3. Wimp paradigm - Freeze out (and beyond)200.145.112.249/webcast/files/SERPICO3_freezeout.pdf ·...

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Geneva Lake shore, February 2012 3. Wimp paradigm - Freeze out (and beyond) Pasquale Dario Serpico (Annecy-le-Vieux, France) School on Dark Matter ICTP-SAIFR (São Paulo, Brazil) - 28/06/2016

Transcript of 3. Wimp paradigm - Freeze out (and beyond)200.145.112.249/webcast/files/SERPICO3_freezeout.pdf ·...

  • Geneva Lake shore, February 2012

    3. Wimp paradigm - Freeze out (and beyond)

    Pasquale Dario Serpico (Annecy-le-Vieux, France) School on Dark Matter ICTP-SAIFR (São Paulo, Brazil) - 28/06/2016

  • RECAP & PLANRecent determination (Planck 2015, 68% CL)

    Ωch2=0.1188±0.0010, i.e. Ωc~0.26

    [Main] Goal: compute value of number to entropy density ratio, Y0�Xh

    2 = 2.74⇥ 108✓MXGeV

    ◆Y0

    ! We shall first provide a heuristic argument for the simplest (yet powerful!) toy-model evolution equation for Y! We shall use this equation in different regimes to elucidate a couple of classes (not all!) of DM candidates! Some generalizations will be briefly discussed.! Later (Lec. 4, most likely) we’ll come back to sketch a “microscopic” derivation/interpretation of the equation we started with

    Caveat: matching ΩX is one condition for a good DM candidate, not the only one! Remember lecture 2 (collisionless, right properties for LSS structures...)

  • DM CLASSIFICATION / PARAMETER SPACE

    mass

    Will discuss different classes based on production mechanisms. However, these are typically linked with masses and couplings as well!

    coupling

    WIMPfreeze-outEW-like

    EW

    asymmetric

    gravitational

    from inflationmisalignement

    (axion-like)

    freeze-in

    MACRO DM (e.g. black holes)

    QCD-like

    gravity-like

    MeV-GeVsub-eV GUT-scale

  • BOLTZMANN EQ. FOR DM DENSITY CALCULATION

    dn

    dt+ 3H n = �h�vi[n2 � n2eq]

    XX $ (thermal bath particles)

    dn

    dt+ 3H n = 0 ) n / a�3

    Assume that binary interactions of our particle X are present with species of the thermal bath

    If interaction rate Γ=n σ v very slow wrt Hubble rate H, # of particles conserved covariantly, i.e.

    If interaction rate Γ>> H, # of particles follows equilibrium, e.g. for non-relativistic particles

    n = g

    ✓mT

    2�

    ◆3/2exp

    ⇣�mT

    ⌘eq

    The following equation has the right limiting behaviours

    for now, symbolic only

    must be quadratic, for binaryprocesses

  • REWRITING IN TERMS OF Y ANDdn

    dt+ 3H n = �h�vi[n2 � n2eq]

    dY

    dt= �sh�vi[Y 2 � Y 2eq]

    dY

    dx= � x sh�vi

    H(T = m)[Y 2 � Y 2eq]

    d

    dt(a3s) = 0 =) d

    dt(a T ) = 0 =) d

    dt(a/x) =

    x� a

    x2ẋ = 0 =) dx

    dt= H x

    dY

    dx= �

    p45�MPl m

    he�(x)h⇥vipge�(x)x2

    ✓1� 1

    3

    d log he�d log x

    ◆(Y 2 � Y 2eq)

    Define x=m/T (m arbitrary mass, either MX or not); for an iso-entropic expansion one has

    More in general (arbitrary s(t) and H(t)):

    M. Srednicki, R. Watkins and K. A. Olive,“Calculations of Relic Densities in the Early Universe,”Nucl. Phys. B 310, 693 (1988)

    P. Gondolo and G. Gelmini,“Cosmic abundances of stable particles: Improved analysis,”Nucl. Phys. B 360, 145 (1991).

    X

    dY

    dt=

    d

    dt

    ⇣ns

    ⌘=

    d

    dt

    ✓na3

    s a3

    ◆=

    1

    s a3d

    dt

    �na3

    �=

    1

    s a3

    ✓a3

    dn

    dt+ 3a2ȧ n

    ◆=

    1

    s

    ✓dn

    dt+ 3H n

    radiation-dominated period

  • FREEZE-OUT CONDITION

    x

    Yeq

    dY

    dx= ��eq

    H

    "✓Y

    Yeq

    ◆2� 1

    #

    �eq(xF ) = H(xF )

    The previous equation is a Riccati equation: no closed form solution exist!

    Approximate analytical solutions exist for different hypotheses/regimes

    If Γeq >> H the particle starts from equilibrium condition at sufficiently small x (high-T), when relativistic. Crucial variable to determine the Yfinal is the freeze-out epoch xF from condition

    (In the following, we shall assume the choice m=MX)

    For heff ~ const., we can re-write

    �eq = h�vineqwith

  • RELATIVISTIC FREEZE-OUT

    Y (xF ) = 0.28g � {1(B), 3/4(F)}

    he�(xF )

    �⌫h2 '

    Pm⌫

    94 eV

    �Xh2 = 0.0762�

    ✓MX

    eV

    ◆g � {1(B), 3/4(F)}

    he�(xF )

    If the solution to this condition yields xF

  • NON-RELATIVISTIC FREEZE-OUT

    gh⇥vi(2�)3/2

    M3X

    x�3/2F

    e�xF =

    r4�3

    45ge�

    M2X

    x2F

    MPl

    x1/2F

    e�xF =

    r4�3

    45ge�

    (2�)3/2

    MPlMX g h⇥vi

    Y (xF

    ) =n(x

    F

    )

    s(xF

    )=

    g

    he�

    45

    2�2(2�)3/2x3/2F

    e�xF

    Y (xF ) =

    r45 ge��

    xFhe�MPl MX �⇥v⇥

    = O(1) xFMPl MX �⇥v⇥

    which also writes

    Thus one obtains

    (Note the important result Y(xF)~ 1/)

    �eq(xF ) = H(xF )to determine xF

  • NON-RELATIVISTIC FREEZE-OUT: INTERPRETATION

    Y (xF ) � O(1)xF

    MPl MX ⇥�v⇤makes sense, in the Boltzmann suppressed tail:The more it interacts, the later it decouples, the fewer particles around.

    =) �Xh2 '0.1 pb

    h�vi

    Also, plugging numbers (typically xF~30), one has

    h⇥vi ⇠ �2

    m2' 1 pb

    ✓200GeV

    m

    ◆2dimensionally, for electroweak scale masses and couplings, one gets the right value!

    But the pre-factor depends from widely different cosmological parameters (Hubble parameter, CMB temperature) and the Planck scale. Is this match simply a coincidence?

    Dubbed sometimes “Weakly Interacting Massive Particle” (WIMP) Miracle

  • EXERCISE, PART 1

    By using any software of your choice (including symbolic ones like Mathematica© , etc.), write a simple code to solve for the relic abundance equation.

    ‣ Compare with the analytical approximations discussed during the lecture.

    ‣ Feel free to explore what happens under different conditions (e.g. different dependences for the cross section; epochs of entropy variations… for which the exercise assigned in Lec. 1 is needed!)

    Have a look at 1204.3622 for comparison and for some “tricks” on how to make the computation more efficient (notably if you find, as you probably

    should, problems of numerical stiffness)

  • CARE SHOULD BE TAKEN WHEN DEALING WITH...

    K. Griest and D. Seckel,“Three exceptions in the calculation of relic abundances,'' Phys. Rev. D 43, 3191 (1991).

    * i.e., whenever σ(s) is a strongly varying function of the center-of-mass energy s(one recently popular example is the “Sommerfeld Enhancement”)

    • cohannihilations with other particle(s) close in mass• resonant annihilations*• thresholds*

    http://lapth.cnrs.fr/micromegas/

    10/3/12 12:15 PMbandeau horizontal MicrOmega

    Page 1 of 2http://lapth.in2p3.fr/micromegas/bandeauhorizontal.html

    MicrOMEGAs: a code for the calculationof Dark Matter Properties

    including the relic density, direct and indirect rates in a general supersymmetric modeland other models of New Physics

    Geneviève Bélanger, Fawzi Boudjema, Alexander Pukhov and Andrei Semenovhttp://www.physto.se/~edsjo/darksusy/

    Nowadays, relic density calculations have reached a certain degree of sophistication and are automatized with publicly available software.But if you have a theory with “unusual” features... better to check!

    J. Edsjo and P. Gondolo,“Neutralino relic density including coannihilations,”  Phys. Rev. D 56, 1879 (1997) [hep-ph/9704361].

    For a pedagogical overviewof generalization in presence ofcoannihilations (and decays), see

    http://lapth.in2p3.fr/micromegas/http://www.physto.se/~edsjo/darksusy/

  • LINK WITH COLLIDERS

    new particle

    • If one has a strong prior for new TeV scale physics (~with ew. strength coupling) due to the hierarchy problem, precision ew data (e.g. from LEP) suggest that tree-level couplings SM-SM-BSM should be avoided!

    we want to avoid!

    • Straightforward solution (not unique!) is to impose a discrete “parity” symmetry e.g.: SUSY R-parity, K-parity in ED, T-parity in Little Higgs. New particles only appear in pairs!

    we want it!

    ➡ Automatically makes lightest new particle stable! ➡ May have other benefits (e.g. respect proton stability bounds...)

    In a sense, some WIMP DM (too few? too much?) is “naturally” expected for consistency of the currently favored framework for BSM physics at EW scale.

    Beware of the reverse induction:

    LHC is now testing this paradigm, but if no new physics is found at EW scale it is at best the WIMP scenario to be disfavored, not the “existence of DM” 


  • WIMP (NOT GENERIC DM!) SEARCH PROGRAM

    W+, Z, γ, g, H, q+, l+

    W -, Z, γ, g, H, q -,l -

    ECM ≈ 
102±2 GeV

    New physics

    X=χ, B(1),…

    Newphysics

    X

    Early universe and indirect detection

    Direct detection 
(recoils on nuclei)

    Collider Searches

    multimessenger approach

    " demonstrate that astrophysical DM is made of particles (locally, via DD; remotely, via ID) " Possibly, create DM candidates in the controlled environments of accelerators

    " Find a consistency between properties of the two classes of particles. Ideally, we would like to calculate abundance and DD/ID signatures → link with cosmology/test of production

    More on tha

    t in the rest

    of the schoo

    l!

  • F(“FEEBLY”)IMPS… AND FREEZE-INWe solved the evolution equation for Y under the assumption of initial equil. abundance, Y(x

  • SOME PROPERTIES OF FREEZE-IN

    Y1 / h�vi

    In the eq., we can then neglect Y wrt Yeq Assuming negligible initial abundance (otherwise it’s not produced via freeze-in!)

    1 100 10000 1e+06 1e+08Temperature [GeV]

    1e-28

    1e-24

    1e-20

    1e-16

    1e-12

    1e-08

    0.0001

    Y =

    n/s

    gs = 1e-5gs = 1e-6gs = 1e-7gs = 1e-8

    Yeq

    sin α = 0.01

    Mχ = 100 GeV

    MH2 = 150 GeV

    Figure 2: The dark matter abundance as a function of the temperature for differentvalues of gs. The solid orange line shows Yeq. In this figure Mχ = 100 GeV, MH2 =150 GeV and sinα = 10−2.

    freeze-in mechanism is easily satisfied for all the values of gs considered in thefigure.

    The dependence of Y on the dark matter mass is illustrated in figure 3,which shows Y as a function of the temperature for different values of Mχ.In it we set gs = 10−8, sinα = 10−2, and MH2 = 150 GeV. This value ofMH2 implies that only for the two smallest values of Mχ (10 GeV and 70 GeV)⟨σv⟩ can be enhanced thanks to the H2 resonance –via diagram (c). From thefigure we see that it makes a huge difference in the value of Y whether thisresonant enhancement can take place or not. In fact, models that benefit fromthe enhancement have an abundance about six orders of magnitude larger thanthose that do not –see the lines for Mχ = 70 GeV and Mχ = 80 GeV. Note thatthe freeze-in temperature does depend on Mχ, and it is given by Tf.i. ∼ Mχat large masses (see the bottom line). For masses below 10 GeV or so, theabundance becomes independent of the dark matter mass so the line for Mχ =10 GeV (upper one) can actually be interpreted as valid for any Mχ < 10 GeV.At the other end of the spectrum, we see that the abundance decreases as thedark matter mass in increased –compare Mχ = 100 GeV (magenta line) withMχ = 1 TeV (blue dotted-line).

    Figure 4 displays the dependence of Y withMH2 . The other parameters weretaken as gs = 10−8, sinα = 10−2 and Mχ = 300 GeV. Again we notice a hugedifference in Y between models where dark matter can be produced resonantly(two upper lines) and those where it cannot (three lower lines). As expected,the freeze-in temperature is determined by Mχ and does not depend on MH2 .We also observed from the figure that Y increases with MH2 .

    In the previous figures, we have considered dark matter masses in the GeV-

    7

    M. Klasen and C. E. Yaguna, “Warm and cold fermionic dark matter via freeze-in,”   JCAP 1311, 039 (2013)

    ! Since it typically requires small couplings, it is harder to test (possible signatures more model dependent)

    dY

    dx' x sh�vi

    H(m)Y 2eq Y1 '

    Z 1

    x0

    dx0x0 s h�viH(m)

    Y 2eq

    inverse dependence wrt WIMP freeze-out

    ! Can also check that Y saturates at smaller x (order 1) wrt xfo~20-30 (early universe history more important)

    ! Note that now

    ! Can be generalized to other production mechanisms, e.g. via decays in the plasma (similarly, Y∞ proportional to decay rate…)

  • EXERCISE, PART 2Use the previously developed code, generalizing to freeze-in regime (It is also worth

    exploring the generalization of the analytical approximations, for comparison)

    ‣ Now change Ti, reducing it to 5 mX, 0.5 mX, 0.1mX. Repeat the above exercise and determine the new values of g for which the DM abundance is matched. Comment the results.

    ‣ Assume mX =100 GeV; Compute the temperature evolution of X abundance as a function of g. Determine the values of g for which X constitutes the DM abundance.

    ‣ Take a stable scalar particle X of mass mX, annihilating with cross section of order g4/mX2, g being its coupling to the SM. Start from an initial temperature Ti=1000 mX and assume that the SM is a thermal bath, but no initial population of X

  • A STRAIGHTFORWARD GENERALIZATIONIf DM not its antiparticle, must follow the evolution of both DM (+) and antiDM (-), via 2 coupled eqs.

    dn±dt

    + 3Hn± = �h�vi[n+n� � n+eqn�eq]

    (assuming that DM-DM→SM SM or antiDM-antiDM →SM SM are not allowed…)

    � = mn P = nT ⌧ �n = g✓mT

    2�

    ◆3/2exp

    ⇣�mT

    ⌘Reminder (from Lec. 1): for non-relativistic species at LTE, no chemical potential

    eq

    Accounting for a possible particle/antiparticle asymmetry yields (exercise: prove it from FD/BE in the low-density limit)

    n±eq = neqe⇠±

    n+eqn�eq = n

    2eq

    ⇠± ⌘ µ±/Tremember the equilibrium condition

    for chemical potentials!

  • A STRAIGHTFORWARD GENERALIZATION, 2Deriving the equation for the difference of DM/antiDM, and introducing again the number densities normalized to the entropy density, Y=n/s, one gets

    The remaining independent equation writes:

    dY+dx

    = �xs(x)h�viH(m)

    [Y 2+ � ⇣ Y+ � Y 2eq]

    For ζ≠0 can be easily solved numerically! Exemple solution from

    Y+ � Y� = const. ⌘ ⇣

    H. Iminniyaz, M. Drees and X. Chen, JCAP 1107, 003 (2011) [1104.5548]

    T. Lin, H. B. Yu and K. M. Zurek, Phys. Rev. D 85, 063503 (2012) [1111.0293]

    Y+

    Y-

    Yeq

    Yh=0±

    20 40 60 80 10010-12

    10-11

    10-10

    10-9

    10-8

    x

    Y±HxL

    Figure 1. Evolution of Y ±(x) illustrating the e↵ect of the asymmetry ⌘. After freeze-out both Y �

    and Y + continue to evolve as the anti-particles find the particles and annihilate. The Y ±⌘=0 curveshows the abundance for ⌘ = 0, a mass m = 10 GeV and annihilation cross-section �0 = 2 pb. Incontrast, with a non-zero asymmetry ⌘ = ⌘B = 0.88⇥ 10�10 and same mass and cross-section, themore abundant species (here Y +) is depleted less than when ⌘ = 0. Also shown is the equilibriumsolution Yeq(x).

    In the above we have also defined req ⌘ e�2⇠(x), where ⇠ is determined by

    2 sinh ⇠ =⌘

    Yeq. (2.13)

    Notice from (2.12) that we have taken into account the temperature dependence in

    he↵ and ge↵. Because ge↵ is monotonically increasing with T , we see that the parentheses

    in the definition of g1/2⇤ is positive definite. In the numerical results that follow we use the

    data table from DarkSUSY [48] for the temperature dependence of g⇤(T ) and he↵(T ).6

    Eq. (2.10) reproduces the well known case ⌘ = 0 for which one finds that r = 1 for

    any x. We will instead focus on scenarios with ⌘ 6= 0 in the following. As shown in Fig. 1,the e↵ect of nonzero ⌘ is to deplete the less abundant species more e�ciently compared to

    ⌘ = 0 for the same annihilation cross section and mass.

    2.2 The relic abundance of asymmetric species

    Equation (2.10) can be solved by numerical methods and imposing an appropriate initial

    condition at a scale x = xi � 10, where the non-relativistic approximation works very well.Although we have chosen xi = 10, we have checked that larger values (10 < xi < xf , where

    xf is defined below) do not alter the final result. From (2.10) one sees that in the early

    6Note that we use the notation of DarkSUSY for the massless degrees of freedom parameters. To

    translate our notation to that of Kolb and Turner [49] one should make the substitutions ge↵ ! g⇤ andhe↵ ! g⇤S .

    – 6 –

    M. L. Graesser, I. M. Shoemaker and L. Vecchi, JHEP 1110, 110 (2011) [1103.2771]

    See also the following for useful analytical approximations

    Note: For ζ=0 same solution as for standard self-conjugated freeze-out applies to each of Y+,Y-!

  • EXERCISE, PART 3

    Apply the freeze-out formalism (analytically and then numerically) to baryons (~i.e. “protons”) with mb~1 GeV & ~ 1/mπ2

    (for the latter, you can also perform a more accurate search e.g. on PDB)

    What is the current energy density of baryons? (Ωbh2~1/5ΩCDM h2~0.02, or look at recent Planck publication…)

    Generalize the previously developed code to allow for a possible asymmetry.

    Find the value of ζb for which the calculation matches the data. Relate the parameter ζb to the commonly used η=nb/nγ, as well as to Ωbh2

    Is freeze-out of a symmetric universe made of protons/antiprotons a plausible mechanism behind their abundance?

    Can you explain why these variables can be used interchangeably, without reference to the cross section, for instance?

  • THE MISLEADING PIE CHART PROBLEM

    Perhaps you heard that we do not know what 95% of the Universe is made of (because of DM and DE)

    The situation is worse! The latter exercise shows you the we do not know where baryons come from either! (and actually for neutrinos we think we know how they were produced, but not the origin & value

    of their mass, hence their contribution to pie)

    The (yet unknown) physical mechanism behind the observed matter-antimatter asymmetry is called

    Baryogenesis

    Apart for “theoretical elegance”, it seems plausible that this is dynamically generated, not some initial condition

    (should have been diluted away via inflation)

  • ANDREI SAKHAROV CONDITIONS

    ! B is violated (obvious, otherwise B does not change with t) ! violation of C (to avoid extra B-production reactions balanced by extra anti-B production reactions) & CP (to avoid excess of left-handed B compensated by right-handed anti-B) ! departure from thermal equilibrium takes place (otherwise processes increasing & decreasing B @ equilibrium by CPT-symmetry, i.e. equil. distributions only depend on mass, equal for part/antipart. due to CPT)

    One can generate baryon asymmetry dynamically in QFT provided that

    Remarkably, these conditions are met in the SM as well... but (to cut a long story short) but “too weakly” to be useful (2nd order EW phase transition, small CP-violation…)

    ! EW baryogenesis: in extended models of TeV scale physics (SUSY or not), the problem mentioned with the SM could be overcome.

    ! Leptogenesis: initial L production, then reshuffled via sphalerons (violating B+L but conserving B-L in the SM at T~100 GeV). L asymmetry linked to neutrino mass generation, but typically (although not necessarily) high mass scale phenomenon

    The two main (not unique!) classes of BSM model trying to explain that are:

    Nobel Peace Prize 1975

  • ASYMMETRIC DM?Not in WIMP paradigm! DM result of thermal freeze-out, baryons from some unknown baryogenesis mechanism. A nice avenue would be some similar process happening in

    the dark sector, too!

    Theoretically, not hard: all classes of models considered for baryogenesis can be considered (actually more, since phenomenologically more freedom in dark sector...)

    K. Zurek “Asymmetric Dark Matter: Theories, Signatures, and Constraints,’'  Phys. Rept. 537 91 (2014) 1308.0338

    Is this relation suggestive of a common origin (co-genesis)?

    �dm�b

    ' 5

    ndm � n̄dm 6= 0! Introduce a DM candidate which is not self-conjugated, allowing for asymmetry in number density ndm � n̄dm / nb � n̄b! Use dynamics to relate it to the baryon asymmetry

    �dm�b

    =|ndm � n̄dm|mdm

    nb mb⇥ �mdm

    mN! Generically one has (κ model dependent!)

  • EXERCISE, PART 4

    Assume that DM is asymmetric, characterized by an asymmetry constant of the same order of the baryon one, say ζX ~ 0.2 ζb

    
Derive the DM mass mX from matching the DM to baryon abundance ratio.

    While keeping these parameters (ζX,mX) fixed, study with your code for which values of the DM annihilation cross section the relic abundance is actually

    dominated by the asymmetric component (If you wish, Y+>>Y-).

    Are the values you found similar to the ones needed in the WIMP paradigm?
If not, using the naive scaling ~ g4/mX2 for which values of g it is ok? 


    What if you use ~ α2/mU2: For which value of mU the calculation matches?

  • SUMMARY OF WHAT WE LEARNED

    ✤ We described heuristically how to derive the relic abundance via freeze-out mechanism

    ✤ We saw why non-relativistic relics seem to work... WIMP cold DM paradigm.

    ✤ WIMPs rich in collider, direct and indirect signatures, hence well studied.

    ✤ We described the “freeze-in” alternative (harder to detect!)

    ✤ Generalization to non-self conjugated DM

    ✤Asymmetric DM alternative. Introduction of Baryogenesis, and links with it

    ✤ Next, we shall have a more microscopic look at the “Boltzmann Eq.” justifying better the above heuristic equation (and explain how to compute RHS)