SLIDE 1 What I learned on Dark Matter while preparing a lecture on it
Andr´ as Patk´
Institute of Physics, E¨
- tv¨
- s University, Budapest
Outline:
- Astrophysical DM evidences: rotation-curve, grav. lensing, CMB, DD-ID detections
- Seminar theme:Rotational curves in standard and modified gravity models
- Neutrinos: excluding SM ν’s; the WIMP paradigm
- Seminar theme:Sterile neutrino DM
- Basic theory tools of WIMP search: Effective Theories, Simplified Models
- Seminar theme:Inflationary Higgs fluctuations might account for DM and explain the
present ground state of SM simultanously
SLIDE 2
Galactic star distribution and rotation curves
mv2 r
= GmM(<r)
r2
F .Zwicky (1930) → V. Rubin (1980): DM content ≈ 80%
X-ray energy distribution measurement in galaxy clusters – in excess to the virial
theorem (2Ekin
T = V T pot) with luminous matter only
Gravitational lensing: map of the whole of the gravitating matter content
(cca. 2003-04)
SLIDE 3
Modeling cosmological Background radiation (WMAP, PLANCK)
Ωbh2 = 0.02226(23), ΩDMh2 = 0.1186(20), h =
H 100km/Mpc/s = 0.678(9)
Claims for direct and indirect DM observations
SLIDE 4 There is enough uncertainty for investigating alternatives! Example: Dark Matter content from rotation curves of 40 galaxies (Almeida, Amendola, Niro, arXiv:1805.11.067) Specific MOND model: 5th force mediated by scalar field Φ(x), assuming different coupling αi to DM and BM T µ
(BM)ν;µ = −αBMT(BM)Φ;ν,
T µ
(DM)ν;µ = −αDMT(DM)Φ;ν
T µ
(Φ)ν;µ = (αBMT(BM) + αDMT(DM))Φ;ν,
T = T µ
µ
Local gravity experiments: αBM 10−2 → MOND-contribution of BM is negligible DM MOND potential: ΨMOND(x) = −Gβ
ρ(x′) |x−x′|e−|x−x′|/λ, β = αBMαDM
DM matter distribution of Navarro-Frenk-White (1996): ρNF W(r) =
ρs
r rs(1+ r rs) 2
v2
circular = rdΨ dr ,
Ψ =
gas + Ψ(N) disk + Ψ(N) bulge
DM + ΨMOND
SLIDE 5
Data set, fitting, results SPARC database: measurements performed with SPITZER satellite (arXiv:1606.09251), 175 galaxies Combined fits to 4 randomly chosen set, consisting of 10 galaxies each Number of parameters β, λ + 2 constants for FRW-profile, 18-19 further constants characterising the structure of BM Result: β = 0.34 ± 0.04, λ = 5.61 ± 0.91kpc 8σ deviation from β = 0, DM density reduced by 20%
SLIDE 6 Dark Matter: non-luminous, at most weakly interacting matter with lifetime longer than the age of the Universe
Textbook material: D.H. Perkins: Particle Astrophysics, Oxford U.P ., 2003,2009 Natural first candidate: relic SM neutrinos (Marx, Szalay 1976) At T > 3MeV in thermal equilibrium through γ ↔ e+ + e− ↔ νi + ¯ νi Decoupling: weak reaction rate Wweak < expansion rate of the universe Hrad Hrad = 2.07 · 105g1/2
∗
(kBT )2, g∗ = 43/2 Wweak =< ρlσweakv >, ρl = 3×2.404
4π2h3c3(kBT )3,
σweak =
G2 F (3.15kBT )2 3π
, v ≈ c Wweak ≈
1 25(kBT )5
→ kBTdec ≈ 3MeV
SLIDE 7 Actual number density of neutrinos/flavor: 113/cm3, neutrinos would represent dark matter fully with
i mνi ∼ 12eV
the closure ratio Ων =
ρν ρcrit for relativistic species increases linearly with mν
Problems which discard this option:
- mν too large compared to existing upper bounds (< 0.5eV)
- Hot dark matter would smooth out density fluctuations, contrary the observed large scale
structure
SLIDE 8 The WIMP option
χχ-annihilation into γ maintains the thermal equilibrium until decoupling: MW IMP Tdec
2π
3/2 e−(MW IMP /Tdec) =
fT 2 dec <σannv>MP L, 1 2MW IMPv2 = 3 2Tdec
Annihilation cross-section dependence on MW IMP: for MW IMP < MEW, σann ∼ G2
FM 2 W IMP
for MW IMP > MEW, σann ∼ g4M −2
W IMP
Number density at present: NW IMP,0 =
Tdec
3 ×
fT 2 dec <σannv>MP L
Closure parameter today: ΩW IMP,0 =
MW IMP NW IMP,0 ρcrit,0
, ρcrit,0 =
3H2 0c2 8πG
Allowed windows for WIMP ”miracle”:
SLIDE 9 Appealing WIMP-candidate: sterile neutrinos Latest review: A. Boyarsky et al. arXiv:1807.07938 Motivations and hints
- Dynamical interpretation of ν oscillations
- Interpretation of very small non-zero mν of active ν’s (<< me)
- Arbitrarily weaker interaction with baryonic matter than that of active ν’s
- Miniboone and LSND oscillation experiments admit the existence of fourth neutrino
generation Construction: n gauge singlet RH neutrinos + 3 SM-neutrinos; all Majorana particles L = LSM + iνR / ∂νR − lLF νR ˜ Φ − ˜ Φ†νRF †lL − 1
2
RMMνR + νRM † Mνc R
(˜ Φ = ǫΦ) νR: n sterile flavors, MM: the n × n mass-matrix of the sterile sector, F : 3 × n matrix of the Yukawa-couplings of the sterile neutrinos to singlets formed from the Higgs-dublet Φ and the SM lL’s.
SLIDE 10 The type-I see-saw mechanism Neutrino oscillation due to distinction between interaction (α) and mass (i) eigenstates: νLα = (Vν)αi νi, Vν = (1 + η)Uν η characterises the unitarity violation due to probability escape into unknown particles. Perturbative mass generation via νR static exchange among ˜ ΦTlc
L and lL ˜
Φ singlets: → 1
2
Φ
M F T
˜ ΦTlc
L
Higgs-effect: Φ → (0, v)T and choosing the ”cut-off” Λ= min{eigenvalues of MM} dimension-5 effective (non-renormalisable) operator is produced: mν = −mDM −1mT
D ∼ O(v2/Λ),
mD = F v
SLIDE 11 The type-I see-saw mechanism Exact diagonalisation of the mass-term
1 2
νc
R
mD mT
D
MM νc
L
νR
N (M) mdiag
ν
M diag
N
ν(M) N (M)
- with the mass-eigenstates expressed through interaction states
ν(M) = νL + νc
L − θνc R − θ∗νR = V † ν ν(W ) L
+ V T
ν ν(W )c L
− U †
νθν(W )c R
− U T
ν θ∗ν(W ) R
, N (M) = νR + νc
R + θTνc L + θ†νL = V † Nν(W ) R
+ V T
N ν(W )c R
+ U †
Nθν(W )c L
+ U T
Nθ†ν(W ) L
. (mixing matrix: θ = mDM −1
M )
Weak interaction of N suppressed by the action of Θ = θU ∗
N:
LN−int = − g
√ 2
µ + eLγµΘNW − µ
g 2 cos ΘW Zµ
gMN mW √ 2
SLIDE 12 Type-I see-saw, remarks, experimental (observational) hints Simplified (less general) construction: – associate exactly 1 heavy neutrino with 1 light flavor – perform see-saw computation for each of them separately. Strategy (Shaposhnikov, 2004): – 2 of them can be used to account for ∆m2
solar, ∆m2 atmospheric,
– 1 is used to arrange BAU via lepton number asymmetry Thermal Θ-suppressed production of N: ultraweak interaction prevents thermalisation → nN << nν Gunn-Tremaine mass lower bound (1979): The Fermi velocity of degenerate Fermi-Dirac gas of mass mmin in a sphere of radius R and mass M should not exceed the escape velocity →
2fdof m4 minR3
1/3 =
R
→ mmin ∼ 100 − 400eV Refined analysis using dwarf spheroidal galaxies: Boyarsky et al., JCAP 0903 (2009) 005
SLIDE 13 Decay-constraint from main channel N → νανβνβ τ −1 = ΓN→3ν = G2
F M5
96π3
τ < τUniverse = 4.4 · 1017s
- α |θα|2 < 3.3 · 10−4 10keV
M
5 Possible clear X-ray signal from 1-loop level N → γ + ν decay ΓN→γν = 9αG2
F M5
256π4
1 128ΓN→3ν
Galactic 3.5 keV line confirmed at 11σ level!!! No accepted interpretation yet.
SLIDE 14 Basic theory tools of DM search Review article: Simplified models vs. Effective Field Theory Approaches,
by A. de Simone and T. Jacques, EPJ C76 (2016) 367
Search strategies:
- DM-production at colliders
- Direct Detection (DD) of scattering of DM-particles off nuclei
- Indirect Detection (ID) of decay products from self-annihilation of
DM-particles WIMP phenomenology should offer a unified treatment of the corresponding
- bservables, keeping the number of parameters minimal
SLIDE 15 Basic theory tools of DM search
Two methods widely used:
- I. Effective Field Theory (EFT): most general contact interactions of DM and SM particles
emerges by ”integrating” over mediating force fields at a characteristic scale, expansion in powers of (energy/scale).
- fits well DD-ID experiments (applicable in several channels),
- questionable validity of the expansion in annihilation channel (E ∼ 2MW IMP),
- misses resonant enhancement, when E ≈ Mforce
- many terms in the effective theory, if no specific model is employed
Scattering experiments Typical characteristics of DAMA, XENON, etc. experiments: v ∼ 10−3c, Erecoil 10MeV Differential event rate:
dR dErecoil = nχ mχmA
dσχA dErecoil
SLIDE 16 Basic theory tools of DM search: the EFT-approach Nuclear cross-section constructed from σχN, computable in Born-approximation of NR quantum mechanics 12 invariant Fourier-transformed non-relativistic χ − N contact interaction potentials si = ξ†
s′ σi 2 ξs
Spin-independent scattering σN
SI; the unit operator dominates it; ONR 1
= 1 (ONR
11
also contributes, velocity dependent terms are subdominant)
SLIDE 17
Basic theory tools of DM search: the EFT-approach Question: which of the 10 effective relativistic χ − N interactions contributes to ONR
1
? Answer: ONR
1
←
N=p,n(cN D1ON D1 + cN D5ON D5),
ON
D1 = ¯
χχ ¯ NN, ON
D5 = ¯
χγµχ ¯ NγµN
SLIDE 18 Basic theory tools of DM search: the EFT-approach Incoherent sum σtotal
SI
=
N µ2
χN
π
D1)2 + (cN D5)2
Nuclear coefficients should be constructed to DM-parton interaction Example: cp
Di = 4mχmp(2cu Di + cd Di),
cn
D5 = 4mχmn(cu Di + 2cd Di)
c(q)
Di ∼ 1 M2
(q)
Quark level coefficients can be constrained at LHC, and RG-evolution is understood when used in NR experiments.
More details: A. De Simone, T. Jacques, EPJ C76 (2016) 367
SLIDE 19 Basic theory tools of DM search: the SimpM-approach
- II. Simplified Models (SimpM): Mediator field between the DM and SM particle
- direct contact to the UV (high scale) completion of the BSM theory,
- restricted number of EFT operators when type of the mediator is fixed,
- Requirements: Lorentz-inv., Gauge inv. (above the mediator scale), minimal flavor SB
Categorization of the models:
SLIDE 20 The SimpM-approach: the case of scalar DM particle + Higgs mediator ”0s0” model L0s0 = 1
2 (∂µΦ)2 − 1 2m2 φΦ2 − λΦ 4 Φ2H†H
→ (broken phase)1
2 (∂µΦ)2 − 1 2m2 φΦ2 − λΦ 2 √ 2vhΦ2
Interesting regime:
mφ < mh/2, Higgs generated in a collider experiment can decay h → ΦΦ Γ(h → ΦΦ) =
λ2
Φv2
32πmh
4m2
φ
m2
h
Contribution to the invisible width of Higgs: Γinv Γtotal 0.2 → λφ 10−2 Γtotal = 4.2MeV, mh = 125.6GeV Self-annihilation to SM-particle-pairs
< σvrel(ΦΦ → h → ¯ ff) >=
λ2
Φm2 f
8π
m2 f m2 Φ
3/2 (m2
h−4m2 φ)2+m2 hΓ2 h,
also through UV-completion:
< σvrel(ΦΦ → hh) >=
λ2
Φ
512πm2
Φ
(mφ >> mh := 0)
SLIDE 21 the SimpM-approach: Dirac-spinor DM particle + scalar/pseudoscalar mediator (0s1/2) L0Ss1/2 = 1
2 (∂µS)2 − 1 2m2 SS2 + ¯
χ(i / ∂ − mχ)χ − gχS ¯ χχ − gSMS
j yf √ 2 ¯
ff, L0As1/2 = 1
2 (∂µA)2 − 1 2m2 AA2 + ¯
χ(i / ∂ − mχ)χ − gχiA¯ χγ5χ − igSMA
j yf √ 2 ¯
ff, S/A − f − f coupling arises from Higgs-effect yf =
√ 2mf v
, gSM assumed universal, no Higgs-S mixing (∼ S2|H|2) is assumed Parameters: mχ, mS/A, gχ, gSM. Collider search strategies: / ET + jet, / ET + t¯ t, / ET + b¯ b:
SLIDE 22 the SimpM-approach: Dirac-spinor DM particle + scalar/pseudoscalar mediator QCD-production rate combined with (for nearly on-mass-shell S/A): Γ(S/A → ¯ χχ) =
g2
χmS/A
8π
4m2
χ
m2
S/A
n/2 , n = 3, for A, n = 1 for S Self-annihilation into SM fermion-pair < σvrel > (¯ χχ → S → ¯ ff) = Nc(f)
g2
χg2 SMy2 f
16π m2
χ
m2 f m2 χ
3/2 (m2
S−4m2 χ)2+m2 SΓ2 S × v2
rel
← p-wave decay < σvrel > (¯ χχ → A → ¯ ff) = Nc(f)
g2
χg2 SMy2 f
16π m2
χ
m2 f m2 χ
3/2 (m2
A−4m2 χ)2+m2 AΓ2 A
SLIDE 23 the SimpM-approach: Dirac-spinor DM particle + scalar/pseudoscalar mediator
Effective four-fermion interaction describing low-energy (non-relativistic) χ − N scattering OS = gχgSMyq
√ 2m2
S (¯
χχ)(¯ qq), OA = gχgSMyq
√ 2m2
A (¯
χiγ5χ)(¯ qiγ5q) Remarks on Higgs as mediator in case of the 0s1/2 model:
- Gauge-invariant effective interaction for E > mh:
Lint = −gχH†H
√ 2v ¯
χχ → gχh¯ χχ
- h replaces S in the general results
- Limitations on gχ arise from allowed invisible width of Higgs
SLIDE 24 Summary
- 1. Uniquness of the DM interpretation of each of the observational evidences still
allows/requires critical reassessment
- 2. 3 mass ranges, where relic density of Weakly Interacting Massive Particles could stand for
DM
- 3. Effective Field / Mediator Field Theory framework for unified phenomenology of
DM particle production in colliders + Direct Detection scattering off heavy nuclei + Indirect Detection of annihilation/decay products
SLIDE 25 SM Higgs-vacuum instability and primordial black hole production
At last
- 4. Possible DM from quantum theory of Standard Fields
The instability of VHiggs = −1
2m2h2 + λeff(h) 24
h4 Lifetime of the metastable ground state of the universe > tUniverse ≈ 13.7 billion years
SLIDE 26
SM Higgs-vacuum instability and primordial black hole production During inflation the Higgs-amplitude performs random walk under quantum kicks: |∆qh| = H
2π
(A. A. Starobinsky and J. Yokoyama, Phys.Rev. D50 (1994) 6357)
SLIDE 27
SM Higgs-vacuum stabilisation during reheating Escape from instability: After tend, high enough Treheat produces thermal mass-term 1
2m2 Th2 c
compensating −λ
4h4 c
Condition: 1
20.12T 2 RH > λ 4h2 end
previous 3 figures: Espinosa, Giudice, Morgante, Riotto, Senatore, Strumia, Tetradis, JHEP 09 (2015) 174
SLIDE 28 SM Higgs-vacuum instability and primordial black hole production The scenario [J.R. Espinosa, D. Racco, A. Riotto, PRL 120, 121301 (2018)]:
- On inflating background classical roll-down of hc starts at t∗: H(t∗ − tend) 20 e-folding
(see green continous curve on the Figure above)
- Fast increase of hc induces long wavelength fluctuations δhk of increasing amplitude:
δhk ∼
H
√
2k3 amplified after horizon crossing at tk < tend: δhk ∼ H
√
2k3 ˙ hc(t) ˙ hc(tk)
- curvature dominated by Higgs-fluctuations: ζh = H δρh
˙ ρh = H2 √ 2k3 ˙ hc(tk)
- δhk returns below horizon in the radiation domination epoch and produces density contrast
∆(x) =
4 9a2H2∇2ζ(x)
- When and where it exceeds ∆c ∼ 0.45 the matteoriginating from Higgs-decay collapses
to form Primordial Black Holes
- With fine tuning ΩP BH(teq)
ΩDM
∼ 10−2. Mergers, matter accretion probably increase this
value to O(1) Primordial gravitational waves generated by the same mechanism (Espinosa, Racco, Riotto, arXiv:1804.07732)
SLIDE 29 Stochastic treatment of long wavelength part of a self-interacting scalar field
L = 1
2
0Φ2 + ξRφ2
− λ
4Φ4
de Sitter metric background: R = −12H2, H = const. Splitting the Heisenberg field operator into short and long wavelength parts: Φ(x, t) = Φ(x, t) +
- k Θ(k − ǫa(t)H)
- akϕk(t)e−ikx + a†
kϕ∗ k(t)eikx
Φ(x, t) coarse grained field with wavelengths longer than the size of the horizon. Mode functions of short wavelength modes ( k2 >> m2) obey: ¨ ϕk + 3H ˙ ϕk + k2
a2ϕk = 0
Solution: ϕk =
H √ 2k
k
− 1
Hη = a0eHt
Slow-roll EoM for Φ(x, t): 3H ˙ Φ(x, t) = −V ′(Φ(x, t)) + ǫ˙ a(t)H
- k δ(k − ǫa(t)H)
- akϕk(t)e−ikx + a†
kϕ∗ k(t)eikx
Interpretable as (operator) diffusion equation with noise term f(x, t) =
1 3Hǫ˙
a(t)H
- k δ(k − ǫa(t)H)
- akϕk(t)e−ikx + a†
kϕ∗ k(t)eikx
Noise correlation: < f(x1, t1)f(x2, t2) >= H3
4π2δ(t1 − t2)sin z z ,
z = ǫa(t)H|x1 − x2| For x1 ≈ x2 conventional diffusion equation with (∆Φ)2 = ”DtH” = H2
4π2
SLIDE 30 Classical slow-roll on the unstable side of the Higgs potential At some t∗ < tend the average Higgs-field jumps over to the unstable side where V (hc) ≈ −λ
4h4 c
Slow classical roll-down starts somewhat later (when ∆hq < ∆hc = ˙ hc/H): 3H ˙ hc ≈ −V ′(hc) = λh3
c
Solution relative to the end point of inflation: hc(t) = hc(tend)
c(tend) 3H
(tend−t) 1/2 Increasing amplitudes of fluctuations are driven by the increase of hc(t): ¨ δhk + 3H ˙ δhk + k2
a2δhk + V ′′(hc)δhk = back reaction of metric perturbations
In the limit of long wavelengths (k2/a2 << H2) the equation of ˙ δhk ≡ vh(k)
- nce more take derivative of the previous equation with respect the time:
¨ vh + 3H ˙ vh + V ′′(hc)vh ≈ 0 Coincides with the equation of hc, therefore δhk = C(k)˙
hc(t)
Matching at t = tk < tend where the mode k leaves the horizon: C(k) =
H
√
2k3 ˙ hc(tk).
From δhk with standard formulae one arrives at the Higgs dominated curvature perturbation!
