New Options for SUSY-kind Dark Matter
Elena Arbuzova
Dubna State University & Novosibirsk State University
based on common works with A.D. Dolgov and R.S. Singh
Symmetry 13 (2021) 5, 877; Eur.Phys.J.C 80 (2020) 11, 1047;
JCAP 04 (2019) 014; JCAP 07 (2018) 019
supported by the RSF Grant 20-42-09010

20th Lomonosov Conference on Elementary Particle Physics
Moscow State University, Moscow, 19-25 August, 2021

E.V. Arbuzova

New Options for SUSY-kind DM

25 August 2021

1 / 15Dark Matter Mystery
invisible form of matter disclosing itself through its gravitational action
electrically neutral, since doesn’t scatter lightproperties are practically unknown

Particles of many different types can be DM candidates
The fractional mass density of dark matter:%DMΩDM =≈ 0.265%crit
The critical energy density of the universe:%crit =

2
3H02 mPl
3≈ 5 keV/cm ,8πmPl = 1.22 · 1019 GeV = 2.18 · 10−5 g

H0 is the present day value of the Hubble parameter:
H0 = 100h km s−1 Mpc−1 ≈ 70 km s−1 Mpc−1

The observed mass density of DM in contemporary universe:%DM ≈ 1 keV/cm3
E.V. Arbuzova

New Options for SUSY-kind DM

25 August 2021

2 / 15SUSY Dark Matter
Low energy minimal SUSY model:
Predicts the existence of stable LSPs with mass MLSP ∼ 100–1000 GeV
No manifestation at LHC =⇒ restricted parameter space open for SUSY

The LSP’s energy density(obs)%LSP ∼ %DM (MLSP /1 TeV )2 ,(obs)%DM ≈ 1 keV /cm3

For MLSP ∼ 1 TeV, %LSP is of the order of the observed DM energy density
For larger masses LSPs would overclose the universe.
LSPs are practically excluded as DM particles in the conventional cosmology.

In (R + R 2 )-gravity the energy density of LSPs may be much lower =⇒it reopens for them the chance to be the dark matter, if MLSP ≥ 1000TeV.
EA, A. D. Dolgov and R. S. Singh, “Dark matter in R + R 2 cosmology,” JCAP 04(2019) 014, arXiv:1811.05399 [astro-ph.CO]
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General Relativity (GR):SEH = −2mPl16πZ√d 4 x −g R

Beyond the frameworks of GR:"#2 Z2pmPlR
Stot = −d 4 x −g R−+ Sm
16π6mR2R 2 -term leads to exponential cosmological expansion (Starobinsky inflation).
(A. A. Starobinsky, Phys. Lett. B91, 99 (1980))

It creates considerable deviation from the Friedmann cosmology in the
post-inflationary epoch. (EA, A.D. Dolgov and R. Singh, “Distortion of thestandard cosmology in R + R 2 theory,” JCAP 1807 (2018) no.07, 019)

Kinetics of massive species and the density of dark matter particles differ
significantly from those in the conventional cosmology.
Curvature R(t) can be considered as an effective scalar field (scalaron) with themass mR and with the decay width Γ.
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Cosmological evolution in R 2 theory: 4 distinct epochs
I. Inflationary stage: the curvature was sufficiently large, and the universe
expanded exponentially with slowly decreasing R(t).
II. Scalaron dominated regime:R(t) approached zero and started to oscillate around it as
R=−4mR cos(mR t + θ),tmR = 3 · 1013 GeV

Hubble parameter:ȧ2=[1 + sin(mR t + θ)]
a3tEnergy density of matter drops down as
22mR3
3H 2 mPl3mPl%R 2 =instead of %GR ==240πt8π32πt 2H≡

III. Transition period from scalaron domination to domination of the usual(relativistic) matter.
IV. After complete decay of scalaron we arrive to the cosmology governed by GR.
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Cosmological energy density for different decay channels
Scalaron decay into 2 massless scalars minimally coupled to gravity
Γs =mR32 ,24mPl%s =mR3240πt

Scalaron decay into a pair of fermions with mass mf :
Γf =mR mf22 ,6mPl%f =mR mf2
240πt

Scalaron decay into gauge bosons induced by the conformal anomaly:Γan =β12 α2 N mR32 ,96π 2 mPl%an =β12 α2 N mR3
4π 2 120πt

β1 is the first coefficient of the beta-function, N is the rank of the gauge groupα is the gauge coupling constant (at high energies it depends upon the theory).

Much slower decrease of the energy density of matter than normally forrelativistic matter is ensured by the influx of energy from the scalaron decay.
Normally for relativistic matter: % ∼ 1/a4 (t) ∼ 1/t 8/3 , since a(t) ∼ t (2/3) at SD.
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Connection of the temperature with time: GR ⇐⇒ R 2
In thermalized plasma with %therm = π 2 g∗ T 4 /30
1/2
290
3mPlπ 2 g∗ T 42=⇒
(tT)
=mPl = const%GR ==GR32πt 230
32π 3 g∗g∗ is the number of relativistic species in the plasma, g∗ ∼ 100.
R 2 -theory:mR3π 2 g∗ T 4=240πt30m3%an = 2.6 · 10−2 αR2 Rt%s ==⇒=⇒mR3= const
8π 3 g∗0.78(tT 4 )an = 2 αR2 mR3 = constπ g∗(tT 4 )s =The coupling αR is taken at the energies equal to the scalaron mass.

CorrespondinglyṪ
T!

E.V. Arbuzova

GR1=−2tṪT
New Options for SUSY-kind DM!
=−s;an14t
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Evolution of X -particles in thermal plasmaFreezing of massive species X =⇒ Zeldovich Eq., 1965 (Lee-Weinberg, 1977):
3/2
MX T
22ṅX + 3HnX = −hσann v i nX − neq , neq = gs
e −MX /T
2πhσann v i is the thermally averaged annihilation cross-section of X-particles
neq is their equilibrium number density, gs is the number of spin states.

For annihilation of the non-relativistic particles:
hσann v i = σann v =hσann v i =πα2 βann(S-wave),
2MX23πα2 βann T(P-wave, Majorana fermions)
MX2MX2

MX is a mass of X -particle, α is a coupling constant, in SUSY theories α ∼ 0.01
βann is a numerical parameter ∼ the number of annihilation channels, β ∼ 10.
We assume that direct X -particle production by R(t) is suppressed in comparison withinverse annihilation of light particles into X X̄ -pair.
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Some comments
Two possible channels to produce massive stable X -particles:
Directly through the scalaron decay into a pair X X̄ ,By inverse annihilation of relativistic particles in thermal plasma.

Direct production of X X̄ -pair by scalaron gives(0)%X ≈ %DM ≈ 1keV/cm3 , if MX ≈ 107 GeV”Catch-22”:
For such small mass thermal production results in too large %X .
(0)For larger masses %X would be unacceptably larger than %DM .
A possible way out:
Since oscillating curvature scalar creates particles only in symmetric state,
the direct production of X -particles is forbidden, if they are Majorana
fermions, which must be in antisymmetric state.
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Scalaron decay into massless non-conformal scalars
Dimensionless Zeldovich equation
2
 a 3Mx
df0.03gs α2 βann mR 3 f 2 − feqinf, x ==−,n=nin
X
35dx
π g∗
MXxaTFor g∗ = 100, α = 0.01, βann = 10, mR = 3 × 1013 GeV, and nγ = 412/cm3

The present day energy density of the X -particles:%X = MX nγ ffin ≈ 1081010 GeV!

MXGeV/cm3

To compare with the observed energy density of DM: %DM ≈ 1 keV/cm3 .
X -particles must have huge mass MX  mR to make reasonable DM density.
If MX > mR , then classical scalaron field can still create X -particles, but theprobability of their production would be strongly suppressed =⇒ such LSP withthe mass somewhat larger than mR could successfully make the cosmological DM.
E.V. Arbuzova

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Scalaron decay into a pair of fermions
The decay width and the energy density:
Γf =mR mf2,26mPl%f =mR mf2
120πt

The largest contribution into the cosmological energy density at scalaron dominatedregime is presented by the decay into the heaviest fermion species.

We assume:
The mass of the LSP is considerably smaller than the masses of the otherdecay products, mX < mf , at least as mX . 0.1mf .
The direct production of X -particles by R(t) can be neglected.

In such a case LSPs are dominantly produced by the secondary reactions inplasma, which was created by the scalaron production of heavier particles.

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Kinetic equation for freezing of fermionic species
2dfα2 βann nin mR mf2 f 2 − feq=− 3dx
2π g∗x5mX6nin = 0.09gs mX3 is the initial number density of X -particles at T ∼ mX .

%X = mX nγnXnrel
now= 7 · 10−9mf3mX mR

cm−3α = 0.01, βann = 10, g∗ = 100, nγ ≈ 412/cm3 , nrel ≈ %rel /3TIf we take mf = 105 GeV and mX = 104 GeV, then %X  %DM .

%X becomes comparable with the energy density of the cosmological DM,%DM ≈ 1 keV/cm3 , if mX ∼ 106 GeV, mf ∼ 107 GeV: m3  106 GeV  keV
f%X = 0.23
107 GeVmX
cm3E.V. Arbuzova

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Scalaron decay into gauge bosons due to conformal anomaly
X , X̄ are Majorana fermions =⇒ direct production by scalaron is forbidden.
X X̄ -pairs are produced through the inverse annihilation of relativistic
particles in the thermal plasma.

Log of the ratio of the energy density of X -particles to the observed energy density ofDM as a function of µ = mR /MX calculated through the Zeldovich equation.

X-particles may be viable candidates for the carriers of the cosmological darkmatter, if their mass MX ≈ 5 · 1012 GeV.
E.V. Arbuzova

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Possible observations
According to our results, the mass of DM particles, with the interaction strength
typical for supersymmetric ones, can be in the range from 106 to 1013 GeV.

Possibilities to make X-particles visible:
1

Annihilation effects in clusters of dark matter in galaxies and galactic halos,
in which, according toV. S. Berezinsky, V. I. Dokuchaev and Y. N. Eroshenko, Small-scale clumps
of dark matter, Phys. Usp. 57 (2014) 1 [arXiv:1405.2204]

the density of DM is many times higher than DM cosmological density.
2The decay of superheavy DM particles, which could have a lifetime longenough to manifest themselves as stable DM, but at the same time lead tothe possibly observable contribution to the UHECR spectrum.

3

Furthermore, instability of superheavy DM particles can arise due toZeldovich mechanism through virtual black holes formation.
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Conclusions
The existence of stable particles with interaction strength typical for SUSY andheavier than several TeV is in tension with conventional Friedmann cosmology.
Starobinsky inflationary model opens a way to save life of such X -particles,
because in this model the density of heavy relics could be significantly reduced.
If the epoch of the domination of the curvature oscillations (scalaron domination)
lasted after freezing of massive species, their density with respect to the plasma
entropy could be noticeably diluted by radiation from the scalaron decay.
The range of allowed masses depends upon the dominant decay mode of scalaron.
Dominant decay channel of the scalaron

Allowed MX to form DM

Minimally coupled scalars mode

MX & MR ≈ 3 · 1013 GeV

Massive fermions modeMX ∼ 106 GeV

Gauge bosons modeMX ∼ 5 · 1012 GeV

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The END
Thank You for Your Attention

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