A note on gravitational dark matter production

Academic Edi o : And ea Lapi
Recei ed: 14 Decembe 2024
Re ised: 29 Janua y 2025
Accep ed: 31 Janua y 2025
Published: 4 Feb ua y 2025
Ci a ion: de Ha o, J.; Pan, S. A No e
on G a i a ional Da k Ma e
P oduc ion. Uni e se 2025,11, 49.
h ps://doi.o g/10.3390/
uni e se11020049
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uni e se
A icle
A No e on G a i a ional Da k Ma e P oduc ion
Jaume de Ha o 1,* and Sup iya Pan 2,3
1Depa amen de Ma emà iques, Uni e si a Poli ècnica de Ca alunya, Diagonal 647, 08028 Ba celona, Spain
2Depa men o Ma hema ics, P esidency Uni e si y, 86/1 College S ee , Kolka a 700073, India;
sup iya.ma hs@p esiuni .ac.in
3Ins i u e o Sys ems Science, Du ban Uni e si y o Technology, P.O. Box 1334, Du ban 4000, Sou h A ica
*Co espondence: jaime.ha [email p o ec ed]
Abs ac : Da k ma e , one o he undamen al componen s o he uni e se, has emained
mys e ious in mode n cosmology and pa icle physics, and hence, his ield is o u mos
impo ance a he p esen momen . One o he ounda ional ques ions in his di ec ion is
he o igin o da k ma e , which di ec ly links o i s c ea ion. In he p esen a icle, we s udy
he g a i a ional p oduc ion o da k ma e in wo dis inc con ex s: i s ly, when ehea ing
occu s h ough g a i a ional pa icle p oduc ion, and secondly, when i is d i en by decay
o he in la on ield. We es ablish a connec ion be ween he ehea ing empe a u e and he
mass o da k ma e , and om he ehea ing bounds, we de e mine he ange o iable da k
ma e mass alues.
Keywo ds: ehea ing; g a i a ional pa icle p oduc ion; cons ain s; da k ma e
1. In oduc ion
Da k ma e , a undamen al componen o he uni e se, emains one o he mos
p o ound mys e ies in mode n cosmology and pa icle physics. Despi e i s g a i a ional
e ec s being obse ed ac oss a a ie y o as ophysical scales, om galaxies o he cosmic
mic owa e backg ound, i s o igin and na u e con inue o elude us. Among he p oposed
mechanisms o da k ma e gene a ion, he g a i a ional p oduc ion o da k ma e s ands
ou as a pa icula ly compelling explana ion, especially wi hin he con ex o he ea ly
uni e se, see o ins ance [1–20].
G a i a ional da k ma e p oduc ion le e ages he unique ole o g a i y, a uni e sal
in e ac ion, as he p ima y mechanism o gene a ing da k ma e pa icles. This p ocess
equi es no addi ional couplings o in e ac ions wi h he S anda d Model, elying solely
on he dynamics o he expanding uni e se. Such p oduc ion is especially ele an du ing
he ehea ing phase ollowing cosmic in la ion, whe e he uni e se ansi s om an ea ly
in la iona y epoch o a adia ion-domina ed e a.
Two p ima y scena ios domina e discussions o g a i a ional da k ma e p oduc ion:
one in which ehea ing occu s h ough he copious p oduc ion o hea y pa icles [
21
]
ha subsequen ly decay in o S anda d Model pa icles, and ano he in which ehea ing
esul s om he decay o he in la on ield di ec ly in o S anda d Model pa icles [
22
].
These pa hways connec he physics o in la ion, ehea ing, and da k ma e , es ablishing
a ela ionship be ween he ehea ing empe a u e and he mass o da k ma e . This
connec ion allows us, h ough ehea ing cons ain s, o iden i y he ange o iable da k
ma e masses.
This sho no e del es in o he heo e ical ounda ions o g a i a ional da k ma e
p oduc ion s udied in se e al wo ks [
1
–
10
,
12
–
20
] emphasizing i s dependence on in la ion-
a y ehea ing dynamics, i s po en ial obse a ional implica ions, and he ange o iable
Uni e se 2025,11, 49 h ps://doi.o g/10.3390/uni e se11020049
Uni e se 2025,11, 49 2 o 16
da k ma e masses wi hin his amewo k. The es o his sho no e is o ganized as
ollows. In Sec ion 2, we p esen he g a i a ional ehea ing o mulas. Sec ion 3desc ibes
he g a i a ional p oduc ion o da k ma e in he con ex o g a i a ional ehea ing. In
Sec ion 4, we discuss he g a i a ional p oduc ion o da k ma e and ehea ing h ough
he decay o he in la on ield. Finally, in Sec ion 5we conclude he p esen no e wi h a
b ie summa y.
Th oughou he a icle, we wo k unde he assump ion o he spa ially la F iedmann–
Lemaî e–Robe son–Walke (FLRW) geome y wi h
a( )
(he ea e
a
) ep esen ing he
expansion scale ac o o he uni e se and we ha e used he ollowing no a ions:
1. “END” deno es he end o in la ion.
2. “0” deno es he p esen ime.
3. “ eh” deno es he ehea ing ime.
4. ρA,END
deno es he ene gy densi y o he p oduced
A
-pa icles, a he end o in la ion.
5. ρB,END =
3
M2
plH2
END
is he ene gy densi y o he backg ound a he end o in la ion
(
Mpl
is he educed Planck mass), ha is, i co esponds o he ene gy densi y o he
in la on ield.
6. ρ co esponds o he ene gy densi y o he adia ion.
7. ΘA=ρA,END
ρB,END is he hea ing e iciency o he A-pa icles.
8. ¯
ΘA=3Γ2
AM2
pl
ρB,END
is he decay e iciency o he
A
-pa icles, whe e
ΓA
is he decay a e o
he A-pa icles.
9. Ω h2
is he densi y pa ame e o adia ion (
h=H0/
100 km/s/Mpc in which
H0
is
he p esen day alue o he Hubble cons an ).
10.
ΩAh2is he densi y pa ame e o he A-pa icles.
Conce ning he obse a ional cons ain s on some o he pa ame e s, we ha e used
he ollowing alues [23]:
1. ΩYh2=0.12 ±0.0012, whe e he Y-pa icles a e he candida e o da k ma e .
2. Ω h2∼
=2.47 ×10−5.
3. h=0.674 ±0.005.
4. T0=2.7255 ±0.006K∼
=2.35 ×10−13 GeV ∼
=9.6 ×10−32 Mpl.
2. G a i a ional Rehea ing Fo mulas
This sec ion p o ides a de ailed e iew o he esul s ecen ly ob ained in [
24
,
25
] (see
also [
21
,
26
–
28
] o ind some o he ecen esul s in he con ex o g a i a ional ehea ing).
To begin wi h, we conside a po en ial, which, nea he minimum
φ=
0, beha es like
φ2n
(
n
is a na u al numbe ). We examine hea y massi e
X
-pa icles, which a e p oduced
g a i a ionally due o he coupling o a
X
- ield wi h he Ricci scala (see, o ins ance, [
21
],
and also [
29
,
30
] o he ounda ions o quan um ield heo y in cu ed spaces and i s
g a i a ional e ec s), and hen hey decay in o S anda d Model (SM) pa icles in o de
o ehea he uni e se. Du ing he oscilla ions, close o he minimum, as he po en ial
beha es like
φ2n
, hence, wi h he use o i ial heo em, he e ec i e Equa ion o S a e
(EoS) pa ame e ,
we
, becomes
we =n−1
n+1
. No e ha
we
is highe han 1
/
3 o
n>
2.
This gua an ees ha he in la on’s ene gy densi y decays as e han he ene gy densi y
o he p oduced pa icles as well as hei decay p oduc s, because, as a unc ion o he
scale ac o , he ene gy densi y o ma e decays as
a−3
, he one o adia ion as
a−4
, and
o he luid wi h EoS pa ame e
we
, i decays as
a−3(1+we )
, which in he case o he
in la on’s ene gy densi y becomes
a−6n
n+1
.
1
As a consequence, he ene gy densi y o he
la e p oduc s will e en ually domina e and inally his will lead o a success ul ehea ing
o he uni e se. We shall concen a e on he class o in la iona y po en ials ha ing simila
Uni e se 2025,11, 49 3 o 16
beha io close o he minimum, e.g., Hype bolic In la ion, Supe con o mal
α
-A ac o
E-models o Supe con o mal α-A ac o T-models [31–33].
In o de o clea ly ealize he mechanism o g a i a ional ehea ing, i is essen ial o
unde s and he decay p ocess. The decay p ocess can be desc ibed wi h he use o he
dynamics go e ned by he Bol zmann equa ions:



dρX( )
d +3HρX( ) = −ΓXρX( )
dρ ( )
d +4Hρ ( ) = ΓXρX( ),
(1)
whe e
ρX( )
s ands o he ene gy densi y o he p oduced
X
-pa icles,
ρ ( )
is he ene gy
densi y o he adia ion ( he ene gy densi y o he decay p oduc s), and
ΓX
is he decay a e
o
X
-pa icles in o SM pa icles and i is assumed o be a cons an . Fo example, conside ing
an in e ac ion o he
X
- ield wi h e mions, his leads o he decay o he
X
-pa icles wi h
he decay a e ΓX=ˆ
h2mX
8π, whe e ˆ
his a dimensionless cons an [34].
We p oceed wi h he solu ion which desc ibes he ene gy densi y o he hea y massi e
pa icles as ollows
ρX( ) = ρX,ENDaEND
a( )3
e−ΓX( − END), ≥ END. (2)
He e, his solu ion ep esen s ha he decay begins a he end o in la ion, as discussed
in [
11
]. This is due o he ac ha he hea y
X
-pa icles a e p oduced a he end o in la ion,
when he in la on s a s o oscilla e. Consequen ly, he decay o hese pa icles commences
immedia ely upon hei c ea ion.
Now, inse ing (2) in o he second equa ion o (1) and conside ing he ac ha decay
s a s a he end o in la ion, one can ob ain
ρ ( ) = ρX,ENDaEND
a( )4Z
END
a(s)
aEND
ΓXe−ΓX(s− END)ds. (3)
We now p oceed by de ining
dec
as he ime whe e decay ends, ha is, when
ΓX( dec − END)∼1
.
This quickly gi es
dec ∼ END +1
ΓX
, (
ΓX=
0), and we ocus on he in es iga ion o he
e olu ion o he decay p oduc s o
≫ dec
. We p esume ha he backg ound domina es
and wi h such an assump ion, we ha e
a(s)∼
=aEND(s/ END)n+1
3n
since
we =n−1
n+1
. Now,
bea ing in mind ha
Z∞
END
(s/ END)n+1
3nΓXe−ΓX(s− END)ds ∼
=¯
Θ−n+1
6n
XΓ4n+1
3n∼
=¯
Θ−n+1
6n
X, (4)
whe e
Γ
deno es Eule ’s Gamma unc ion, we each he conclusion ha o
> dec
, he
ollowing app oxima ion can be made
ρ ( )∼
=ρX,END ¯
Θ−n+1
6n
XaEND
a( )4
, (5)
whe e we ha e used he de ini ion o he decay e iciency o
X
-pa icles. Now, since,
we =n−1
n+1
, he ene gy densi y o he backg ound, ha is, he ene gy densi y o he in la on
ield, sa is ies he equa ion ˙
ρB= (1+we )ρB, and hus, i e ol es as
ρB( ) = ρB,ENDaEND
a( )6n
n+1. (6)
Uni e se 2025,11, 49 4 o 16
The e o e, since he uni e se becomes ehea ed when ρB∼ρ , we ob ain
ΘX=¯
Θ
n+1
6n
XaEND
a eh 2(n−2)
n+1, (7)
and as a esul o which, he ene gy densi y o he decay p oduc s a he ime o ehea ing
is gi en by he ollowing
ρ , eh ∼
=ρB,END ¯
Θ−n+1
2(n−2)
XΘ
3n
n−2
X. (8)
Now, using he S e an–Bol zmann law
T eh =30
π2g eh 1/4ρ1/4
, eh
(he e
g eh =
106.75 de-
no es he e ec i e numbe o deg ees o eedom in he SM), one ob ains he ollowing
ehea ing empe a u e,
T eh =90
π2g eh 1/4 ¯
Θ−n+1
8(n−2)
XΘ
3n
4(n−2)
XqHENDMpl. (9)
I is essen ial o calcula e he ange o alues o
ΓX
. We i s no e ha
ΓX≪HEND
because he decay occu s well a e he end o in la ion. Addi ionally, we ha e assumed
ha he ene gy densi y o he backg ound domina es a he end o decay, ha means,
ρ ,dec ≪ρB,dec ∼
=3M2
plΓ2
X. The e o e, using he ela ion ρB,dec ∼
=3M2
plΓ2
X, we ob ain,
aEND
adec 4∼
=¯
Θ
2(n+1)
3n
X, (10)
and inse ing he abo e ela ion in o ρ ,dec ≪3M2
plΓ2
X, one a i es a
Θ
n
n−1
X≪p¯
ΘX≪1. (11)
Mo eo e , combining (11) wi h he ollowing bound o he ehea ing empe a u e,
5
×
10
−22Mpl ≤T eh ≤
5
×
10
−10Mpl
, which s a es ha he ehea ing empe a u e emains
in an in e al consis en wi h he Big Bang Nucleosyn hesis (BBN), which occu s a abou
∼
1 MeV scale, and a ains an uppe bound a abou
∼
10
9
GeV in o de o mi iga e he
issues ela ed o he g a i ino p oblem [
35
–
38
] one ob ains ou dis inc cases. Howe e ,
he only iable case is he ollowing
Θ
n
n−1
X≪p¯
ΘX≤1084(n−2)
n+1 HEND
Mpl !2(n−2)
n+1
Θ
3n
n+1
X, (12)
p o ided ha he inequali y
10−42(n−1)
nMpl
HEND n−1
n
≪ΘX≪10−28(n−2)
nMpl
HEND 2(n−2)
3n
, (13)
holds. Finally, we commen on he maximum alue o he ehea ing empe a u e. The
maximum ehea ing empe a u e is ob ained a he epoch when he decay coincides wi h he
end o he in la on’s domina ion, which means, when
ρ , eh ∼
3
Γ2
XM2
pl
. Now, conside ing
Equa ion (8), one ob ains
p¯
ΘX∼Θ
n
n−1
X, (14)
Uni e se 2025,11, 49 5 o 16
and inse ing his in o (9), he maximum ehea ing empe a u e becomes
Tmax
eh ∼
=5.4 ×10−1Θ
n
2(n−1)
XqHENDMpl, (15)
whe e ΘX, due o he bounds o he ehea ing empe a u e, is cons ained as ollows
10−42(n−1)
nMpl
HEND n−1
n
≤ΘX≤10−18(n−1)
nMpl
HEND n−1
n
. (16)
3. G a i a ional P oduc ion o Da k Ma e
This sec ion deals wi h he g a i a ional p oduc ion o da k ma e in he amewo k
o g a i a ional ehea ing. We conside wo quan um scala ields,
X
and
Y
, which a e
con o mally coupled o he Ricci scala . The
X
- ield p oduces hea y
X
-pa icles wi h mass
mX
, which will decay in o SM pa icles and ehea he uni e se. The
Y
- ield p oduces
Y-pa icles wi h mass mY, and hey co espond o he p esen -day da k ma e candida e.
3.1. Maximum Rehea ing Tempe a u e
The case when he
X
-pa icles decay close o he onse o adia ion leads o he
maximum ehea ing empe a u e. We begin by conside ing he ene gy densi y o he da k
ma e a he p esen ime which is gi en by
ρY,0 =ρY,ENDaEND
a03
=ΘYρB,ENDaEND
a03
. (17)
Now, wi h he use o he ollowing
aEND
a03
=aEND
a eh 3a eh
a04a0
a eh
=aEND
a eh 3a eh
a04Tmax
eh
T0
, (18)
whe e he adiaba ic expansion o he uni e se a e ehea ing has been conside ed,
i.e., a0T0=a ehTmax
eh , we now calcula e
ρ ,0a0
a eh 4a eh
aEND 3
=ρB, eha eh
aEND 3
=ρB,ENDΘX, (19)
whe ein we make use o
aEND
a eh 3=Θ
n+1
n−1
X
[
25
], which a ises unde he assump ions o he
backg ound e ol ing as
ρB( ) = ρB,END(aEND/a( ))6n
n+1, (20)
and he decay o he
X
-pa icles is a he onse o adia ion. Wi h hese, we ind he
ollowing ela ion be ween he ene gy densi y o he Y-pa icles and he maximum
ehea ing empe a u e
ρY,0 =ΘY
ΘX
ρ ,0
Tmax
eh
T0
, (21)
ha is:
ΩYh2=Ω h2ΘY
ΘX
Tmax
eh
T0
. (22)

Uni e se 2025,11, 49 6 o 16
Now, inse ing he alue o he maximum ehea ing empe a u e and he obse a ional
alues o ΩYh2and Ω h2, one a i es a he ollowing bound
8.63 ×10−28sMpl
HEND
=ΘYΘ
2−n
2(n−1)
X(23)
Nex , we deal wi h scala pa icles con o mally coupled o g a i y ( o he non-
con o mally coupled case, specially he minimally coupled case, see, o ins ance, [39,40]),
when
mA≪HEND
, because in his case one can use he Wen zel–K ame s–B illouin (WKB)
me hod in he complex plane in o de o analy ically calcula e he
β
-Bogoliubo coe icien s,
which is he key piece o inding he ene gy densi y o he p oduced pa icles. Using he
esul s ob ained in [25], we ob ain
ΘA=1
12π3 mA
Mpl !5/2sMpl
√2HEND ∼
=2.26 mA
Mpl !5/2
, (24)
whe e we ha e used he usual alue o he Hubble a e a he end o in la ion,
i.e., HEND ∼10−6Mpl. Then, we ob ain:
mY
Mpl ∼
=1.7 ×10−10Θ
n−2
5(n−1)
X∼
=1.7 ×10−10
2.26 mX
Mpl !5/2

n−2
5(n−1)
, (25)
and aking in o accoun he cons ain , 5
×
10
−22Mpl ≤Tmax
eh ≤
5
×
10
−10Mpl
, coming om
he BBN success, a e subs i u ing (15) in o i , we ob ain:
10−21sMpl
HEND ≤Θ
n
2(n−1)
X≤10−9sMpl
HEND
=⇒10−18 ≤Θ
n
2(n−1)
X≤10−6, (26)
ha is:
0.7 ×10−72(n−1)
5n≤mX
Mpl ≪10−6, (27)
whe e we ha e used ha
mX≪HEND ∼
=
10
−6Mpl
. Finally, aking in o accoun (15) and (24),
he ehea ing empe a u e can be w i en as a unc ion o he mass
mX
and he pa ame e
n
,
as ollows:
Tmax
eh =5.4 ×10−4
2.26 mX
Mpl !5/2

n
2(n−1)
Mpl. (28)
Finally, we close his sec ion wi h Figu es 1and 2. In Figu e 1, we display he depen-
dence o
mY
wi h
mX
as gi en in Equa ion (25) o di e en alues o
n
. This clea ly exhibi s
a pa e n o inc easing
n
. Simila ly, in Figu e 2we show how he maximum ehea ing
empe a u e,
Tmax
eh
, depends on
mX
o di e en alues o
n
. One can clea ly no ice ha
o a speci ic alue o
n
, i
mX
inc eases,
Tmax
eh
also inc eases. Addi ionally, one can u he
no ice ha o a pa icula alue o
mX
, i
n
inc eases, hen
Tmax
eh
inc eases and i assumes
he maximum alue o n=∞.
Uni e se 2025,11, 49 7 o 16
Figu e 1. The dependence o
mY
masses wi h
mX
has been shown o di e en alues o
n
. We wo k
wi h he uni s whe e Mpl =1.
Figu e 2. The maximum ehea ing empe a u e e sus he mass o he p oduced pa icles,
mX
o
di e en alues o nhas been depic ed. We wo k wi h he uni s whe e Mpl =1.
Quin essen ial In la ion
In his sec ion, we discuss he bounds on he masses o
X
and
Y
pa icles in a special
cosmological scena io, namely, he Quin essen ial In la ion [
21
,
26
,
27
,
41
–
56
]—a uni ied
cosmological model, whe e a e in la ion, he uni e se en e s a kina ion phase, i.e., all he
ene gy densi y is kine ic, which means ha
we =
1, and hus, i is equi alen o he case
n=∞
. Using he alues o
ΘX
and
ΘY
, we ha e he ollowing ela ion be ween he masses
mY
Mpl ∼
=5.64 ×10−11Mpl
HEND 1/10smX
Mpl
, (29)
which o HEND ∼
=10−6Mpl, leads o
mY
Mpl ∼
=2×10−10smX
Mpl
. (30)
Con e sely, he bound o he maximum ehea ing empe a u e leads o he
ollowing cons ain
10−18 ≤pΘX≤10−6, (31)
Uni e se 2025,11, 49 8 o 16
whe e we con inue aking
HEND ∼
=
10
−6Mpl
. Inse ing he alue o
ΘX∼
=
2.26
mX
Mpl 5/2
,
we a i e a he ollowing
5.51 ×10−15 ≤mX
Mpl ≤2.2 ×10−5, (32)
which mus be imp o ed due o he ac ha we ha e assumed
mX≪HEND ∼
=
10
−6Mpl
,
leading o,
5.51 ×10−15 ≤mX
Mpl ≪10−6. (33)
Now, using his las cons ain , we ind he ange o iable alues o he mass o da k
ma e lies in he ollowing egion:
1.48 ×10−17 ≤mY
Mpl ≪10−13. (34)
3.2. Gene al Case: Decay Be o e he Onse o Radia ion
When he decay is be o e he end o he domina ion o he in la on, using (7) and (8),
one has,
ρB, eha eh
aEND 3
=ρB,ENDΘ
3(n−1)
2(n−2)
X¯
Θ−n2−1
4n(n−2)
X. (35)
The e o e, om (17), (18) and (19), he ene gy densi y o he Ypa icles becomes
ρY,0 =ΘYΘ−3(n−1)
2(n−2)
X¯
Θ
n2−1
4n(n−2)
Xρ ,0
T eh
T0
, (36)
and aking in o conside a ion he o mula o he ehea ing empe a u e (9) one a i es a ,
ρY,0 =90
π2g eh 1/4
ΘYΘ−3/4
X¯
Θ
n+1
8n
Xρ ,0 qHENDMpl
T0
, (37)
which in e ms o he densi y pa ame e s, akes he o m
ΩYh2=90
π2g eh 1/4
ΘYΘ−3/4
X¯
Θ
n+1
8n
XΩ h2qHENDMpl
T0
. (38)
This coincides wi h he p e ious case when
p¯
ΘX=Θ
n
n−1
X
. Now, inse ing he obse a ional
alues, one ob ains,
8.67 ×10−28sMpl
HEND
=ΘYΘ−3/4
X¯
Θ
n+1
8n
X, (39)
and om he exp ession o ΘY, we de i e
mY
Mpl ∼
=1.7 ×10−10Θ3/10
X¯
Θ−n+1
20n
X, (40)
wi h he ollowing cons ain s (see o de ails [25]):
10−36(n−1)
n≪ΘX≪10−24(n−2)
n, (41)
Uni e se 2025,11, 49 9 o 16
and
Θ
n
n−1
X≪p¯
ΘX≤1072(n−2)
n+1Θ
3n
n+1
X, (42)
whe e we con inue using
HEND ∼
=
10
−6Mpl
. F om his las cons ain , i is also possible o
ob ain he bound o mYas,
1.7 ×10−1010−36(n−2)
5n≤mY
Mpl ≪2×10−10Θ
n−2
5(n−1)
X, (43)
wi h ΘX∼
=2.26mX
Mpl 5/2 and
10−72(n−1)
5n≪mX
Mpl ≪min(10−48(n−2)
5n; 10−6). (44)
Quin essen ial In la ion
In he case wi h
n=∞
, he o mulas simpli y as ollows. The ehea ing empe a u e
in his case is gi en by
T eh =90
π2g eh 1/4 ¯
Θ−1
8
XΘ
3
4
XqHENDMpl ∼
=5.4 ×10−4¯
Θ−1
8
XΘ
3
4
XqHENDMpl. (45)
The ela ion be ween di e en hea ing e iciencies and he decay e iciency is gi en by
8.67 ×10−25 =ΘYΘ−3/4
X¯
Θ
1
8
X, (46)
which leads o
mY
Mpl ∼
=1.7 ×10−10Θ3/10
X¯
Θ−1
20
X, (47)
wi h he cons ain s:
10−36 ≪ΘX≪10−24, (48)
and
ΘX≪p¯
ΘX≤1072Θ3
X. (49)
Finally, we ha e he ollowing bound o he masses:
4×10−14 ≪mX
Mpl ≪3×10−10, (50)
and
1.07 ×10−17 ≤mY
Mpl ≪2×10−10Θ
1
5
X, (51)
wi h
ΘX∼
=
2.26
mX
Mpl 5/2
. Taking in o accoun he bound o
ΘX
, one can conclude ha he
iable masses o da k ma e a e o he o de (10−17 −10−16)Mpl ∼(101−102)GeV.
Uni e se 2025,11, 49 16 o 16
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