7.2 Chemical decoupling

Transcription

1 Sprng term 2014: Dark Matter lecture 5/9 orsten Brngmann readng: chapter 4 of J. Edsjö, Aspects of neutrno detecton of neutralno dark matter, PhD thess, Uppsala (1997) [hep-ph/ ]. P. Gondolo and G. Gelmn, Cosmc abundances of stable partcles: Improved analyss, Nucl. Phys. B 360, 145 (1991).. Brngmann, Partcle Models and the Small-Scale Structure of Dark Matter, New J. Phys. 11, (2009). [arxv: [astro-ph.co]]. 7.2 Chemcal decouplng Let us now have a closer, and more detaled, look at the chemcal decouplng process. We have already encountered the collson-less Boltzmanuaton n Eq.(34). Includng a collson-term C[f] todescrbeanykndofnteractons that the partcles may experence, t can be wrtten as ˆL[f] =C[f], (88) where the Louvlle operator ˆL s the covarant generalzaton of the convectve dervatve: ˆL[f] = df d = dxµ d = @x + dpµ µ µ (90) µ p µ = p 0 (@ t H p r p ) f. (92) Here, s an a ne parameter along a geodesc and n the last step, we assumed a flat FRW geometry and changed to local (or comovng) momenta p ap (n the followng, we wll, however, always consder comovng momenta and drop the bars over the p for smplcty). he phase-space dstrbuton functon f(x µ,p µ )snormalzedsuchthatthenumberdenstyofa speces wth g nternal degrees of freedom s gven by n = g R d 3 p/(2 ) 3 f. he collson term for the annhlaton process s gven by C = 1 d 3 p d 3 k d 3 k 2g (2 ) 3 2Ẽ (2 ) 3 2! (2 ) 3 2! (2 )4 (4) ( p + p k k) h M 2 ff g(!)g(!) M 2! ff f(e)f(ẽ), (93) where k µ =(!, k) and k µ =(!, k) arethe4-momentaofthesmpartclesf and g = g eq = e!/ 1 ± 1 ther dstrbuton functons (wth a mnus-sgn 1

2 for bosons and a plus-sgn for fermons). 12 In ths expresson, M 2 refers to the matrx element squared, summed over all possble SM partcles f, ncludng ther nternal (e.g. spn) degrees of freedom, and also summed over the nternal degrees of freedom of the DM partcles. CP nvarance mples M 2! ff = M 2 ff and n thermal equlbrum annhlaton and creaton processes should happen wth the same frequency. hs means that we can replace g(!)g(!) ntheaboveequatonwth f eq (E)f eq (Ẽ): C = 1 2g M 2! ff = g E d 3 p d 3 k d 3 k (2 ) 3 2Ẽ (2 ) 3 2! (2 ) 3 2! (2 )4 (4) ( p + p k k) hf eq (E)f eq (Ẽ) f(e)f(ẽ) (94) d 3 p (2 ) v 3 Møl! hf ff eq (E)f eq (Ẽ) f(e)f(ẽ), (95) where v Møl (EẼ) 1 q(p p) 2 m 4 s the velocty of one DM partcle n the rest frame of the other. Integratng the full Boltzmanuaton (88) over R d 3 pg / [(2 ) 3 E]thenresultsn ṅ +3Hn = h v n n, (96) where the thermal averaged annhlaton cross secton defned as h v! ff = g ḡ d 3 p d 3 p (2 ) 3 (2 ) v 3 Møl! fff eq (E)f eq (Ẽ). (97) In arrvng at ths result, we had to assume that f(e) / f eq (E), wth a factor of proportonalty that descrbes an e ectve chemcal potental whch may depend on (but not E); ths s motvated by the fact that the much more abundant scatterng processes of DM wth SM partcles stll keep the DM partcles n knetc, but not chemcal equlbrum (see the followng secton). Very often, DM s assumed to be ts own antpartcle, =,nwhchcase we recover the famlar expresson Eq. (86); note that there s no addtonal factor of 1 for dentcal ntal state partcles In-medum e ects can be taken nto account by addng Paul blockng (or Bose enhancement) factors: M 2! ff = M 2(vac)! ff [1 g(!)][1 g(!)] and M 2 ff = M 2(vac)! ff [1 f(e)][1 f(ẽ)]. For non-relatvstc DM, however, E m +p2 /(2m ) and thus f 1; momentum conservaton then also enforces!,! >m and thus g 1. 2

3 Coannhlatons In general, one expects to fnd many new, e.g. supersymmetrc, partcles rather close n mass to the lghtest partcle whch wll be the DM. Snce these partcles wll eventually decay to the DM partcle (whch s assumed to be protected aganst decay by an nternal symmetry), ther ntal abundances also contrbute to the DM relc densty today and we have to study the full set of (coupled) Boltzmanuatons that govern ther evoluton. o do so, we need to compute both the total annhlaton and nclusve scatterng cross sectons, but also the nclusve decay rates and the relatve veloctes : j = X X 0 Xj = X Y ( j! X) (98) ( X! j Y ) (99) j = X (! j X) (100) X q (p p j ) 2 m 2 m2 j v j =. (101) E E j In the above expressons, X and Y denote all (sets of) standard model partcles that appear n the nteractons. In complete analogy to the case of a sngle annhlaton mode, one can now derve the followng set of Boltzmann equatons for so-called coannhlatons: ṅ +3Hn = NX h j v j n n j j=1 X j6= j X X X [h Xjv 0 j (n X j6= ) h 0 Xjv j n j j ] [ j (n ) j n j j ]. (102) Relatvely shortly after freeze-out, all partcles wll decay nto the DM partcles P and we are thus only nterested n the total number densty n N =1 n.summngeq.(102)overthen results n ṅ +3Hn = NX h j v j n n j,j=1 j. (103) Note that the terms on the second and thrd lnes n Eq. (102) do not change the total number n of (heavy) partcles and thus cancel n the sum. here s, 3

4 however, an mportant consequence of nclusve scatterng: Snce n X n, these processes happen su cently often to keep the dstrbutons n (or very close to) chemcal equlbrum wth respect to each other,.e. all should have a very smlar chemcal potental. As a consequence, we can to agoodapproxmatonassumethat n n ' neq, (104) neq where X ' X g dp (2 ) 3 e E = 2 2 X g m 2 m K 2. (105) hs results n ṅ +3Hn = h e v n 2 n 2 eq (106) whch s of the same form as for the case of no co-annhlatons, Eq. (86), but wth the annhlaton rate replaced by an e ectve annhlaton rate h e v = X j h j v j neq j. (107) neq Because of the Boltzmann suppresson factors of / exp[ (m m )/ ], we thus expect that only partcles rather close n mass to the dark matter partcle contrbute to the e ectve annhlaton rate. Integratng the Boltzmanuaton Let us now brng the Boltzmanuaton nto a form whch s more convenent for ntegraton. Introducng the rato of the number densty to the entropy densty, and t (a 3 s) = 0, the left-hand sde of Eq. (106) becomes Y = n s, t a 3 (a 3 s)y +3HsY = sẏ. (109) Next, we change the ndependent varable from t to the dmensonless quantty x m /. (110) Exlotng agan entropy conservaton, and s =(2 2 /45)g s e 3,wethenfnd 0=a t a 3 s =3Hs + ġs e g s e s +3 apple s =3s H 1+ 3g s e dge s ẋ d x. (111) 4

5 By usng the frst Fredmanuaton, ths allows us to rewrte Eq. (106) as r dy dx = g 1/2 m 45G x 2 h e v Y 2 Yeq 2, (112) where Y eq s gven by and Y eq = s = 45x g s e ( ) X g m m g 1/2 gs e p 1+ ge 3ge s 2 K 2 x m m (113) dge s. (114) d here exst very nce and e cent approxmate ways of solvng Eq. (112) analytcally whch work well as long as h e v( ) takes a relatvely smple form. A thorough dscusson of soluton strateges, be t analytcal or numercal, goes beyond the scope of these lectures, so let us smply denote by Y 0 the result of ntegratng Eq. (112) from x =0tox 0 = m / 0, where 0 s the photon temperature of the Unverse today. he DM relc densty s then obtaned as 7.3 hermal decouplng = 0 / c (115) = m s 0 Y 0 / c (116) = h 2 m Y 0. (117) 100 GeV As already mentoned, WIMPs stay n thermal contact wth the heat bath of SM partcles qute long after chemcal decouplng,.e. after number densty changng processes have ceased (whch typcally happens around cd m /25: scatterng events between WIMPs and SM partcles are much more frequent than WIMP annhlatons smply because SM partcles are much more abundant (SM partcle annhlatons to WIMPs, on the other hand, are knematcally possble only n the heavly suppressed tal of the energy dstrbuton). Only when even these processes stop to be e cent after knetc, or thermal, decouplngtherearenolongeranynteractonsbetween WIMPs and SM partcles and the former have completely decoupled from the thermal bath. o descrbe knetc decouplng, t su ces to look at the second moment of the Boltzmanuaton because hgher orders n p 2 /m 2 are heavly suppressed. Integratng the left-hand sde of Eq. (88) over R d 3 pg p 2 / [(2 ) 3 E] results n d 3 p (2 ) g 3 p2 (@ t Hp r p ) f = t ( n )+15m H n (118) = 3m n +2H, (119) 5

6 where n the last step we have assumed ṅ = 3Hn,.e.chemcalfreeze-out has already taken place. We have also ntroduced a WIMP temperature defned by d 3 p g (2 ) 3 p2 f(p) 3 m n. (120) Note that ths defnton does not requre any assumptons about the form of f(p), but smply provdes a convenent means of characterzng the devaton from thermal equlbrum (for whch = holds). he collson term for elastc scatterng f $ f s gven by C = 1 2g d 3 k (2 ) 3 2! d 3 k (2 ) 3 2! d 3 p (2 ) 3 2Ẽ (2 )4 (4) ( p + k p k) M 2 (1 g ± )(!) g ± (!)f( p ) (1 g ± )(!) g ± (!)f(p), (121) where the summaton over all SM partcles n thermal equlbrum s agan not shown explctly. Whle we wll assume that the g are thermal dstrbutons as before, no assumptons about the dstrbuton functon f(p) arenecessary; as long as the WIMPs are much less abundant than ther scatterng partners, however, Paul suppresson factors for f can safely be neglected as has been done n n the above expresson. After chemcal freezeout, one typcally has! m. For knematcal reasons, the average momentum transferred durng the scatterng events s thus small, compared to the rest mass, so the collson term can be expanded as C = P 1 j=0 Cj,wherethecoe centsare defned by a aylor expanson n ( k k). Formally, ths can be wrtten by replacng the 3D Drac delta-functon n Eq. (121) by (3) ( p p + k 1X 1 h j k) = ( k k) r p (3) ( p p), (122) j! j=0 whch s defned as usual n terms of ntegraton by parts. After a lengthy calculaton 13,ncludngonlytermsofthelowestnon-vanshngordernp 2 /E 2 and!/m,onefndsthefollowngexpressonforthecollsonntegral: C ' C 1 + C 2 ' c( )m hm 2 p + p r p +3 f(p), (123) where c( )= X 1 12g (2 ) 3 m 4 dk k 5! 1 g ± 1 g ± M 2 t=0 s=m 2 +2m!+m 2`. (124) 13 For detals, see the thrd reference lsted above. 6

7 For clarty, the sum over all SM scatterng partners has here been made explct. For relatvstc SM partcles, the above ntegral can be solved analytcally f (as usually s the case) M 2 / (!/m ) n n ths expresson, resultng n c( ) / 4+n. Note also that the frst moment of Eq. (123) just vanshes as t should snce we are consderng scatterng processes that do not change the number densty of dark matter partcles. he second moment of the Boltzmanuaton, on the contrary, reads +2H = g d 3 p 3m n (2 ) 3 E p2 C[f(p)] ' g 3n c( ) d 3 p h (2 ) 3 p2 m p + p r p +3 f(p) = g 3n c( ) d 3 p h 6m 2p 2 f(p) (2 ) 3 = 2m c( )[ ]. (125) Analogous to the treatment of the frst moment of the Boltzmanuaton, one may now ntroduce the dmensonless quantty y m s 2/3 / a 2 (126) and re-wrte the left-hand sde of Eq. (125) as +2H = m 1 s 2/3 ẏ. In exactly the same fashon as n Eqs. ( ), we then arrve at 1 dy 2m 1+ dg s y dx = 3ge s e c( ) d 1. (127) xh Inspecton of ths equaton shows the expected asymptotc behavour for the WIMP temperature: at early tmes, or large, thetermnfrontoftherght- hand sde s much larger than unty (recall that H / 2 and c( ) / 4+n, typcally wth n =2);thesoluton = thus provdes a strong attractor of the d erental equaton,.e. the system s very e cently kept n thermal equlbrum. When becomes small, the WIMPs fnally decouple completely from the thermal bath: the pre-factor becomes neglgbly small and y stays constant,.e. / s 2/3 / a 2 whchsmplyreflectstheredshftofthe WIMP momenta. Snce the transton between the two regmes happens on a rather short tmescale, the knetc decouplng temperature s naturally defned by equatng these two asymptotc behavours, as f the decouplng process were ndeed to occur nstantaneously: x kd = m kd y x!1 s2/3 2 = kd. (128) 7

8 Wth the formalsm presented above, one can calculate the knetc decouplng temperature to an accuracy of O(x 1 kd ), for any WIMP canddate that was non-relatvstc at chemcal freeze-out and for whch we have x cd x kd.he knetc decouplng temperature for, e.g., neutralno DM les between a few MeV and a few GeV, wth x kd between 200 and almost 10 5 amuchlarger range than for x cd whch falls nto the range 20. x cd. 28 for neutralnos wth the correct relc densty. 8