Production of WIMPS in early Universe
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1 Production of WIMPS in early Universe Anjishnu Bandyopadhyay Universität Bonn July 5, 2013 Anjishnu Bandyopadhyay Production of WIMPS in early Universe 1 / 30
2 1 2 Generalized Equation Equation for relics Anjishnu Bandyopadhyay Production of WIMPS in early Universe 2 / 30
3 Abstract What are we calculating? Anjishnu Bandyopadhyay Production of WIMPS in early Universe 3 / 30
4 Abstract What are we calculating? Relic Abundance of WIMPS Anjishnu Bandyopadhyay Production of WIMPS in early Universe 3 / 30
5 Abstract What are we calculating? Relic Abundance of WIMPS How do we calculate? Anjishnu Bandyopadhyay Production of WIMPS in early Universe 3 / 30
6 Abstract What are we calculating? Relic Abundance of WIMPS How do we calculate? We solve the Boltzmann equation for relic WIMPS Anjishnu Bandyopadhyay Production of WIMPS in early Universe 3 / 30
7 Departure from equilibrium have produced important relics Light elements Neutrino Background Net baryon number Relic WIMPS Anjishnu Bandyopadhyay Production of WIMPS in early Universe 4 / 30
8 Criterion for decoupling Γ H(coupled) Γ H(decoupled) To treat decoupling with better accuracy we solve the Boltzmann equation. Anjishnu Bandyopadhyay Production of WIMPS in early Universe 5 / 30
9 Generalized Equation Equation for relics Boltzmann equation ˆL[f ] = C[f ] (1) where f is the phase space distribitution function. ˆL is the Liouville operator. C is the Collision operator. Anjishnu Bandyopadhyay Production of WIMPS in early Universe 6 / 30
10 Generalized Equation Equation for relics The non relativistic Liouville operator for a particle species of mass m subjected to a force F is ˆL NR = d dt = t + v x + F m v (2) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 7 / 30
11 Generalized Equation Equation for relics The non relativistic Liouville operator for a particle species of mass m subjected to a force F is ˆL NR = d dt = t + v x + F m v (2) The covariant relativistic generalisation of L is Relativistic Liouville operator ˆL R = d dλ = dx α x α dλ + dp α p α dλ ˆL R = p α x α Γα βγ pβ p γ p α (3) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 7 / 30
12 Generalized Equation Equation for relics Boltzmann equation for FRW metric For FRW metric the phase space density f is homogeneous and isotropic. So we get ˆL[f (E, t)] = E f t Ṙ R p 2 f E (4) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 8 / 30
13 Generalized Equation Equation for relics Boltzmann equation for FRW metric For FRW metric the phase space density f is homogeneous and isotropic. So we get ˆL[f (E, t)] = E f t Ṙ f p 2 (4) R E We can express number density n in terms of f (E, t) n(t) = g (2π) 3 d 3 pf (E, t) (5) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 8 / 30
14 Generalized Equation Equation for relics Boltzmann equation for FRW metric For FRW metric the phase space density f is homogeneous and isotropic. So we get ˆL[f (E, t)] = E f t Ṙ f p 2 (4) R E We can express number density n in terms of f (E, t) n(t) = g (2π) 3 d 3 pf (E, t) (5) We integrate by parts to get dn dt + 3Ṙ R n = g (2π) 3 C[f ] d 3 p E (6) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 8 / 30
15 Collision term Generalized Equation Equation for relics For a process ψ + a + b + i + j + the collision term is given by g (2π) 3 C[f ] d 3 p ψ E ψ = dπ ψ dπ a dπ b dπ i dπ j x(2π) 4 δ 4 (p ψ + p a + p b p i p j ) x[ M 2 ψ+a+b+ i+j+ faf b f ψ (1 ± f i )(1 ± f j ) M 2 i+j+ ψ+a+b+ fif j (1 ± f a )(1 ± f b ) (1 ± f ψ )] (7) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 9 / 30
16 Generalized Equation Equation for relics We use two assumptions to simplify the collision term. CP invariance Absence of Bose condensation or Fermi degeneracy Anjishnu Bandyopadhyay Production of WIMPS in early Universe 10 / 30
17 Generalized Equation Equation for relics We use two assumptions to simplify the collision term. CP invariance Absence of Bose condensation or Fermi degeneracy The first condition implies M 2 ψ+a+b+ i+j+ = M 2 i+j+ ψ+a+b+ fif j = M 2 (8) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 10 / 30
18 Generalized Equation Equation for relics We use two assumptions to simplify the collision term. CP invariance Absence of Bose condensation or Fermi degeneracy The first condition implies M 2 ψ+a+b+ i+j+ = M 2 i+j+ ψ+a+b+ fif j = M 2 (8) The second condition implies 1 ± f 1 (9) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 10 / 30
19 Generalized Equation Equation for relics Rewriting in terms of different parameters We rewrite the Boltzmann equation in terms of Y Y = n ψ (10) s where s is the entropy density Conservation of entropy per comoving volume (sr 3 = constant) implies ṅ ψ + 3Hn ψ = sẏ (11) We define another variable x = m T (12) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 11 / 30
20 Generalized Equation Equation for relics During radiation dominated era t = 0.301g 1/2 m Pl m 2 x 2 Thus the generalised Boltzmann equation becomes Generalized Boltzmann equation dy dx = x dπ ψ dπ a dπ b dπ i dπ j (2π) 4 M 2 H(m)s xδ 4 (p ψ + p a + p b p i p j )[f a f b f ψ f i f j ] (13) where H(m) = 1.67g 1/2 m Pl m 2 H(x) = H(m)x 2 Anjishnu Bandyopadhyay Production of WIMPS in early Universe 12 / 30
21 Equation for relics Generalized Equation Equation for relics We assume that The species is stable. Thus only annihilation and inverse annihilation processes can change its number density. ψ + ψ X + X Anjishnu Bandyopadhyay Production of WIMPS in early Universe 13 / 30
22 Equation for relics Generalized Equation Equation for relics We assume that The species is stable. Thus only annihilation and inverse annihilation processes can change its number density. ψ + ψ X + X X and X have thermal distribitution with zero chemical potential. So we have f X = exp( E X /T ) f X = exp( E X /T ) (14) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 13 / 30
23 Generalized Equation Equation for relics The δ function in the equation implies E ψ + E ψ = E X + E X. f X f X = exp[( E X + E X )/T ] = exp[( E ψ + E ψ Thus we get EQ )/T ] = fψ f ψ EQ (15) [f ψ f ψ f X f X ] = [f ψ f ψ f EQ ψ f ψ EQ ] (16) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 14 / 30
24 Generalized Equation Equation for relics So we get dn ψ dt + 3Hn ψ = σ ψ+ ψ X + X v [nψ 2 (neq ψ )2 ] (17) dy x σ dx = ψ+ ψ X + X v [(Y 2 YEQ 2 H(m) ) (18) where ( σ ψ+ ψ X + X v = n EQ ψ ) 2 dπ ψ dπ ψ dπ X dπ X (2π)4 xδ 4 (p ψ + p ψ p X p X M2 exp( E ψ /T )exp( E ψ /T ) (19) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 15 / 30
25 Generalized Equation Equation for relics Summing over all the channels we get Equation for relics x dy Y EQ dx = Γ A H [ ( Y Y EQ ) 2 1] (20) where Γ A = n EQ σ A v (x 3) we have Y EQ (x) = g eff g s x 3/2 e x (21) (x 3) we have Y EQ (x) = g eff g s (22) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 16 / 30
26 The term hot implies that the species freezes out when it is relativistic (x f 3). Since Y EQ is constant the final value of Y is quite insensitive to precise value of x f. Relic Abundance where s 0 = 2970cm 3 g eff Y = Y EQ (x f ) = g s (x f ) (23) [ ] nψ 0 = s geff 0Y = 825 cm 3 g s(x f ) (24) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 17 / 30
27 Results for hot relics Mass density and mass bound ρ 0 ψ = s 0Y m = 2.97x10 3 Y (m/ev )evcm 3 (25) Ω ψ h 2 = 7.83x10 2 [g eff /g s(x f )](m/ev ) (26) From WMAP results we know for hot relics Ω 0 h m 0.178eV [g s (x f )/g eff ] (27) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 18 / 30
28 Mass bound Light neutrinos decouple when T MeV. So we have g s = Anjishnu Bandyopadhyay Production of WIMPS in early Universe 19 / 30
29 Mass bound Light neutrinos decouple when T MeV. So we have g s = For a single two component neutrino species g eff = 1.5. Therefore g eff /g s = 0.14 Anjishnu Bandyopadhyay Production of WIMPS in early Universe 19 / 30
30 Mass bound Light neutrinos decouple when T MeV. So we have g s = For a single two component neutrino species g eff = 1.5. Therefore g eff /g s = 0.14 Cosmological mass bound m ν 1.27eV (28) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 19 / 30
31 Mass bound Light neutrinos decouple when T MeV. So we have g s = For a single two component neutrino species g eff = 1.5. Therefore g eff /g s = 0.14 Cosmological mass bound m ν 1.27eV (28) For a species ψ which decouples early on i.e. at T 300GeV we have g s Anjishnu Bandyopadhyay Production of WIMPS in early Universe 19 / 30
32 Mass bound Light neutrinos decouple when T MeV. So we have g s = For a single two component neutrino species g eff = 1.5. Therefore g eff /g s = 0.14 Cosmological mass bound m ν 1.27eV (28) For a species ψ which decouples early on i.e. at T 300GeV we have g s Mass bound m ψ eV (29) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 19 / 30
33 Here freeze out happens when the species is non relativistic. Anjishnu Bandyopadhyay Production of WIMPS in early Universe 20 / 30
34 Here freeze out happens when the species is non relativistic. We also consider the velocity dependence of the cross section. σ v = σ 0 x n (30) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 20 / 30
35 Here freeze out happens when the species is non relativistic. We also consider the velocity dependence of the cross section. Thus the Boltzmann equation takes the form σ v = σ 0 x n (30) where dy dx = λx n 2 (Y 2 Y 2 EQ ) (31) λ = 0.264(g s /g 1/2 )m Pl mσ 0 (32) Y EQ = 0.145(g/g s )x 3/2 e x (33) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 20 / 30
36 Solving the equation where = Y Y EQ = Y EQ λx n 2 (2Y EQ + ) (34) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 21 / 30
37 Solving the equation = Y EQ λx n 2 (2Y EQ + ) (34) where = Y Y EQ At early times Y tracks Y EQ closely, so both and are small. = 0 implies x n+2 2λ (35) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 21 / 30
38 Solving the equation = Y EQ λx n 2 (2Y EQ + ) (34) where = Y Y EQ At early times Y tracks Y EQ closely, so both and are small. = 0 implies x n+2 (35) 2λ At late times Y tracks Y EQ poorly, so Y Y EQ. Thus we have = λx n 2 2 (36) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 21 / 30
39 Solving the equation = Y EQ λx n 2 (2Y EQ + ) (34) where = Y Y EQ At early times Y tracks Y EQ closely, so both and are small. = 0 implies x n+2 (35) 2λ At late times Y tracks Y EQ poorly, so Y Y EQ. Thus we have = λx n 2 2 (36) Integrating we get Y = = n + 1 λ x n+1 f (37) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 21 / 30
40 Freeze out Anjishnu Bandyopadhyay Production of WIMPS in early Universe 22 / 30
41 Freeze out (x f ) = cy EQ (x f ) (38) c is constant of order unity Anjishnu Bandyopadhyay Production of WIMPS in early Universe 22 / 30
42 Freeze out (x f ) = cy EQ (x f ) (38) c is constant of order unity (x f ) x n+2 f λ(2 + c) (39) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 22 / 30
43 Freeze out (x f ) = cy EQ (x f ) (38) c is constant of order unity (x f ) x n+2 f λ(2 + c) (39) x f ln[(2 + c)λac] ( n + 1 ) ln[ln(2 + c)λac)] 2 (40) where a = 0.145(g/g s ) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 22 / 30
44 Relic Abundance We choose c(c + 2) = n + 1 since it gives best result. [ ] x f = ln 0.038(n + 1)(g/g 1/2 )m Pl mσ 0 (41) ( n + 1 ) { ln[ln 0.038(n + 1)(g/g 1/2 )m Pl mσ 0 }] (42) 2 Anjishnu Bandyopadhyay Production of WIMPS in early Universe 23 / 30
45 Relic Abundance We choose c(c + 2) = n + 1 since it gives best result. [ ] x f = ln 0.038(n + 1)(g/g 1/2 )m Pl mσ 0 (41) ( n + 1 ) { ln[ln 0.038(n + 1)(g/g 1/2 )m Pl mσ 0 }] (42) 2 Relic Abundance Y = 3.79(n + 1)x n+1 f (g s /g 1/2 )m Pl mσ 0 (43) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 23 / 30
46 Relic density Results for cold relics nψ 0 = s 0Y = 1.13x10 4 (n + 1)x n+1 f (g s /g 1/2 cm 3 )m Pl mσ 0 (44) Ω ψ h 2 n+1 9 (n + 1)xf GeV 1 = 1.07x10 (g s /g 1/2 (45) )m Pl σ 0 Anjishnu Bandyopadhyay Production of WIMPS in early Universe 24 / 30
47 WIMP annihilation depends upon whether it is Dirac or Majorana type. We consider the Dirac case where σ A v is velocity independent. Using g = 2 g = 60 c 2 5 we get σ 0 = c 2 G 2 F m2 /2π (46) x f ln(m/GeV ) + ln(c 2 /5) (47) ( Y 6x10 9 m ) [ ln(m/GeV ) + (c ] 2/5) GeV (48) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 25 / 30
48 Mass bound Anjishnu Bandyopadhyay Production of WIMPS in early Universe 26 / 30
49 Mass bound We have x Ω ψ ψ h2 = 3(m/GeV ) 2 [ 1 + 3ln(m/GeV ) ] 15 (49) Using Ω ψ ψh 2 = ± we get Mass bound m = 5.81 ± 0.04GeV Anjishnu Bandyopadhyay Production of WIMPS in early Universe 26 / 30
50 WIMP miracle Model independent σ A v For s-wave annihilation we get σ A v = (1.85 ± 0.04)x10 9 GeV 2 (50) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 27 / 30
51 WIMP miracle Model independent σ A v For s-wave annihilation we get σ A v = (1.85 ± 0.04)x10 9 GeV 2 (50) The cross section is in the weak scale!! Anjishnu Bandyopadhyay Production of WIMPS in early Universe 27 / 30
52 WIMP candidates We saw heavy neutrino cannot be a possible WIMP. Anjishnu Bandyopadhyay Production of WIMPS in early Universe 28 / 30
53 WIMP candidates We saw heavy neutrino cannot be a possible WIMP. Sneutrinos have large annihilation cross sections, masses have to exceed several hundred GeV to be a good candidate. Anjishnu Bandyopadhyay Production of WIMPS in early Universe 28 / 30
54 WIMP candidates We saw heavy neutrino cannot be a possible WIMP. Sneutrinos have large annihilation cross sections, masses have to exceed several hundred GeV to be a good candidate. Lightest neutralino is the best candidate. Anjishnu Bandyopadhyay Production of WIMPS in early Universe 28 / 30
55 WIMP candidates We saw heavy neutrino cannot be a possible WIMP. Sneutrinos have large annihilation cross sections, masses have to exceed several hundred GeV to be a good candidate. Lightest neutralino is the best candidate. Detailed calculations show that the lightest neutralino will have the desired thermal relic density in at least four distinct regions of parameter space ψ could be (mostly) a bino or photino. Anjishnu Bandyopadhyay Production of WIMPS in early Universe 28 / 30
56 We assume adiabtic expansion (sr 3 = const) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 29 / 30
57 We assume adiabtic expansion (sr 3 = const) For hot relics Y depends upon g s (x f ) Anjishnu Bandyopadhyay Production of WIMPS in early Universe 29 / 30
58 We assume adiabtic expansion (sr 3 = const) For hot relics Y depends upon g s (x f ) Possible hot relics are light neutrino, light gravitino. Anjishnu Bandyopadhyay Production of WIMPS in early Universe 29 / 30
59 We assume adiabtic expansion (sr 3 = const) For hot relics Y depends upon g s (x f ) Possible hot relics are light neutrino, light gravitino. For cold relics Y is inversely proportional to m σ v Anjishnu Bandyopadhyay Production of WIMPS in early Universe 29 / 30
60 We assume adiabtic expansion (sr 3 = const) For hot relics Y depends upon g s (x f ) Possible hot relics are light neutrino, light gravitino. For cold relics Y is inversely proportional to m σ v Neutralinos are the best WIMP candidates. Anjishnu Bandyopadhyay Production of WIMPS in early Universe 29 / 30
61 Thanks for your attention Anjishnu Bandyopadhyay Production of WIMPS in early Universe 30 / 30
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