 -> origin of CMB!

Notice that recombination and photon decoupling are two different processes, although they happen at nearly the same time. Photons decouple because ne drops due to recombination (else zdec ~ 40!).

η

‘equilibrium’ BBN

•  T > 1MeV: p and n in equilibrium through weak interactions •  T ~ 1MeV: weak interactions too slow, ν freeze-out •  Light element binding energies: a few MeV -> when is it favourable to create the light elements? same game as before: for species with mass number A = #n

+#p and charge Z = #p, assumed in equilibrium with p & n

nA = gA

✓mAT

2�

◆3/2

e(µA�mA)/T = gAA3/2

2A

✓mNT

2�

◆3(1�A)/2

nZp nA�Z

n eBA/T

where we used again µA=Zµp+(A-Z) µn. With XA= nA A/nN mass fraction:

XA = . . . = (const)

✓T

mN

◆3(A�1)/2

XZp XA�Z

n �A�1eBA/T

BBN II: NSE

-> System of equations for ‘nuclear statistical equilibrium’:

1 = Xn + Xp + X2 + X3 + . . .

Xn/Xp = e�Q/T

X2 = C

✓T

mN

◆3/2

XpXn�eB2/T

X3 = . . .

etc … •  Q = 1.293 MeV •  B[2H] = 2.22 MeV •  B[3H] = 6.92 MeV •  B[3He] = 7.72 MeV •  B[4He] = 28.3 MeV

(Kolb & Turner)

log(

X(e

q))

BBN III: actual BBN

In reality the reactions drop out of equilibrium eventually, and one needs to use the Boltzmann equation. Results:

•  T ~ 10 MeV+ : equilibrium, Xn=Xp=1/2, rest X << 1 •  T ~ 1 MeV: n<->p freeze-out, Xn ~ 0.15, Xp ~ 0.85, NSE

okay for rest (with X << 1) •  T ~ 0.1 MeV: neutrons decay, n/p~1/8, NSE breaks down

because 4He needs Deuterium (2H) which is delayed until 0.07 MeV because of high η and low B2.

•  T ~ 65 keV: now synthesis of 4He can proceed, gets nearly all neutrons that are left:

Rest is hydrogen, with some traces of 2H, 3He, 7Li and 7Be.

X4He =4n4

nN= 4

nn/2nn + np

= 2nn/np

1 + nn/np= 2

1/81 + 1/8

⇡ 0.22

(figures from A. Weiss, Einstein Online Vol. 2 (2006), 1018)

0.1 MeV 0.01 MeV

(Dodelson)

Timeline summary

Energy  (γ)   *me   event  

1  MeV   7s   neutrino  freeze-­‐out  

0.5  MeV   10s   e+/e-­‐  annihila.on,  Tγ  ~  1.4  Tν  

70  keV   3  minutes   BBN,  light  elements  formed  

0.77  eV   70’000  yr   onset  of  maMer  domina.on  

0.31  eV   300’000  yr   recombina.on  

0.26  eV   380’000  yr   photon  decoupling,  origin  of  CMB  

0.2  meV   14  Gyr   today  

Comparison to observations

3 classical ‘pillars’ of big-bang model: 1.  Hubble law -> Monday 2.  Cosmic Microwave Background 3.  BBN and element abundances CMB: we expect an isotropic radiation with

thermal spectrum to fill the universe

(COBE / NASA)

Nobel  prizes:  1978  CMB  discovery  2006  CMB  proper.es  2011  accelerated  exp.    

Comparison II : BBN

BBN tests: •  baryon/photon ratio η •  effective # of relativistic degrees of freedom •  consistency of different abundances

The CMB anisotropies (Valeria) also depend on the baryon abundance (even-odd peak heights). Results are consistent with BBN! Ωb ≈ 0.05 (what is all the rest??!!)

Non-equilibrium treatment Looking at equilibrium quantities is very useful, but: •  when / how does freeze-out really happen? •  what to do when NSE breaks down •  … -> Treatment with Boltzmann equation L[f] = C[f]/2

L: Liouville operator ~ d/dt C: collision operator (à particle physics!)

relativistic / covariant form: f=f(E,t) -> only α = 0 relevant, p0=E and Γ from FRW metric =>

L = p↵ �

�x↵� �↵

��p�p� �

�p↵

L[f ] =E

�

�t� a

ap2 �

�E

�f(E, t)

Liouville operator

We only want to know the abundance -> integrate L[f]/E over

3-momentum p to get n: •  first term is just dn/dt •  second term: rewrite in terms of p and integrate by parts

(d/dE = E/p d/dp)

L[f ] =E

�

�t� a

ap2 �

�E

�f(E, t)

g

2�2

Zdpp2 L[f ]

E= n + 3Hn

1.  no collisions: L[f]=0 -> n ~ a-3 as expected! 2.  deviations from full equilibrium will be encoded in µ

Collision operator Roughly: (# of particles ‘in’) – (# of particles ‘out’) of phase space volume d3p d3x

For (reversible) scattering of type 1+2 <-> 3+4 and with

g1

(2⇥)3

Zd3p1

C1[f ]2E1

=Z

d�1d�2d�3d�4(2⇥)4�(4)(p1 + p2 � p3 � p4)|M|2

[f3f4(1± f1)(1± f2)� f1f2(1± f3)(1± f4)]

d�i ⌘gi

(2�)3d3pi

2Ei

f(E) ⇡ e(µ�E)/Tkinetic equilibrium & low temperature: [. . .]! e�(E1+E2)/T

he(µ1+µ2)/T � e(µ3+µ4)/T

i

ni = eµi/T n(eq)i , n(eq)

i =gi

(2�)3

Zd3p e�Ei/T

⇤⇤v⌅ ⇥ 1

n(eq)1 n(eq)

2

Zd�1d�2d�3d�4e

�(E1+E2)/T (2⇥)4�(4)(p1 + p2 � p3 � p4)|M|2

with and

) n1 + 3Hn1 = n(eq)1 n(eq)

2 h�vi

n3n4

n(eq)3 n(eq)

4

� n1n2

n(eq)1 n(eq)

2

!

Annihilation and freeze-out

Consider annihilation processes: assume Y in thermal equilibrium 1)  <σv> large -> n -> n(eq)

2)  <σv> small -> n ~ a-3

Introduce x=m/T and Y=n/T3 (Y~n/s, constant for passive evol.) some algebra… => freeze-out governed by Γ/H (Γ = n(eq)<σv>)

A + A$ Y + Y

nA + 3HnA = h�vi⇣(n(eq)

A )2 � n2A

⌘

x

Y(eq)A

Y 0A = � �A

H(x)

2

4

YA

Y(eq)A

!2

� 1

3

5

(with Y(eq) = 0.09 g (for fermions) if x<<1 and Y(eq) = 0.16 g x3/2 e-x if x>>1)

Hot and cold relics Hot relics: freeze-out when still relativistic (xf<1) ->

YA(x!1) = Y

(eq)A (xf ) = 0.278gA/g⇤S(xf )

Cold relics: freeze out when xf >> 1 => YA suppressed by e-m/T

Abundance generically proportional to 1/σ We can compute ρA,0 = mA nA,0 ΩA = ρA,0/ρcrit numerically, weak cross- sections lead to Ω ~ 1 -> WIMP miracle

(Kolb & Turner)

Abundance estimate

•  Assume roughly constant ->

•  Y(eq) << Y from freeze-out onwards ->

•  Integrate from freeze-out to late times ->

•  Usually Y∞ << Yf ->

•  This depends only linearly on xf, so take e.g. xf = 10

•  particle density scales like a3 after freeze out

⇒  (extra factor ~30 from entropy generation in later annihilation processes)

⇒  insert Y∞ … final:

dY

dx= � �

x2

⇣Y 2 � Y (eq)2

⌘

dY/dx ' ��Y 2/x2

1/Y1 � 1/Yf = �/xf

Y1 ⇡ xf/�

� = mn0 = mn1

✓a1

a0

◆3

= mY1T 30

✓a1T1

a0T0

◆3

' mY1T 30

30

�X ⇠⇣xf

10

⌘ ✓g⇤(m)100

◆1/2 10�39cm2

h�vi

� =m3h⇥viH(m)

reasons for decoupling / freeze-out

•  neutrinos decouple from thermal equilibrium because interactions become too weak (~ T5 scaling of interaction rate)

•  photons (CMB) decouple from equilibrium because e- disappear (recombination)

•  baryons freeze-out from annihilation because of baryon-antibaryon asymmetry (no more antibaryons to annihilate with)

•  WIMP’s (dark matter) freeze out from because their density drops due to e-m/T Boltzmann factor (if they are WIMP’s)

Summary

•  Methods –  distribution function f(t,x,v) –  conservation of entropy –  full / kinetic equilibrium –  Boltzmann equation (evolution of f)

•  Results –  evolution of particle number, pressure and energy in

equilibrium –  T ~ 1/a (except when g*S changes, e.g. particle annihil.) –  thermal history, freeze-out of particles when interactions

become too slow, WIMPs –  origin of CMB –  abundances of light elements

observa*onal  “pillars”  of  the  cosmological  standard  model