Analytical expressions for the kinetic decoupling of WIMPs. Luca Visinelli University of Bologna Physics and Astronomy Department

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1 Analytical expressions for the kinetic decoupling of WIMPs Luca Visinelli University of Bologna Physics and Astronomy Department

2 Based on: Visinelli and Gondolo, PRD 91 8 (2015) Kinetic decoupling of WIMPs: analytic expressions Luca Visinelli and Paolo Gondolo Department of Physics and Astronomy, University of Utah, 115 South 1400 East #201, Salt Lake City, Utah , USA (Dated: January 10, 2015) We present a general expression for the values of the average kinetic energy and of the temperature of kinetic decoupling of a WIMP, valid for any cosmological model. We show an example of the usage of our solution when the Hubble rate has a power-law dependence on temperature, and we show results for the specific cases of kination cosmology and lowtemperature reheating cosmology. PACS numbers: d

3 WIMP freeze-out Regulated by Boltzmann equation dn dt +3Hn = h vi n 2 n 2 eq h vi thermal average:! SM SM Figure: Kolb & Turner Late-time behavior n n eq, so a 3 n constant Present DM density h cm 3 /s h vi The WIMP miracle : weak interactions h 2 = O 10 1

4 Freeze-out vs. kinetic decoupling WIMP annihilation into SM particles Scattering of WIMP and SM particles Annihilation rate Changes n Process ends at ann H(T freeze ) with ann T freeze M /20 Momentum exchange rate Does not change n Process ends at exc H(T kd ) with T kd T freeze exc

5 Temperature of kinetic decoupling T kd is a key parameter in cosmological models. determines the cutoff in the density spectrum links the size of the smallest halos with WIMP nature The mass of the smallest halo M cut is the largest between: from WIMP free-streaming from acoustic oscillations M fs = 4 3 k fs M ao = H(T kd ) 3

6 More on kinetic decoupling The scattering of WIMPs of mass M off a plasma at temperature (random walk) T M is a Brownian motion p Ne p = p p M T p : WIMP momentum p : WIMP momentum spread N e : number of collisions required to change the momentum by. p

7 Fokker-Planck equation The WIMP occupation number f = f (p) follows a Fokker-Planck H(T = apple p f (1 + f ) + Hubble Rate Relaxation Rate Statistics We solve FP for any and, assuming f 1 (T ) H(T )

8 For f 1, we rewrite FP as the Master Equation a dt da +2[1+ (T )] T = 2 (T ) T where we defined the kinetic temperature T = 2 3 Z p 2 f (p ) d3 p 2M WIMP scattering off the plasma is regulated by the ratio (T )= (T ) H(T )

9 The general solution to the Master Equation is the sum of a homogeneous and an inhomogeneous part T (a) =T i ai a 2 e G(a,a i ) a 2 Z a e G(a,a 0) (a 0 ) T (a 0 ) a 0 da 0 a i where we introduced T i = T (a i ) and G(a, a 0 )=2 Z a a 0 (a 00 ) da00 or a 00 G(t, t 0 )=2 Z t t 0 (t 00 ) dt 00

10 Limits at early- and late-time (T ) H(T ) or (T ) 1 : WIMPs are tightly coupled to the plasma at = constant and T T (T ) H(T ) or (T ) 1 : WIMPs decouple from the plasma and a 2 T = constant T / T 2 T i

11 Applications: power-law cosmology (1) We choose the parametrization for the cosmology: H(T )=H i T T i a T = constant 4+n T We also set (T )= i and i = i T i H i

12 Applications: power-law cosmology (2) The solution to the Master Equation is with s = T = Ts e s [ (1, s)+ (, s i )] 2 (4 + n ) = 2 (4 + n ) i T T i 4+n If WIMPs are initially tightly coupled to the plasma, T = Ts e s (1, s) (s i! +1) For =1, =2 (RD), and for n =2 (p-wave) T = Ts 1/4 e s 3 4,s (Berschinger, PRD )

13 Temperature of kinetic decoupling (I) We define (T kd )=H(T kd ) In power-law models T kd = T i i 1 4+n Standard RD: T kd,std = T i H(Ti ) i 1 2+n

14 Temperature of kinetic decoupling (II) The late-time behavior of T = T 2 T i 2 i 2+n T 1 2+n during RD is 1+n 2+n Compare with (Bringmann and Hofmann, JCAP 0407, ) T = T 2 T kd,std 2 2+n 1 2+n 1+n 2+n gives T kd = T n+2 kd,std T 2 i! 1 4+n T kd = T kd,std for =2(RD) T 2 for =4,n=2 kd,std = T Low Temperature RH Reheating model Gelmini and Gondolo, JCAP 0810,

15 Summary and conclusions T kd is a key parameter in cosmological models, since it sets the size of the smallest protohalos. T kd is related to the kinetic temperature T through the Fokker-Planck equation; The Fokker-Planck equation is solved in terms of the kinetic temperature T ; The solution is in terms of T, and can be implemented in numerical models like DarkSUSY; In power-law models (RD, LTR, Kination), generic expression for T and are obtained. T kd