Boltzman equation

1.3 Possible resolutions of drawbacks of the Big bang cosmology and the SM of par-

1.3.2 Dark Matter

1.3.2.3 Boltzman equation

Before we describe the DM evolution through different phases of the Universe, let us define energy density (ρ) and entropy density (s) of the Universe. The energy density of all species in thermal equilibrium can be well approximated by the contribution of relativistic species [11].

The simplified expression for energy density is given by [11]

ρ= π²

30g^∗T⁴, (1.49)

where g^∗s are the number of effective massless degrees of freedom (dof) provided by g^∗ = X

bosons

gi

Ti

T 4

+7 8

X

fermions

gi

Ti

T 4

(1.50) where g_i is the number of dof of individual species and T_i indicates the temperature of i’th species of fermions or bosons. On the other hand, entropy density (s) in an expanding Universe is an important quantity to measure. Similar to ρ, sis also dominated by the contributions of

1.3. Possible resolutions of drawbacks of the Big bang cosmology and the SM of particle

physics 19

relativistic dofs which can be approximated as [11]

s= 2π²

45 g^∗_sT³, (1.51)

where g^∗_s is the entropy dof given by, g_s^∗ = X

bosons

g_iTi

T 3

+7 8

X

fermions

g_iTi

T 3

. (1.52)

Therefore, both g^∗ andg_s^∗ are functions of time. As for most of the time in the early Universe, maximum number of species had uniform temperature, it can be approximated that g^∗ ' g^∗_s [11].

The evolution of any particle, supposeX, in Universe is governed by the Boltzman equation.

This is generally written as

L(fˆ ) = ˆC(f), (1.53)

where ˆC is the collision operator and ˆL is the Liouville operator [11]. f → f(x^µ, p^µ) is the phase space distribution of the concerned particleX. According to FRW model, the Universe is spatially flat and homogeneous which corresponds,f(x^µ, p^µ)→f(E, t). Thereafter the number density of the X particle can be obtained as

n_X(t) = g (2π)³

Z

d³pf(E, t), (1.54)

wheregis the number of dof ofXparticle. Now one can find the simplified form of the Boltzman equation using the standard form of Liouville operator [11]. The evolution of number density of the particle 6nX therefore reads,

dn_X

dt + 3Hn_X = g (2π)³

Z C(fˆ )

E d³p. (1.55)

A generalized collision process of X particle with other species can be identified as X+a₁ + a₂+...↔i₁+i₂+i₃+.... Then the R.H.S of Eq.(1.55) can be converted to [11]

g (2π)³

Z C(fˆ ) E d³p=−

Z

dΠ_XdΠa₁dΠa₂...dΠi₁dΠi₂dΠi₃..(2π)⁴δ⁴(p_X+pa₁+pa₂..−pi₁−pi₂−pi₃...)

×h

|M|²_X+a

1+a2+...→i₁+i2+i3+...fXfa1fa2..(1±fi₁)(1±fi₂)..

− |M|²_i

1+i₂+i₃+...→X+a₁+a₂+...fi1fi2..(1±fa₁)(1±fa₂)(1±fX)...

i

, (1.56)

20 Chapter 1. Introduction wherefin is the phase space density ofn’th species anddΠj =gj d³p

(2π)³2E. The “+” sign applies for bosons while “−” applies to fermions. We will use Maxwell-Boltzman distribution statistics for all the particle species in oder to simplify the Boltzman equation further.

Freeze out mechanism: According to freeze-out mechanism, the particle we are concerned with (X) was in thermal equilibrium at early Universe. Later as the Universe cools down, its interaction rate with thermal plasma (Γ_X) slowly declines. The condition Γ_X ∼ H ( H is the expansion rate of the Universe) determines the decoupling point of Xfrom thermal plasma and then the relic abundance of DM freezes out.

Suppose X is stable (or long lived) in cosmological time scale and hence could be a viable DM candidate. At early stage,Xwas in thermal equilibrium with other particles in the thermal plasma. Hence during that time annihilation and inverse annihilation processes ofXe.g. XX ↔ Y Y were active where Y represents the other particles in the thermal plasma. Let us also consider X has no decay mode. During thermal equilibrium, phase space distribution for any species (f_in) can be written as following Maxwell Boltzman distribution,

f_in =e^−EinT . (1.57)

We define the number density of X during thermal equilibrium n_X ' n^eq_X. We also use some standard redefinitions of variables x= ^m_T^X,Y_X = ⁿ_s^X wheresis the entropy density (Eq.(1.51)) of the Universe. In that case after few intermediate steps [11], the Eq.(1.55) can be simplified to the conventional form,

dYX

dx =−hσvis

Hx (Y_X² −Y_X^eq2), (1.58)

where Y_X^eq= ⁿ

eq X

s and hσvi is thermally averaged interaction cross section of X which is related with the interaction rate of X as Γ_X = n_Xσv. The parameter v is identified as the relative velocity between two DM particles in the centre of mass frame (XX ↔Y Y). To obtain Eq.(1.58) from Eq.(1.55) the change of variable from time ttox has been performed using T ∝ ¹_t.

Now in the relativistic (x << 3) and non-relativistic limit (x << 3) one can obtain from Eq.(1.54),

Y_X^eq = 0.145g

g^∗_sx^3/2e^−x for x >>3, (1.59)

= 0.278g_eff

g_s^∗ for x <<3, (1.60)

1.3. Possible resolutions of drawbacks of the Big bang cosmology and the SM of particle

physics 21

where geff =g for bosons and ^3g₄ for fermions. The value ofx at which ΓX = H, is the freeze out value conventionally denoted by x_f.

Note that, Eq.(1.58) is not solvable analytically. It requires to invoke some physics intuition to simplify Eq.(1.58). The L.H.S. of Eq.(1.58) is proportional to O(Y_X), while R.H.S. is of

Ξ

x²O(Y_X²) with Ξ>>1 as we will shortly see. Hence forx <<1, the factor (Y_X²−Y_X^eq2) in R.H.S of Eq.(1.58) must have been very small. Then we can safely assumeY_X ∼Y_X^eqduring relativistic regime. On the other hand for x >> 1, Y_X^eq falls as e^−x. Therefore in view of Eq.(1.58), Y_X should stop following Y_X^eq during non-relativistic regime.

Freezeout may occur when X is in both (a) relativistic and (b) non-relativistic regimes.

Below we discuss the related phenomenology of these.

(a) In this case X decouples from thermal plasma during relativistic regime. The particles which decouples at relativistic mode is known as “hot relic”. For this type of particles, we can approximate Y_X(x_f) ' Y_X^eq=constant where x_f denotes the decoupling point of X. (see Eq.(1.60) where xf >>3). Hence at present Universe the value ofYX would be

Y_X^∞'0.278g_eff

g_s^∗. (1.61)

Therefore, present energy density of relic X can be obtained as ρ⁰_X = s₀Y_X^∞m_X ' 3×10³m_X

eV

g_eff(x_f)

g^∗_s(xf)cm⁻³, (1.62) where s₀ '3000 cm⁻³ is the present entropy density. Then the relic abundance of X particle (ΩX = _ρ^ρ

C) will be provided as

Ω_Xh² '7.9×10⁻²geff(xf) g_s^∗(x_f)

m_X eV

, (1.63)

whereρC = 1.88h²×10⁻²⁹gm cm⁻³is the critical density of the Universe [39]. One can wonder whether light neutrinos in the SM could be a DM candidate. They decouple at temperaturex_f ∼ O (1 MeV). Then considering three generations of neutrinos, roughlyg_s^∗(xf) =g^∗(xf) = 10.75 andg_eff = 3×

3 4

. Hence, to satisfy the correct relic abundance of DM, Ω_X=νl '0.11, neutrino mass should satisfy mX=νl ' 0.95 eV using Eq.(1.63). However the recent Planck [39] and WMAP [124] data put a stringent upper limit on sum of the neutrino masses asP

imνi <0.22 eV. It confirms that neutrinos in the SM can not provide correct relic abundance of DM.

22 Chapter 1. Introduction (b) The species which decouples at non-relativistic regime is often called as“cold relic”. Theo- retically velocity dependence of annihilation cross section can be parametrized as

hσvi ∝v^p, (1.64)

wherep= 0 corresponds to s-wave annihilation whilep= 2 indicates p-wave annihilation. Now as v∝T^1/2, it is possible to writehσvias

hσvi=σ0x⁻ⁿ, (1.65)

where n= ^p₂.

The Boltzman equation of Eq.(1.58) can be further reduced to dYX

dx =−Ξx⁻ⁿ⁻²(Y²−Y_X^eq2), (1.66) where Ξ = _75.15^{2π2 √g}_g^∗s∗m_Pσ₀m_X and we have also usedH= 1.67√

g^∗_m^T2

P. Now in order to find the relic density of X, let us attempt to solve Eq.(1.66) analytically. First we assumehσvihas only s-wave contribution (n= 0).

dY_X

dx =−Ξ

x²(Y²−Y_X^eq2), (1.67)

After freeze out, it is also safe to consider Y >> Y_X^eq. Therefore, Eq.(1.66) turns out to be, dYX

dx =−Ξ

x²Y_X². (1.68)

The solution is obtained by differentiating both sides of Eq.(1.68) from x = x_f to x = ∞ yielding.

Y_X^∞= x_f

Ξ. (1.69)

In Fig. 1.3, general behaviour of YX has been shown. It can be viewed that after certain value of x =x_f, Y_X stops following Y_X^eq and after sometime it freezes out. Now it is very trivial to calculate the relic abundance of X as

Ω_X = ρ_X

ρ_C, (1.70)

= s0Y_X^∞mX

ρ_C , (1.71)

1.3. Possible resolutions of drawbacks of the Big bang cosmology and the SM of particle

physics 23

Figure 1.3: Evolution ofY_X as a function ofx= ^m_T^X in freeze out scenario [160] considering different magnitudes of interaction cross section ofX field. We also show the evolution ofY_X^EQ.

whereρC = 1.88h²×10⁻²⁹gm cm⁻³ is the critical density of the Universe ands0 ' 3000 cm⁻³ is present entropy density [39]. Putting the values of ρ_C,s₀ and substitutingY_X^∞ in Eq.(1.71) we find,

Ω_Xh² '1.07×10⁹ x_f (g^∗_s/g^1/2∗ )m_Pσ₀

GeV⁻¹. (1.72)

As an example, assuming g^∗_s =g∗ '100 and xf '20, we can obtain an estimate of ΩX as ΩXh²= 2×10⁻¹⁰GeV⁻²

hσvi , (1.73)

where we substituteσ₀byhσvifrom Eq.(1.65) Hence to achieve correct order of relic abundance

∼0.1,hσvi has to be around 10⁻⁹ GeV⁻². This is the order of weak scale interaction. Hence, any massive particle having this order of interaction cross section can satisfy the relic abundance bound. Theoretically, these kind of weakly interacting DM candidates are commonly categorized as WIMP.

Freeze in: Recently another kind of dark matter production mechanism is proposed known as feebly interacting massive particle (FIMP) [125, 126]. In this scenario, due to very weak coupling, the DM never stays in thermal equilibrium. Initial density of DM is assumed to be zero, and at later epoch it can be produced thermally (annihilation) or non thermally (decay from some heavier particles).

24 Chapter 1. Introduction Suppose X is the DM candidate which is having interaction with two other particles B1

and B₂ with m_B1 > m_B2. Then the B₁→B₂+X process governs the freeze in of DM. In this case the collision factor ( ˆC(f)) of Boltzman equation (Eq.(1.55)) turns out to be

g 2π³

Z C(f)ˆ

E d³p=− Z

dΠ_XdΠ_B1dΠ_B2(2π)⁴δ⁴(p_B1 −p_X−p_B2)

×h

|M|²_B

1→B₂+Xf_B1(1±f_X)(1±f_B2)

− |M|²_B1→X+B2f_B1(1±f_X)(1±f_B2)...i

. (1.74)

To solve the Boltzman equation one needs to make few important assumptions: (i) initial abundance of X particle is zero (fX = 0), (ii) dilute gas limit i.e. f2±1 ' f1±1 = 1, (iii) the other two particle are in thermal equilibrium f₁ =e^−ET1 , f₂ = e^−ET2. Next to obtain the relic abundance of X particle in this case, one has to solve Eq.(1.55) using Eq.(1.74). We will not discuss it further since our focus in the thesis will be mainly on the freeze out scenario. We refer Refs. [125, 126] for further study on FIMP mechanism.