#### PeV IceCube signals and Dark Matter relic abundance in modified cosmologies

Eur. Phys. J. C
PeV IceCube signals and Dark Matter relic abundance in modified cosmologies
G. Lambiase 1 2
S. Mohanty 0
An. Stabile 1 2
0 Physical Research Laboratory , Ahmedabad 380009 , India
1 INFN - Gruppo Collegato di Salerno , Baronissi (SA) , Italy
2 Dipartimento di Fisica “E.R. Caianiello”, Università di Salerno , I-84084 Fisciano (SA) , Italy
The discovery by the IceCube experiment of a high-energy astrophysical neutrino flux with energies of the order of PeV, has opened new scenarios in astroparticles physics. A possibility to explain this phenomenon is to consider the minimal models of Dark Matter (DM) decay, the 4-dimensional operator ∼ yαχ L Lα H χ , which is also able to generate the correct abundance of DM in the Universe. Assuming that the cosmological background evolves according to the standard cosmological model, it follows that the rate of DM decay χ ∼ |yαχ |2 needed to get the correct DM relic abundance ( χ ∼ 10−58) differs by many orders of magnitude with respect that one needed to explain the IceCube data ( χ ∼ 10−25), making the four-dimensional operator unsuitable. In this paper we show that assuming that the early Universe evolution is governed by a modified cosmology, the discrepancy between the two the DM decay rates can be reconciled, and both the IceCube neutrino rate and relic density can be explained in a minimal model.
1 Introduction
The IceCube Collaboration, in its 4-year dataset [
1,2
], had
reported three neutrino-induced cascade events with energies
∼ 1 PeV [2]. First candidates for the generation of such
neutrino high energy events were various astrophysical sources
[
3–6
]. However after the analysis of seven years of data of
muon tracks, the IceCube reports [7] that there is no clear
correlations with the known astrophysical hot-spots like the
known supernova remnants (SNR) or Active Galactic Nuclei
(AGN). In the light of no identification of the astrophysical
sources of IceCube neutrinos it has been proposed that the
neutrinos could arise from the decay of PeV mass Dark
Matter (DM) [
8–23
] (the DM if boosted could may be of lower
mass also [
24,25
]).
Unitarity bounds on the cross section of the DM [
26,27
]
ruled out the possibility of a thermal relic density of PeV scale
DM whose annihilation or decay could produce the IceCube
neutrinos. Other production mechanisms which have been
invoked for the PeV mass dark matter relic density are a
secluded sector [53], freeze-out with resonantly enhanced
annihilations [
28
], or freeze-in [
13,21,29,30
]. To explain
IceCube events with DM of mass of ∼ PeV with lifetime
of τ ∼ 1028 sec is required [
8,9
]. If the neutrino flux at
IceCube were to be from DM annihilation into neutrinos
then the same then the same decay rate nχ /τχ must be
equal to the annihilation rate n2χ σ v which implies that
σ v (nχ τχ )−1 10−17cm3/s which is again ruled out
from unitarity constrains [
26,27
]. This implies that PeV DM
decay with lifetime 1028 s is the prefered mechanism for
explaining the IceCube neutrinos, if at all they originate from
DM.
Chianese and Merle [
31
] raised the question of whether
it is possible to explain both the PeV DM relic density and
the decay rate required for IceCube with one operator. The
minimal DM-neutrino dimension four interaction y L¯ · H χ
of DM decay is able to produce the correct DM abundance
by freeze-in for y ∼ 10−12 [
31
] but to get a decay lifetime
of 1028 s the value of y ∼ 10−29 [
22,23,31
]. This implies
that the minimal dimension four operator fails to account for
both the PeV dark matter relic abundance and the decay rate
required to explain IceCube.
The DM models analyzed in [
31
], however, are based on
General Relativity and the standard cosmological inflation
followed by a radiation dominated era. The signal of
inflation is in the cosmic microwave anisotropy spectrum while
the only experimental evidence of the radiation era is the
successful predictions of big-bang nucleosynthesis (BBN)
which occurs around T∼ 1 MeV at t ∼ 1 s. Observations
from Type Ia Supernovae [
32,33
], CMB radiation [
34,35
],
and the large scale structure [
36,37
], suggest that there are
strong evidences that the present cosmic expansion of the
Universe is accelerating. The latter is ascribed to the
existence of Dark Energy (DE), an exotic form of energy
characterized by a negative pressure that at late times dominates
over the cold and dark matter, driving the Universe to the
observed accelerating phase, and new ingredients, such as
DM and Dark Energy (DE), are required [
38–50
]. Both
inflation and dark energy have motivated the extension of
Einstein’s theory to a general f (R) theory [51]. For example
the most successful model of inflation which is consistent
with the low tensor to scalar ratio r favored by experiment
is the Starobinsky model [
52
] L = M P2 R + (1/M 2)R2. The
R + R2 Starobinsky model can be generalized to a R + Rn
model [
53,54
] which predicts a larger tensor to scalar ration
than that allowed by the Starobinsky model and which may be
accessible to experimental efforts to measure the primordial
B-mode polarization in the CMB. The Starobinsky model
and its generalization can also be derived from Supergravity
[
55,56
]. In addition to inflation another motivation for f (R)
gravity models is to explain dark energy model [
57,58
]. In
addition to these cosmologically motivated generalizations
of Einstein’s gravity there is been many classical attempts
to generalize Einstein’s gravity by adding a scalar
component to the tensor metric theory as in the Brans–Dicke model
[
59,60
].
From a cosmological point of view, one of the
consequences of dealing with the cosmology based on modified
gravity is that the thermal history of particles gets
modified. This means that if the cosmological background is
described by modified cosmologies, the expansion rates H
of the Universe can be written in terms of the expansion rate
HG R of GR, i.e. H (T ) = A(T )HG R (T ). Here the factor
A(T ) encodes the information about the underlying model
of gravity that extend/modify GR. Typically, the factor A(T )
is defined in a way that the successful predictions of the
BBN are preserved, so that A(T ) = 1 at early time, i.e. at
the pre-BBN epoch, an epoch of the Universe not directly
constrained by cosmological observations, while A(T ) → 1
when (or before) BBN starts.
In this paper we consider generalizations of the standard
cosmology with the aim to get a consistent minimal model
of PeV DM and IceCube neutrinos. In particular we
calculate the freeze-in abundance of DM in a modified
cosmology and show that the couplings required for obtaining the
required relic abundance depend upon the modified gravity
parameters. By choosing the cosmological parameters
appropriates such that there is no deviation from the predictions
of BBN, we can get the minimal model of decaying DM
L = yα L¯ α · H χ (α indicates the mass eigenstates of the
three active neutrinos, χ the DM particle, H the Higgs
doublet, L Lα the left-handed lepton doublet, and yαχ the Yukawa
couplings) satisfy relic abundance and the IceCube
requirements with single value of a combination of the couplings
α |yα2|.
The paper is organized as follows. In Sect. 2 we recall
the main topic of DM relic abundance and the IceCube data,
pointing out that they cannot be consistently explained by
using the 4-dimensional operator. In Sect. 3 we show that
the latter allows to explain both the DM relic abundance and
the IceCube experiment if it is assumed that after Inflation,
the Universe evolution is described by modified
cosmologies (instead of the standard cosmological model) at least
till the BBN starts. As example of modified cosmologies, we
shall consider scalar tensor theories, Brans–Dicke theory and
f (T ) theories, where T is the scalar torsion. Conclusions are
given in the last Sect. 4.
2 PeV neutrinos and Ice Cube data
In this Section, we recall the main features related to DM
relic abundance and IceCube data [
31
]. The simplest
4dimensional operator able to explain the IceCube high energy
signal, is given by the Lagrangian density
Ld=4 = yαχ L Lα H χ , α = e, μ, τ,
(2.1)
where χ is the DM particle that transforms as χ ∼ (1, 1, 0)
of SM, H ∼ (1, 2, +1/2) is the Higgs doublet, L Lα ∼
(1, 2, −1/2) is the left-handed lepton doublet
corresponding to the generation α(= e, μ τ ), and finally yαχ are the
Yukawa couplings.
Following [
31,61
], we confine ourselves to freeze-in
production, i.e. the DM particles are never in thermal equilibrium
since they interact very weakly, but are gradually produced
from the hot thermal bath. This occurs owing to a feeble
coupling to particles of the SM (at T mχ ), allowing to DM
particles to remain in the Universe because of the smallness
of the back-reaction rates and the slowness of the decay to
occur. Therefore a sizable DM abundance is allowed in this
model, at least until the temperature falls down to T ∼ mχ
(temperatures below mχ are such that DM particles
phasespace is kinematically difficult to access).
The evolution of the DM particle is governed by the
Boltzmann equation. Denoting with Yχ = nχ /s the DM
abundance, where nχ is the number density of the DM particles
and s = 24π52 g∗(T )T 3 the entropy density (g∗ denotes the
degrees of freedom), from the Boltzmann equation one gets
dYχ
d T
1 gχ
= − H T s (2π )3
C
d3 pχ
Eχ
(2.2)
where H is the expansion rate of the Universe and C the
general collision term. For cosmological models in which is
assumed that the relativistic degree of freedom are constant,
i.e. dg∗/d T = 0, the DM relic abundance assumes the form
DM h2 =
2m2χ s0h2
ρcr
0
∞ d x
x 2
dYχ
− d T T = mxχ
,
(2.3)
where x = mχ / T , s0 = 24π52 g∗T03 2891.2/cm3 is the
present value of the entropy density, and ρcr = 1.054 ×
10−5h2GeV/cm3 the critical density. Equation (2.3) must
reproduce the observed DM abundance [
62
]
and at the same time, explain the IceCube data.
In the case of the 4-dimensional operator (2.1), the
dominant contributions to DM production are a) the inverse decay
processes να + H 0 → χ and lα + H + → χ , that occurs when
mχ > m H + mν,l proportional to factor |yαχ |2, and b) the
Yukawa production processes, such as t + t¯ → ν¯α + χ is
proportional to |yαχ yt |2, where t represents the quark top.
For the 4-dimensional operator one gets
dYχ dYχ dYχ
d T
whose explicit expressions are
= d T inv.dec. + d T Y uk. prod.
,
dYχ m2χ χ K1
d T inv.dec. = − π 2 H s
dYχ 1
d T Y uk. prod. = − 512π 6 H s
,
×
ds˜d
α
Wtt¯→ν¯αχ + 2Wtνα→tχ
√s
˜
K1
√s
˜ , (2.7)
T
with s˜ the centre-of-mass energy, χ the interaction rate
given by
χ =
α
2
|yαχ | mχ ,
8π
α = e, μ, τ,
and K1(x ) is the modified Bessell function of the second kind.
Since ddYTχ inv.dec. is dominant1 with respect to ddYTχ Y uk. prod.,
one finds that the relic abundance induced by inverse decay
term is
DM h2|inv.dec. = 0.1188
106.75 3/2
g∗
α |yαχ |2
7.5 × 10−25 .
From (2.9) immediately follows that to have the correct DM
relic abundance (2.4) one has to require
α=e,μ,τ
|yαχ |2 = 7.5 × 10−25.
1 It can be shown [
31
] that for the range of values of yt ∈ [0.5, 1] (such
values of yt covers all possible values obtained by running its value with
t1h0e−e2neDrgMyh)2a|nindv.mdeχc., ∈hen[1c0e4, D10M8h]G2|eoVbs, one gDeMtsh2|iDnMv.dhe2c|.,Yiu.ek..ptrhoedD.M
relic abundance is mainly generated by inverse decay processes.
(2.4)
(2.5)
(2.6)
(2.8)
(2.9)
(2.10)
χ
α
However, Eq. (2.9) is in conflict with the value of
α=e,μ,τ |yαχ |2 needed to explain the IceCube data. To see
that, first note that the DM lifetime τχ = χ−1 has to be larger
that the age of the Universe, τχ > tU 4.35 × 1017s.
Moreover, IceCube spectrum sets a constraints on lower bounds
of DM lifetime τχb 1028s, i.e. τχ τχb, which is
(approximatively) model-independent (see [
31
]). Inserting (2.10) into
(2.8) one obtains
However, the observations of IceCube require the dark matter
decay lifetime τχ =∼ 1028 s which implies
|yαχ |
2
10−58,
(2.11)
and which is ∼ 33 order of magnitudes smaller than the value
of α=e,μ,τ |yαχ |2 ∼ 10−25 needed to explain the DM relic
abundance, see (2.10). As a consequence, the IceCube high
energy events and the DM relic abundance are not compatible
with the DM production if the latter is ascribed to the
4dimensional operator L Lα H χ .
3 PeV neutrinos in modified cosmologies
As discussed in the previous Sections, the 4-dimensional
operator fails to explaining both the IceCube data and DM
relic abundance. This is also a consequence of the
assumption that the early cosmological background evolves
according to GR. The characteristics of the Universe expansion,
such as the expansion rate and the composition, affects the
relic energy density of DM, as well as their velocity
distributions before structure formation. According to the standard
cosmological model, the computation of the relic density of
particles relies on the assumption that the radiation
dominated era began before the main production of relics (and that
the entropy of matter is conserved). However, any
contribution to the energy density (in matter and geometrical sector)
modifies the Hubble expansion rate, hence the relic density.
In modified cosmologies (MC), the expansion rate of the
Universe can be rewritten in the form [
63–70
]
HMC (T ) = A(T )HG R (T ),
where A(T ) is the so called (de)amplification factor. To
preserve the successful predictions of BBN, one refers to the
pre-BBN epoch since it is not directly constrained by
cosmological observations. This means A(T ) = 1 at early
time, and A(T ) → 1 before BBN begins. Typically the
(de)amplification factor can be parameterized as
(3.1)
3.32π 2 g3/2η
45 ∗
mχ
T∗
ν
1
x 5−ν
.
By inserting (3.4) and (3.5) into (2.3) and using
∞ d x x 3+ν K1(x ) = 22+ν 5 + ν 3 + ν , where
0 2 2
(z) are the Gamma functions, one obtains
where
HMC s =
DM h2
45h2
= 1.66π 2g3/2
ν
T∗
mχ
×
0.1188
s0 MPl χ 22+ν
ρcr
mχ
η
,
A(T ) = η
where T∗ is a reference temperature, and {η, ν} free
parameters that depend on the cosmological model under
consideration.2 Investigations along these lines have been
performed in different cosmological scenarios [
63–72
], where
The parameter ν labels cosmological models: ν = 2 in
Randall-Sundrum type II brane cosmology [73], ν = 1 in
kination models [
74–77
], ν = 0 in cosmologies with an
overall boost of the Hubble expansion rate [63], ν = − 0.8
in scalar-tensor cosmology [
63,78
], ν = 2/n − 2 in f (R)
cosmology, with f (R) = R + α Rn [
79,80
].
In terms of the modified expansion rate (3.2), it then
follows that the inverse decay processes (2.6) takes the form
dYχ
d T inv.dec. = − π 2 HMC s
m2χ χ
K1
A comment is in order. The general analysis performed
in [
63
] provides upper bound on η for the
cosmological models with ν = − 0.8, 0, 1, 2, i.e. η 10 ÷ 106
for DM masses mχ ∼ (102 ÷ 104) GeV. However, these
bounds were derived to explain the PAMELA experiment
on the observed electron/positron excess. Relaxing them, the
parameters {η, ν, T∗} are arbitrary and may be choose such
that the condition (3.9) is fulfilled. Therefore we may have
− 3 < ν < 0 for T∗ < Mχ or ν > 0 for T∗ > mχ .
4 Examples of modified cosmologies
As pointed out in the Introduction, cosmological
observations have provided evidences of cosmic acceleration of the
present Universe. Instead to invoke the existence of DE,
modifying hence the matter sector of GR, an alternative
possibility is to modify/generalize the geometrical sector of GR. This
approach leads to ETG, and one of the consequences of
dealing with alternative cosmologies is that the thermal history
of particles turns out to be modified as compared with GR,
Eq. (3.1).
We shall assume that the Universe is described by a flat
Friedman-Robertson-Walker metric
ds2 = dt 2 − a2(t )(d x 2 + d y2 + d z2) ,
(4.1)
where a(t ) is the scale factor. We refer to a Universe radiation
dominated, so that the energy density is given by ρ = π43g0∗T 4 ,
g∗ = 106, while the pressure is p = ρ/3 (the adiabatic index
is w = 1/3). The dot will stand for the derivative with respect
to the cosmic time t .
4.1 Scalar tensor theories (STTs)
The total action of a STT of gravity is given by S = SST T +
Sm [
63
], where
1
SST T = 16π
d4x
−g˜
2 R˜ (g˜)
+ 4ω( )g˜μν ∂μ ∂ν
− 4V˜ ( ) ,
(4.2)
and Sm = Sm [ , g˜μν ] is the matter action (the matter
fields m couple to the metric tensor g˜μν ). The action (4.2)
encodes the Brans-Dicke theory of gravity for ω( ) = ω =
const ant . In the form (4.2), the STT action is refereed as
Jordan frame. By means of the conformal transformation g˜μν =
AC (φ)gμν ( AC is the conformal factor that depends on φ (x ))
and setting 2 = 8π M∗/ AC2 , V (φ) = AC4 (φ)V˜ (φ)/4π , and
α(φ) = d logdAφC (φ) (= (ω( ) + 3)−1, the action (4.2) can be
casted in the so-called Einstein Frame (EF)
where accounts for all corrections induced by modified
cosmology
23+ν
≡ 3π η
T∗
mχ
ν
5 + ν
2
3 + ν
2
.
The above result implies ν > −3 and we have used
25 23 = 32π . To explain the DM relic abundance and
the IceCube data, we have to require
7.5 × 1034 .
2 For example, in [
63
] the enhancement function A(T ) is parameterized
as
A(T ) =
1 + η TTf ν tanh T −TrTere for T > TB B N
1 for T ≤ TB B N
where TB B N ∼ 1MeV. In the regime T TB B N , the function (3.3)
behaviors as (3.2). T f is the temperature at which the WIMPs DM
freezes-out, T f 10 GeV.
SST T =
where Vφ = ∂ V /∂φ. The Bianchi identity (conservation of
the energy momentum tensor) d(ρa3) + pd(a3) = (ρ −
3 p)d log AC (φ) implies T a = const ant for w = 1/3. From
Eqs. (4.4)-(4.6), one gets [
63
]
H 2 ≡ H M2C =
AC2 (φ)[1 + α(φ)φ ]2 HG2 R .
1 − φ 2/6
The prime indicates the derivative with respect to N ≡ ln a
(φ ≡ ddNφ = ddlnφa = −T φT , where φT ≡ dφ/d T ), while
for the scalar field equation one gets (setting λ = V (φ)/ρ)
2(1 + λ)
3(1 − φ 2/6) φ + [(1 − w) + 2λ]φ
√
Vφ
+ 2α(φ)(1 − 3w) + 2λ V = 0 .
The form of the factor A(T ) for a STT follows from (4.7)
(4.8)
A(T ) ≡
AC (φ)[1 + α(φ)φ ]
(1 − φ 2/6)1/2
Assuming that A(T ) is of the form (3.2), Eq. (4.9) can be
rewritten in the form
AC2 α2T 2 +
+ AC2 − η
2
η2T 2
∗
6
T
T∗
T
T∗
2ν
2(ν+1)
= 0 .
φT2 − 2 AC2 αφT
(4.4)
(4.6)
(4.7)
(4.10)
Vρ (1+log Vφ ) = 0. Writing the energy density in terms of
the field φ, ρ = K∗e4φ/√6 with K∗ ≡ π23g0∗T∗4 , Eq. (4.8)
allows to derive the potential V (the integration constant
is√set equal to zero) V (φ) = γ −1 K∗eγ φ , where γ ≡
√ 6 . The potential V is suppressed for temperatures
6+4
T < T∗.
• α 1 - In this case Eq. (4.10) becomes
αT 2φT − 2 = 0 ,
that gives α(φ)dφ = −2/ T , which implies AC (φ (T ))
= A0e−2Tr /T = A1e−2a/a1 , where ( A0, A1) and (Tr , ar )
are integration constants. Therefore the conformal factor
diminishes for decreasing (increasing) temperature (scale
factor). Consistently with our assumption α 1, we
must require ddφa = − 2αTr 1, so that from (4.8) it
follows that V (φ) ∼ V0, where V0 is a constant.
In these examples, results are independent on ν and η. To
obtain the correct DM relic abundance and explain the
IceCube results, hence to fulfill the condition (3.9), the
parameters {η, ν, T∗} must fine tuned. Setting η ∼ O(1) and T∗ =
34
10q GeV, and for T∗ mχ , Eq. (3.9) implies ν = q − 6 . For
example, if the transition temperature occurs at ∼ 1012GeV,
i.e. q ∼ 12, then it follows ν ∼ 5 − 6.
4.2 Brans–Dicke theory
In this Section we consider Brans–Dicke (BD) theory of
gravity. The BD action follows from the most general action
[
59,60
]
(4.9)
S =
d4x √−g φ R −
∇μφ∇μφ − V (φ) , (4.11)
ω(φ)
φ
when ω → const ant and V → 0. Here ω(φ) is an arbitrary
function (the coupling parameter). In Refs. [
59,81,82
] it was
shown that during the radiation dominated era, the solutions
of the field equations are of the form
To solve Eq. (4.10) one has to specify the form of AC (φ).
We study some particular cases:
• α, AC
dφ 2
d z
φ(0) +√6 log z
Noting that φ
1 - In this regime Eq. (4.10) reduces to the form
6 T
= z2 , where z ≡ T , whose solution is φ (z) =
= 0φ,(1E)q−. (√4.68∗N) a,swsiutmhφes(0t)h,eφf(o1)r mco−ns√ta36nt+s.
(4.12)
(4.13)
(4.14)
(4.15)
a(τ ) = a0 (τ + τ−)2 + τ+2 e−β ,
φ (τ ) = φ0β ,
for ω < −3/2. Here τ is the conformal time, related to the
cosmic time by the relation t = a(τ )dτ , τ+, τ−, a0 and φ0
are arbitrary integration constants such that 83πaρ02rφad00 = 1, and
α ≡
1
2ω
2 1 + 3
,
β ≡
1
The interesting aspect of these solutions is that for late time
the scale factor becomes a(τ ) ∼ τ ∼ t 1/2, φ → φ0, i.e. the
standard cosmological model is recovered. As an example to
explain the IceCube data and the DM relic abundance, we
shall consider the solution (4.12). Writing the expansion rate
in the form (3.1), HMC (τ ) = A(τ )HG R (τ ) (HG R = τ1 ), we
get
A(τ ) =
1
2 + α
τ
τ + τ+
+
Notice A(τ ) → 1 as τ τ±. To make some estimations, we
assume hence that in the early time τ < τ± (for example, we
can set τ− ∼ τB B N and τ+ = τ∗ the transition time), so that
A(τ ) = η
τ (T )
τ+
,
η ≡
1
2 + α
−
α − 21 ττ+− .
To apply the above result to (3.5) we should determine the
relation between the conformal time τ and the temperature
T . This task cannot be solved analytically. However, we note
that whatever is the relation τ = τ (T ), since ∼ η−1, to
fulfill the condition (3.9) we can also look at values of
parameters for which η 1. The latter condition implies τ+
2α+1 τ−, which requires ω > 0. Of course, the solutions here
a2nαa−l1yzed are just a subclass of solutions. More general
solutions and a richer phenomenology follow, for example, for the
general cases in which ω(φ) = 0 and the potential V (φ) = 0.
4.3 f (T ) cosmology
Tμλν = ˆ νλμ − ˆ μλν = eiλ(∂μeνi − ∂ν eμi) ,
where eμi(x ) are the vierbein fields defined as gμν (x ) =
ηi j eμi(x )eνj (x ). The action is given by SI0 = 16π1 G d4x eT ,
where T = Sρ μν T ρ μν is the torsion scalar, e = det (eμi) =
√−g, and
Sρ μν
= 21 41 (T μν ρ − T νμρ − Tρ μν )
μ θν
+ δρ T
ν θμθ .
θ − δρ T
We shall consider the simplest generalization of the action
SI0 to construct gravitational modifications based on torsion,
i.e.
1
SI = 16π G d4x e [T + f (T )], (4.19)
where f (T ) is a generic function of the torsion. For
homogeneous and isotropic geometry (4.1), the vierbein fields
assume the form e A
μ = diag(1, a, a, a). By using Eqs. (4.17)
and (4.18) one infers a relation between the torsion and the
expansion rate of the Universe T = −6H 2. The
cosmological field equations read [
109
]
12H 2[1 + fT ] + [T + f ] = 16π Gρ ,
48H 2 fT T H˙ − (1 + fT )
[12H 2 + 4H˙ ] − (T − f ) = 16π G p ,
where fT = d f /dT . The equations close by taking into
account the equation of continuity ρ˙ + 3H (ρ + p) = 0. We
consider the power-law f (T ) model [
111,112
]
f (T ) = βT |T |nT ,
(4.17)
(4.18)
(4.20)
(4.21)
Another interesting model able to explain the accelerated
iudptsehyusacasbrleaibLsoeeefddvtibho-yCentihvUtheitneatiovcWresoriensoienntze(iecsintnibpsotörneoc)av,kdiadocnefoddtnhtnbheeyeccutgtirhrovaenavtitu(thairenetoisottreenynaasdoloffiro)egf.lrTdathhviees- iHHs TT2of ≡the6Hffo,−romnTe(3f3gT.e3t)s=wHi6Ttnh−≡1βHT M(2CnT= +A(1T))HH2Gn,Ra,nwdhaesrseuAm(iTng)
torsion tensor is construct in terms of the first derivatives of 2
tetrad fields (no second derivatives appear). This model is η = 1 , ν = n − 2 ,
referred as the Teleparallel Equivalent of General Relativity T 1
(oTf EfiGelRd)e,qtuhaattioisnseq[8u3iv]a.TlehnetsteomGoedneelrsarleRperelasteinvtitaynaatlttehrenlaetviveel T∗ ≡ 24π453g∗ 4 (2nT + 1) 4(1−1nT )
to inflationary models, as well as to effective DE models, in 1 1
[w8h3i–c1h0t7h]e (Ufonrivaerdseetaaiclceedlerreavtiieown,issederiv[1e0n8b,y10th9e])t.oIrtsihoanstebremens × GeVβ2(T1−nT ) 4(1−nT ) GMePVl 2 GeV . (4.22)
recently discussed in [110], in the framework of possible It is straightforward to show that for the above solution and
future measurement in advancing gravitational wave astron- a(t ) = a0t δ, i.e. H = δt , it follows T (t )a(t ) = const ant .
omy, possible tests able to distinguish among modified f (T ) The transition temperature T∗ given in (4.22) (following from
gravity. HT (T∗) HG R (T∗)) has to be used into Eqs. (3.8) and (3.9).
In teleparallel gravity, one adopts the curvatureless Weitzen- In Fig. 1 we plot vs n. The value ∗of∼βT10i1s1obtained by
böck connection (that encompasses all the information about fixing the transition temperature at T − 109 GeV
the gravitational field) (see Fig. 2), that is T∗ mχ . The parameter δ enters into
8π
By rewriting (4.20) in the form H 2 + HT2 = 3M P2l ρ, where
the expression of p (we shall not present explicitly being not
relevant for our analysis).
For completeness, we also discuss the possibility to use the
best fit of the parameters {βT , nT } for explain the observed
accelerated phase of the Universe. This is obtained from the
CC + H0 + S N e I a + B AO observational data [
113
] and
give βT = (6H02)1−nT 2nTm−01 and nT = 0.05536, where
m0 = 8π3HG0ρ2m is the matter density parameter at present,
and H0 = 73.02 ± 1.79km/(s Mpc) ∼ 2.1 × 10−42GeV is
the current Hubble parameter. In Fig. 3 are reported results
bByBNfixienrga,nTT∗ ∼ 00. 0.15M5.eWV,epgreotviadetrdanβs˜iti≡on GteemVpβ2eT(1r−antTur)e ∼at
10−56 − 10−60. These values do not match the best fit for
βT ∼ (H0/GeV)2(1−nT ) 0m by ∼ 20 order of magnitudes.
5 Conclusions
The IceCube collaboration has reported several neutrino
events with energies varying from TeV to PeV. A
possible explanation for these events is ascribed to DM
particle physics. A viable mechanism for the DM production in
Fig. 3 T∗ vs nT . The range of nT 0.05 is taken from the best fit of
CC + H0 + S N eI a + B AO data. Here β˜ ∼ (H0/GeV)2(1−nT ) 0m .
The transition temperature occurs at BBN temperature T∗ ∼ 0.1 MeV
the early universe is the freeze-in mechanism. In [
31
] it was
shown that the minimal dimension four interaction y L · H χ
fails to explain both DM decay rate required for IceCube and
the correct DM abundance. The lowest dimensional operator
which can explain the IceCube decay rate and relic abundance
is 6-dimensional operator λαMβ λ2γ L Lα C i σ 2 L Lβ (lRγ χ )
S
[
23
]. However, all results are obtained assuming that the
cosmological background evolves according to GR fields
equations.
In this paper, to reconcile the current bound on DM relic
abundance with IceCube data in terms of the 4-dimensional
operator, we have followed a different perspective that relates
the existence of DM hypothesis with modified theories of
gravity. Motivated by cosmological observations by Type Ia
Supernovae, CMB radiation, and the large scale structure,
according to which the present Universe is in an
accelerating phase, new theories beyond GR have been proposed. We
have shown modified gravity models can explain the
IceCube outputs and at the same time the DM relic abundance
observed today in a minimal particle physics model. This
because the cosmological field equations based on modified
gravity models change the thermal history of particles, so
that the expansion rate of the Universe can be written in the
form H (T ) = A(T )HG R (T ), encoding in A(T ) the
parameters characterizing the model of gravity. Using the particular
form of the factor A(T ) derived from different cosmological
models (we have considered STTs, BD gravity and models
related to torsion f (T )), we have solved the Boltzmann
equation to get the abundance of DM particles. The latter turns
out to be modified by a quantity that does only depend on
parameters of the modified cosmological models, and allows
to explain, consistently, both the IceCube data and the correct
DM abundance D M h2 ∼ 0.11.
Finally, we notice that results derived in this paper allow
to exclude some models of modified gravity. For example,
considering the f (R) gravity, with f (R) = R + a Rn, we
have found that IceCube data and DM relic abundance can
be explained provided n < 1, However, such a value is not
favored by recent Planck release, which require n > 1 (and
in fact the Starobisnky model n = 2 is one of the favorite
candidate for Inflation) [
114, 115
]. Results here discussed do
not allow to distinguish between f (T ) and BD theories of
gravity.
Acknowledgements SM thanks INFN for support.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
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