KIAS-P22070 CMB imprints of high scale non-thermal leptogenesis Anish Ghoshal1Dibyendu Nanda2and Abhijit Kumar Saha3 4

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KIAS-P22070

CMB imprints of high scale non-thermal leptogenesis

Anish Ghoshal,1, ∗Dibyendu Nanda,2, †and Abhijit Kumar Saha3, 4, ‡

1Institute of Theoretical Physics, Faculty of Physics,

University of Warsaw,ul. Pasteura 5, 02-093 Warsaw, Poland

2School of Physics, Korea Institute for Advanced Study, Seoul 02455, South Korea

3School of Physical Sciences, Indian Association for the Cultivation of Science,

2A &2B Raja S.C. Mullick Road, Kolkata 700032, India

4Institute of Physics, Bhubaneswar, Sachivalaya Marg, Sainik School, Bhubaneswar 751005, India

We study the imprints of high scale non-thermal leptogenesis on cosmic microwave background

(CMB) from the measurements of inﬂationary spectral index (ns) and tensor-to-scalar ratio (r),

which otherwise is inaccessible to the conventional laboratory experiments. We argue that non-

thermal production of baryon (lepton) asymmetry from subsequent decays of inﬂaton to heavy

right-handed neutrinos (RHN) and RHN to SM leptons is sensitive to the reheating dynamics in the

early Universe after the end of inﬂation. Such dependence provides detectable imprints on the ns−r

plane which is well constrained by the Planck experiment. We investigate two separate cases, (I)

inﬂaton decays to radiation dominantly and (II) inﬂaton decays to RHN dominantly which further

decays to the SM particles to reheat the Universe adequately. Considering a class of α−attractor

inﬂation models, we obtain the allowed mass ranges for RHN for both cases and thereafter furnish

the estimates for nsand r. The prescription proposed here is general and can be implemented in

various kinds of single-ﬁeld inﬂationary models given the conditions for non-thermal leptogenesis

are satisﬁed.

I. Introduction

Observation of neutrino oscillations at various neu-

trino reactor experiments manifests that neutrinos are

massive and have non-zero mixings [1–8]. On the cos-

mological front, the BBN (Big bang nucleosynthesis),

CMBR (Cosmic microwave background radiation) and

LSS (Large scale structure) measurements favor neutrino

mass to remain in the sub-eV range. The simplest mech-

anism to ﬁt the neutrino oscillation data and explain the

origin of the Standard Model (SM) neutrino masses is

the type-I seesaw mechanism [9–12] where the SM is ex-

tended with three right handed neutrinos (RHN), singlet

under SM gauge symmetry. Remarkably, such minimal

extension can also explain the cosmic matter-antimatter

asymmetry which is dubbed as the baryon asymmetry of

the Universe (BAU). The RH neutrinos present in the

type-I seesaw model [9,10] with lepton number violat-

ing (LNV) Majorona mass are inherently unstable and

decay to SM Higgs plus leptons. This out-of-equilibrium

decay process at 1-loop picks up CP violation (CPV) via

the complex Yukawa couplings, leading to asymmetric

decay into the leptons than in the anti-lepton counter-

part [13–16]. Thus the three Sakharov conditions being

fulﬁlled, we can have successful baryogenesis in the early

universe when afterwards the lepton asymmetry partially

gets converted to the positive baryon asymmetry that we

observe today [13–16].

∗anish.ghoshal@fuw.edu.pl

†dnanda@kias.re.kr

‡psaks2484@iacs.res.in;abhijit.saha@iopb.res.in

The production of lepton asymmetry at the early

stages of the Universe can be thermal [13,17] or non-

thermal [18–32] in nature. In the case of thermal lepto-

genesis, the reheating temperature has to be larger than

the RH neutrino mass scale (MN< TR) such that a non-

zero initial abundance of the RH neutrino can be created

eﬃciently from the thermal bath. In the non-thermal

case, the condition MN< TRis not a necessity. The

required initial abundance of RHN can be alternatively

created non-thermally from a heavy scalar decay present

in the early Universe. The scalar ﬁeld can be identiﬁed

with the inﬂaton which leads to an accelerated expan-

sion at the beginning of the universe, in order to solve

the horizon and the ﬂatness problems. The same ﬁeld

could also be responsible for the quantum generation of

the primordial ﬂuctuations seeding the large scale struc-

ture (LSS) of the Universe (see [33] for a review).

Despite the elegant explanation for the tiny SM neu-

trino masses and the generation of matter-antimatter

asymmetry, the seesaw mechanism is excruciatingly diﬃ-

cult to test in laboratories, since in order to successfully

drive leptogenesis, the right-handed neutrino mass scale

has to be above ≳109GeV (see, e.g., [34])1 2. The

indirect tests for high scale leptogenesis, of course, ex-

ist based on neutrino-less double beta decay and lepton

ﬂavor and CP violating decays of mesons [37], via CP

violation in neutrino oscillation [38,39], by the struc-

1This bound can be evaded in case of resonant production of lep-

ton asymmetry in presence of nearly degenerate RH neutrino

species [35]

2With some ﬁne tuning, it is also possible to lower the scale of the

non-resonant thermal leptogenesis to as low as 106GeV [36].

arXiv:2210.14176v3 [hep-ph] 27 Jan 2024

ture of the mixing matrix [40], or from theoretical con-

straints stemming from the demand of Higgs vacuum re-

maining meta-stable in the early universe [41,42]. And

more recently Gravitational Waves (GW) of primordial

origins like that from cosmic strings [43], domain walls

[44], nucleating and colliding vacuum bubbles [45,46] or

other topological defects [47] and primordial blackholes

[48] have been proposed to constrain as well shed some

light on high-scale leptogenesis scenarios. Under these

circumstances, it is necessary, although highly challeng-

ing to ﬁnd new and complementary tests of such heavy

neutrino sectors and consequently the leptogenesis mech-

anism.

Motivated by this, in this paper, we envisage the scope

of tracing the ﬁngerprints of high scale non-thermal lep-

togenesis at CMB experiments. If the lepton asymmetry

is produced via the transfer of energy density from inﬂa-

ton sector to the lepton sector, then the amount of ﬁnal

lepton asymmetry yield is dependent on the reheating

history of the Universe. On the other hand, for a given a

model of inﬂation the predictions of inﬂationary observ-

ables namely the spectral indices and tensor to scalar

ratio also inﬂuenced by the post-inﬂationary [49] physics

e.g. number of e-folds during reheating era as specu-

lated in ref. [49] and subsequently studied in details in

Refs. [50–52]. The aforementioned two observations sug-

gest that the non-thermal leptogenesis at early Universe

is expected to leave non-negligible imprints in the CMB

predictions for inﬂationary observables which we pursue

in this paper3.

We revisit the inﬂatonary reheating epoch [56,57] in

α-attractor models of inﬂation [58–63]. Interestingly,

when we take into account the non-thermal production

of RHN (that subsequently yields lepton asymmetry)

from tree level inﬂaton decay, the reheating tempera-

ture of the Universe cannot be arbitrary, rather guided

by the observed amount of baryon asymmetry of the

Universe. This in turn provides a distinct prediction

of the non-thermal leptogenesis on the ns−rplane,

which turns out to be stronger than the one provided

by PLANCK/BICEP experiment [64].

In our analysis, we have assumed that the inﬂaton de-

cays perturbatively. We have separately discussed two

possible sub-cases: (I) inﬂaton decays dominantly to ra-

diation and (II) tree level interaction between inﬂaton

and radiation is absent. In the latter case, the reheating

of the Universe is realized solely from the decay of RHNs.

We have followed a detailed numerical approach consid-

ering the ﬁnite epoch of the perturbative reheating era.

We point out the allowed ranges of RHN mass in order

to realize the non-thermal leptogenesis for both Case I

and Case II. From these two case studies, we have found

that the occurrence of a successful non-thermal lepto-

3The impact of dark matter production at the very early Universe

on the inﬂationary observables have been studied by the authors

of [53–55].

genesis scenario indeed imposes stringent and a generic

restriction in the ns−rplane, irrespective of whether a

radiation dominated Universe is obtained from inﬂaton

or RHN decay.

II. α-attractor inﬂation model

A general form of the α-attractor inﬂaton potential

(known as E model) is read as [62],

V(ϕ)=Λ41−e−√2

3α

MP2n

,(1)

where MPstands for the reduced Planck scale and Λ

represents a mass scale that determines the energy scale

of the inﬂation. A special case of Eq.(1) with α= 1 and

n= 1 mimics the standard Higgs-Starobinsky inﬂaton

potential [65]. We ﬁrst calculate the spectral index (ns)

under the slow roll approximation to express it in terms

of the model parameters αand n.

ns= 1 −

8ne√2

3α

ϕk

MP+n

3αe√2

3α

ϕk

MP−12,(2)

where ϕkis the inﬂaton ﬁeld value at horizon exit. This

can be simply translated to write ϕkas a function of ns.

ϕk=r3α

2MPln (1 + ∆(ns)) ,(3)

where ∆(ns) = 4n+√16n2+24αn(1−ns)(1+n)

3α(1−ns). Next, we

make an estimate for the inﬂaton ﬁeld value at the end

of inﬂation by equating one of the slow roll parameters

(max[ϵ, η]) to unity as given by,

ϕend =r3α

2MPln 2n

√3α+ 1.(4)

Utilizing Eq.(3) and Eq.(4), we compute quantities

namely tensor to scalar ratio and the number of e-fold

analytically which are given by,

r=64n2

3αe√2

3α

ϕk

MP−12

=192αn2(1 −ns)2

h4n+p16n2+ 24αn(1 −ns)(1 + n)i2,(5)

Nk=3α

4n"e√2

3α

ϕk

MP−e√2

3α

ϕend

MP−r2

3α

(ϕk−ϕend)

(6)

The scalar potential at the end of inﬂation and the scalar

perturbation spectrum are obtained as,

Vend = Λ42n

2n+√3α2n

,(7)

As=3V(ϕk)

4π2r.(8)

The observed value of Aobs

s= 2.2×10−9[66] precisely

ﬁxes one of the model parameters Λ as,

Λ =MP3π2rAobs

21/4

×"2n(1 + 2n) + p4n2+ 6α(1 + n)(1 −ns)

4n(1 + n)#n/2

(9)

It is worth mentioning that each of the quantities

(r, Nk, Vend and Λ) have been expressed exclusively in

terms of three model parameters (n, α and ns). This will

be immensely useful in connecting the lepton asymmetry

from the RHN decay to the inﬂationary observables as

we will describe in a while.

III. Non-thermal Leptogenesis from inﬂaton decay

To embed the non-thermal leptogenesis in the inﬂa-

tionary framework we propose the following Lagrangian

in a model independent manner:

−L ⊃yNϕ NCN+yRϕ X X +YνlLe

HN (10)

+MN

2NCN+h.c., (11)

where Xrepresents a Dirac fermion which is part of the

radiation bath after the completion of reheating era 4.

The N(assumed to be of Majorona nature and having a

bare mass MN) is the RH neutrino which decays to SM

leptons (lL) and Higgs (H) and subsequently contribute

to the yield of lepton asymmetry. We assume all the cou-

pling coeﬃcients yN,yRreal and positive. The neutrino

Yukawa coupling Yνcould be complex and source the CP

violation in the SM lepton sector. In the present analy-

sis we deal with suﬃciently small Yukawa couplings yN

and yRsuch that they do not disturb the shape of the

inﬂationary potential through radiative corrections.

The ﬁrst term in Eq.(11) leads to the non-thermal pro-

duction of Nfrom tree level inﬂaton decay. Whereas,

4Here we remain agnostic about an UV complete framework and

only consider Xto be a sample particle which is part of radiation

bath. As an instance one can consider an U(1)Xgauged exten-

sion of the SM where X, being a gauge singlet vector fermion is

coupled to SM sector via U(1)Xgauge boson and having same

temperature as of SM radiation bath.

the second term is responsible for inﬂaton decay to ra-

diation at tree level. The third term triggers the out-of-

equilibrium decay of RHN to SM leptons. In principle it

is essential to add at least two SM gauge singlet RH Ma-

jorona neutrinos in order to satisfy the neutrino oscilla-

tion data [67]) in the usual type-I seesaw framework. We

consider here N2being larger than the inﬂaton mass and

hence their production from inﬂaton decay is expected to

be suppressed. On the other hand, N1is considered to be

lighter than the inﬂaton and can be produced eﬃciently.

In view of this, we ignore the dynamics of N2and track

the evolution of N1only which is same as Nin Eq.(11).

As mentioned in the introduction section, we will ex-

amine two distinct cases: (I) inﬂaton dominantly decays

to radiation and reheats the Universe i.e. Brϕ→XX >

Brϕ→NN and (II) inﬂaton decays to RHN ﬁrst and its

further decay to SU (2)Ldoublets reheats the Universe

which implies yR≪yN. In the ﬁrst case, we also need to

ensure that the produced RH neutrino is not long-lived

and radiation energy density (ρR) always remains much

larger than the energy density of RH neutrino (ρN) such

that the RHN never dominates the energy budget of the

Universe. In that case, the decay of RHN contributes

negligibly to the ρRand the comoving entropy density

sa3remains almost constant before and after the decay

of N. Thus the reheating temperature of the Universe

is controlled by the inﬂaton decay to radiation only. For

simplicity, in the present work we have assumed that both

the inﬂaton and RHN decay perturbatively.

We deﬁne the decay widths of ϕto both radiation and

Nas given by [30],

ΓN

ϕ=|yN|2mϕ

16π 1−4M2

ϕ!3/2

,ΓR

ϕ≃|yR|2mϕ

8π,

(12)

while the decay width for the RHN is ΓN≃|Yν|2MN

16π.

In the above, we have considered the SM particles to be

much lighter than the inﬂaton. The set of Boltzmann

equations that govern the evolution of energy densities

of various species, number densities for Nand the yield

of lepton asymmetry is given by [30]:

dρϕ

dt + 3H(pϕ+ρϕ) = −ΓN

ϕρϕ−ΓR

ϕρϕ,(13)

dρR

dt + 3H(pR+ρR)=ΓR

ϕρϕ+ ΓNρN,(14)

dρN

dt + 3H(pN+ρN) = ΓN

ϕρϕ−ΓNρN,(15)

dnB−L

dt + 3HnB−L=−ερNΓN

,(16)

where εis the CP asymmetry parameter which is a func-

tion of the neutrino Yukawa couplings and RHN mass

scales (see Eq.(A4)). We have evaluated the neutrino

Yukawa couplings using the Casas Ibara (CI) parametri-

sation [68] using the best-ﬁt values of the neutrino oscil-

lation parameters. This helps us further to write the

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KIAS-P22070CMBimprintsofhighscalenon-thermalleptogenesisAnishGhoshal,1,∗DibyenduNanda,2,†andAbhijitKumarSaha3,4,‡1InstituteofTheoreticalPhysics,FacultyofPhysics,UniversityofWarsaw,ul.Pasteura5,02-093Warsaw,Poland2SchoolofPhysics,KoreaInstituteforAdvancedStudy,Seoul02455,SouthKorea3SchoolofPhysicalSc...

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KIAS-P22070 CMB imprints of high scale non-thermal leptogenesis Anish Ghoshal1Dibyendu Nanda2and Abhijit Kumar Saha3 4

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