Cent. Eur. J. Phys. • 1-37
DESY 13-173
Author version

Central European Journal of Physics

A Minimal Supersymmetric Model ofParticle Physics and the Early Universe

arXiv:1309.7788v1 [hep-ph] 30 Sep 2013

Review Article

W. Buchmüller1∗ , V. Domcke1† , K. Kamada1‡ , K. Schmitz2§
1 Deutsches Elektronen-Synchrotron DESY,
22607 Hamburg, Germany
2 Kavli IPMU,TODIAS, University of Tokyo, Kashiwa 277-8583, Japan

Abstract:
We consider a minimal supersymmetric extension of the Standard Model, with right-handed neutrinos and local B−L, thedifference between baryon and lepton number, a symmetry which is spontaneously broken at the scale of grand unification.
To a large extent, the parameters of the model are determined by gauge and Yukawa couplings of quarks and leptons.
We show that this minimal model can successfully account for the earliest phases of the cosmological evolution: Inflation
is driven by the energy density of a false vacuum of unbroken B−L symmetry, which ends in tachyonic preheating, i.e.
the decay of the false vacuum, followed by a matter dominated phase with heavy B−L Higgs bosons. Nonthermal andthermal processes produce an abundance of heavy neutrinos whose decays generate primordial entropy, baryon asymmetry
via leptogenesis and dark matter consisting of gravitinos or nonthermal WIMPs. The model predicts relations between
neutrino and superparticle masses and a characteristic spectrum of gravitational waves.
Keywords: Supersymmetry, grand unification, inflaton, dark matter, leptogenesis© Versita sp. z o.o.

1.

Introduction

Today we have the Standard Model of particle physics as well as the ΛCDM model of cosmology, which describe
a wealth of experimental and observational data with an accuracy far beyond expectation [1]. On the other hand,
despite this success, it is obvious that both standard models do not represent a final theory. The symmetry
structure of the Standard Model, the smallness of neutrino masses and the discovery of a Higgs boson [2, 3] witha mass in between the vacuum stability and the triviality bound point towards grand unification as the next stepbeyond the Standard Model. Similarly, the parameters of the ΛCDM model, the abundance of matter and dark∗

E-mail: buchmuwi@desy.deE-mail: valerie.domcke@desy.de‡E-mail: kohei.kamada@desy.de§E-mail: kai.schmitz@ipmu.jp†

1

A Minimal Supersymmetric Model ofParticle Physics and the Early Universe

matter, the apparent cosmological constant and the temperature anisotropies of the cosmic microwave backgroundask for an explanation, which requires physics beyond the Standard Model.
Supersymmetry is an attractive framework to extrapolate the Standard Model of particle physics to the energy
scale of grand unification, ΛGUT ∼ 1016 GeV. It also introduces natural dark matter candidates [4–6] and scalar
fields which can realize inflation, thereby providing an important link between particle physics and cosmology.
Moreover, neutrino masses require right-handed neutrinos whose large Majorana masses can account for the tiny
masses of the known neutrinos via the seesaw mechanism. These Majorana masses break the symmetry B−L, thedifference between baryon and lepton number, and their decays can generate a baryon asymmetry via leptogenesis[7]. Extrapolating the feature of the Standard Model that all masses are generated by spontaneous symmetry
breaking, suggests that also B−L is a spontaneously broken local symmetry.
Following these arguments we arrive at a minimal supersymmetric extension of the Standard Model, which is
described by the superpotential
W =

√
λΦ


 2
vB−L
1
c c
ν ∗ c
− S1 S2 + √ hn
i ni ni S1 + hij 5i nj Hu + WMSSM ,
2
2

(1)

where S1 and S2 are the chiral superfields containing the Higgs field responsible for breaking B−L, and nci denotethe superfields containing the charge conjugates of the right-handed neutrinos. The symmetry-breaking sector of
Eq. (1), involving the superfields S1 , S2 and Φ, is precisely the superpotential of F-term hybrid inflation, with Φ
being a singlet whose scalar component φ acts as the inflaton [8, 9]. vB−L is the scale at which B−L is broken.
The B−L charges are qS ≡ qS2 = −qS1 = 2, qΦ = 0, and qnci = 1. h and λ denote coupling constants, andWMSSM represents the MSSM superpotential,
WMSSM = huij 10i 10j Hu + hdij 5∗i 10j Hd .

(2)

For convenience, all superfields have been arranged in SU (5) multiplets, 10 = (q, uc , ec ) and 5∗ = (dc , l), andi, j = 1, 2, 3 are flavour indices. We assume that the colour triplet partners of the electroweak Higgs doublets Hu
and Hd have been projected out. The vacuum expection values vu = hHu i and vd = hHd i break the electroweakp
symmetry. In the following we will assume large tan β = vu /vd , implying vd  vu ' vEW = vu2 + vd2 . We
will restrict our analysis to the case of a hierarchical heavy (s)neutrino mass spectrum, M1  M2 , M3 , whereM i = hni vB−L . Furthermore we assume the heavier (s)neutrino masses to be of the same order of magnitude as
the common mass mS of the particles in the symmetry-breaking sector, for definiteness we set M2 = M3 = mS .
Key parameters of the analysis are then the B−L breaking scale vB−L , the mass of the lightest of the heavy(s)neutrinos M1 , and the effective light neutrino mass parameter me 1 (cf. [10]),vB−L '
Here, mν =

2
vEW
,
mν

M1  vB−L ,

m
e1 ≡

2
(hν† hν )11 vEW
.
M1

(3)

√
m2 m3 ∼ 3 × 10−2 eV is the geometric mean of the two light neutrino mass eigenvalues m2 andm3 , and characterizes the light neutrino mass scale. In addition to the chiral superfields, the model also contains

2

W. Buchmüller, V. Domcke, K. Kamada, K. Schmitz

a vector supermultiplet V ensuring invariance under local B−L transformations and the gravity supermultiplet
consisting of the graviton G and the gravitino G̃.
In the following sections we shall show that this Minimal Supersymmetric Model (MSM), whose parameters arelargely fixed by low energy experiments, provides a consistent description of the transition from an inflationary
phase to the hot early universe. During this ‘pre- and reheating’ process the matter-antimatter asymmetry andthe dark matter abundance are generated. Most of our discussion will be based on Refs. [10–12].
Our work is closely related to previous studies of thermal leptogenesis [13, 14] and nonthermal leptogenesis via
inflaton decay [15–18], where the inflaton lifetime determines the reheating temperature. In supersymmetric
e1 of the lightest heavy Majorana neutrino N1 canmodels with global B−L symmetry the scalar superpartner N
play the role of the inflaton in chaotic [19, 20] or hybrid [21, 22] inflation models. Local B−L breaking inconnection with hybrid, shifted hybrid and smooth hybrid inflation has been considered in Ref. [23]. One of themain motivations for nonthermal leptogenesis has been that the ‘gravitino problem’ for heavy unstable gravitinos[24–28] can be avoided by means of a low reheating temperature. In the following we shall assume that thegravitino is either the lightest superparticle (LSP) or very heavy, m3/2 & 10 TeV. In the first case, gravitinos,
thermally produced at a reheating temperature compatible with leptogenesis, can explain the observed dark
matter abundance [29]. For very heavy gravitinos, thermal production and subsequent decay into a wino or
higgsino LSP can yield nonthermal WIMP dark matter [30–32].
The MSM, defined in Eq. (1), postdicts the earliest phases of the cosmological evolution. The energy density ofa false vacuum with unbroken B−L symmetry drives inflation. Consistency with the measured amplitude of thetemperature anisotropies of the cosmic microwave background fixes vB−L , the scale of B−L symmetry breaking,to be the GUT scale. Inflation ends by tachyonic preheating [33], i.e. the decay of the false vacuum, which sets
the stage for a phase dominated by nonrelativistic matter in the form of heavy B−L Higgs bosons. The further
development is described by Boltzmann equations. Nonthermal and thermal processes produce an abundance ofheavy neutrinos whose decays generate primordial entropy, baryon asymmetry via leptogenesis and gravitino dark
matter from scatterings in the thermal bath. This whole pre- and reheating process is imprinted on the spectrumof primordial gravitational waves [34]. It is remarkable that the initial conditions of the radiation dominatedphase are not free parameters of a cosmological model. Instead, they are determined by the parameters of a
Lagrangian, which in principle can be measured by particle physics experiments and astrophysical observations.
The consistency of hybrid inflation, leptogenesis and dark matter entails interesting relations between the lightest
neutrino mass m1 , the gravitino mass and possibly wino or higgsino masses.
The paper is organized as follows. In Section 2 we discuss F-term hybrid inflation. Corrections from supersymmetry breaking lead to a two-field model which can account for all results deduced from the recently released
Planck data. Section 3 deals with tachyonic preheating and the important topic of cosmic string formation, withemphasis on the current theoretical uncertainties. The description of the reheating process by means of Boltzmann equations and the resulting relations between neutrino masses and superparticle masses are the subject of3

A Minimal Supersymmetric Model ofParticle Physics and the Early Universe

Section 4. The predictions of the gravitational wave spectrum due to inflation, cosmic strings, pre- and reheating
are reviewed in Section 5. Finally, observational prospects are addressed in Section 6.

2.

Inflation

The superpotential of the MSM, cf. Eq. (1), allows for a phase of F-term hybrid inflation. For |φ|  vB−L , the
B−L Higgs fields are fixed at zero, B−L is unbroken and the energy density of the universe is dominated by the4false vacuum energy, ρ0 ' (λ/4) vB−L
≡ V0 , generated by the non-vanishing vacuum expectation value (vev) of

the auxiliary field Fφ and inducing spontaneous supersymmetry breaking. Here, we briefly review the dynamics
and predictions of this inflation model, with particular focus on the status of F-term hybrid inflation in the lightof the recent Planck results [35].
The scalar potential
At the high energy scales involved in inflation, supergravity corrections to the Lagrangian become important,resulting in a tree-level scalar F- and D-term potential given by2F
VSUGRA= eK/MP 

X
αβ̄

K

αβ̄


|W |2 ,Dα W Dβ̄ W − 3
MP2
∗

1
D
VSUGRA
= g2
2

!2
X

qα Kα zα

,

(4)

α

where Dα W = Wα + Kα W/MP2 ; the subscript α (ᾱ) denotes the derivative with respect to the (complex conjugateof the) scalar component zα of the superfield Φα carrying the U (1) gauge charge qα , K αβ̄ is the inverse Kähler
metric and MP = 2.4 × 1018 GeV is the reduced Planck mass. For a canonical Kähler potential,K=X|zα |2 ,(5)αthe D-term scalar potential reduces to the expression familiar from global supersymmetry, but an importantF
supergravity contribution arises from the F-term potential, VSUGRA⊃ |zα |2 ρ0 /MP2 . This yields large contributions

to the masses of the scalar fields zα of the theory. For the superpotential (1), this stabilizes the singlet sneutrinos
and the MSSM scalars at a vanishing field value. The B−L Higgs boson masses also obtain various supergravity
contributions. However, these are suppressed by factors of (vB−L /MP )2 or (φ/MP )2 compared to the leading
order terms, which match the result found in global supersymmetry,22(mS± ) = λ(|φ| ±1 2vB−L ) ,222(mSf ) = λ|φ| .

(6)

The F-term supergravity contribution discussed above does not give a mass-term to the inflaton φ because
2

after expanding eK/MP in Eq. (4), the term in question is cancelled by the corresponding term in Dφ W Dφ̄ W ∗ .
The leading order supergravity contribution to the inflaton mass thus stems from the term proportional to4|φ|4 vB−L/MP4 in the scalar potential [36]. In addition, since supersymmetry is broken during inflation, we need to4

W. Buchmüller, V. Domcke, K. Kamada, K. Schmitz

take the one-loop Coleman-Weinberg (CW) potential for the inflaton field into account, obtained by integratingout the heavy B−L Higgs bosons,
V1l =

1
64π 2

"
STr M

4

ln

M2
Q2

!

1
−
2

!#
'

4
λ2 vB−L

"
ln

64π 2

2|φ|2
2
vB−L

!
+O

4
vB−L

!#
,

(7)

4|φ|4

Here STr denotes the supertrace running over all degrees of freedom of S1 and S2 . M is the corresponding mass2
matrix, cf. Eq. (6), and Q an appropriate renormalization scale, which we have set to Q2 = λvB−L
/2.

√
F +D
From the resulting scalar potential, V = VSUGRA+ V1l , we find the following picture: For |φ| > vB−L / 2, the√2Higgs fields s1,2 are fixed at zero and the inflaton slowly rolls towards the origin. At |φ| = vB−L / 2, (mS−)
becomes negative, triggering a tachyonic instability. The Higgs fields acquire a vev and B−L is broken. Boththe Higgs (which, now that B−L is broken, is best parametrized in unitary gauge as a radial degree of freedom
dubbed σ 0 in Ref. [10], see also Ref. [37] for the explicit relation between σ 0 and the symmetry-breaking Higgs
mass eigenstate in arbitrary gauge) and the inflaton field then quickly fall into their true vacuum, |φ| → 0 and√σ 0 → 2 vB−L , eliminating the vacuum energy contributions of the scalar potential and ending inflation.
Slow-roll inflation
In the slow-roll approximation, the dynamics of the homogeneous inflaton field is governed by 3H φ̇ = −∂V /∂φ∗ ,where H denotes the Hubble parameter. For a scalar potential only depending on the absolute value of φ, cf.
Eq. (7), we can rewrite this in terms of the radial and angular component of φ = √12 ϕe−iθ ,3H ϕ̇ = −V 0 (ϕ) ,θ̇ = 0 .

(8)Turning to the quantum fluctuations of the inflaton field which are visible today in the CMB, we now evaluate
the scalar potential and its derivatives at ϕ = ϕ∗ , the value of ϕ at N∗ ≈ 55 e-folds before the end of inflation,
when the reference scale commonly used to describe the CMB fluctuations left the horizon. With ϕf denoting
the value of the inflaton at the end of inflation,1 ϕ∗ is given byϕ2∗ ≈ ϕ2f +λ
MP2 N∗
4π 2(9)Of particular interest in the following will be the predictions from F-term hybrid inflation for the amplitude of
the scalar fluctuations As , the scalar spectral index ns and the tensor-to-scalar ratio r:
As =

H2
1
≈
8π 2 MP2 ϕ∗
3



vB−L
MP

4

1
,
N∗
At
λ 1
≈
r=
= 16
,
2 N
As
2π
∗
ϕ∗

ns = 1 − 6 + 2η|ϕ∗ ≈ 1 −

N∗ ,
(10)

1

Here, ϕf is determined by either mSof the slow-roll condition, i.e.
− (ϕc ) = 0, cf. Eq. (6), or by√the violation√|η(ϕη )| = 1, cf. Eq. (11), whatever occurs earlier: ϕf ≈ max{vB−L , λMP /( 8π)}.

5A Minimal Supersymmetric Model ofParticle Physics and the Early Universe

2.0
N = 20

N=0

0.5
0.0

N=0

¬

0.5

1.0

V,j < 0®

1.0

V,j > 0VHΦr ,0LV0 - 1

1.5

N = 55

-0.5
-2.0

-1.5

-1.0

-0.5

0.0

1.5

2.0

Φr @vB-L D

Figure 1. Scalar potential for inflation along the real axis in the complex φ field space after adding a constant term W0 to thesuperpotential. Slow-roll inflation is possible both for θ = 0 and θ = π. Here λ = 4.5 × 10−6 , vB−L = 2.9 × 1015 GeV
and mGf = 47.5 TeV.

where At = 2H 2 /(π 2 MP2 )|ϕ∗ denotes the amplitude of the tensor fluctuations, and  and η are the so-calledslow-roll parameters,
=

MP2
2



V0
V

2
,

η = MP2

V 00
.
V

(11)

Moreover, in Eq. (10) we have employed the the approximation2 ϕ2∗  ϕ2f .
F-term hybrid inflation in the light of Planck
Comparing these results with the recently published Planck data [35, 38],

As = (2.18 ± 0.05) × 109 ,ns = 0.963 ± 0.0007 ,r < 0.26 ,(12)

we find that the B−L breaking scale is fixed to vB−L ≈ 8 × 1015 GeV by requiring the correct normalization of
As , the spectral index ns ≈ 0.98 is rather large and the tensor-to-scalar ratio is easily below the current bound.
In particular the large value for ns has raised the question whether F-term hybrid inflation is still viable in viewof the Planck results. To answer this question, we must go beyond the approximations leading to Eq. (10). First,
we will drop the approximation ϕ2∗  ϕ2f , leading to corrections of the predictions listed in Eq. (10). Second,taking into account soft supersymmetry breaking, the superpotential receives a constant term W0 = mGe MP2
proportional to the gravitino mass [39], leading to an additional contribution to the scalar potential, studied e.g.
in Refs [40, 41],
√ 2
VmG
= −2 λvB−L
mGe |φ| cos θ .
f
2

(13)

Note that for small values of λ, the two terms in Eq. (9) can be of similar importance, leading to a slightdeviation from the results listed in Eq. (10).

6W. Buchmüller, V. Domcke, K. Kamada, K. Schmitz®

®

N = 55HΦrf ,Φi L
f

®

0
16

-1 ´ 10®

Φi @GeVDHΦ*r ,Φ*i L

V,j > 05´1015N=0-5 ´ 1015

H-vB-L,0L

Φr @GeVDV,j < 0VHΦr ,ΦiL

HvB-L,0L
0

5 ´ 1015

Figure 2. Inflationary trajectories in the full two-field inflation model. Selection of possible trajectories (solid green lines) in thescalar potential V (φr , φi ) depicted by the dot-dashed orange contour lines and the shading. Lines of constant N aremarked by the dashed blue contours, with the beginning and end of inflation (N = N∗ and N = 0, respectively) marked
by thicker contours.

This term breaks the degeneracy appearing in Eq. (7), which only depends on the absolute value |φ| of the inflaton
field but not on its phase θ. As a result, the inflationary predictions found in Ref. [40] assuming θ = π differ
from those in Ref. [41], which uses θ = 0.3 In particular, for sufficiently large VmG, we find a hill-top potentialffor θ = 0, while for θ = π, one still finds a monotonously decreasing potential along the inflationary trajectory(along the arrows in Fig. 2), cf. Fig. 1.
Indeed, these are only two extreme cases for possible inflationary trajectories in the full two-field inflation model
resulting from Eqs. (7) and (13). In the two-field model, the ending point of inflation φf becomes a ‘critical line’in the complex φ plane, with each point on this line representing a possible ending point for inflation. Henceas opposed to the single-field case there is an additional degree of freedom, i.e. the choice of the inflationary
trajectory labeled by θf . This is visualized in Fig. 2 which shows a selection of possible inflationary trajectories(in green) in the scalar potential (dot-dashed orange countour lines and shading). Contour lines denoting constantnumbers of e-folds N are shown as dashed blue lines, with the ‘critical line’, N = 0, and the onset of the lastN∗ e-folds, N = N55 emphasized. To demonstrate the dependence of the model predictions on the choice of the
trajectory, Fig. 3 shows the predictions for the amplitude As and the spectral index ns as functions of the final
phase θf for λ = 4.5 × 10−6 , vB−L = 2.9 × 1015 GeV and mGe = 47.5 TeV. For this parameter example, we see that
θf ' 16◦ reproduces the correct amplitude, cf. Eq. (12), while simultaneously yielding a value for the spectral
index of ns = 0.965 in very good agreement with the data.
3

Note that Refs. [40, 41] use a different sign convention in the superpotential, implying θ → θ + π.

7

A Minimal Supersymmetric Model ofParticle Physics and the Early Universe

1.00
-9
Aobs
s > 2.18 ´ 10

0.5

0.1
0.05

0.99
0.97
ns > 0.965
0.96Θ f > 16 °Scalar spectral index ns

0.99

1.00

Θ f > 16 °Scalar amplitude As A10-9 E

5.00

0.95
0.94

0.01
0

Π8

Π4

3Π8

Π2

5Π8

3Π4

7Π8

0

Π

Π8

Final inflaton phase Θ f

Π4

3Π8

Π2

5Π8

3Π4

7Π8

Π

Final inflaton phase Θ f

Figure 3. Amplitude and spectral index of the scalar primordial fluctuations for λ = 4.5 × 10−6 , vB−L = 2.9 × 1015 GeV andmG
f = 47.5 TeV. The phase of the endpoint of inflation, θf , labels different inflationary trajectories.

A third possibility of manipulating Eq. (10) is by resorting to a non-minimal Kähler potential [42]. This inparticular introduces a term quadratic in φ in the scalar potential. Tuning the expansion coefficients of such aKähler potential, the spectral index can be tuned to lower values, achieving accordance with the Planck data evenfor θ = π. However, the quadratic term then comes with a negative sign, implying, together with the positive|φ|4 term the existence of a hill-top potential and a local minimum at |φ| =6 0 where the inflaton can get trapped.
Avoiding this requires some fine-tuning in the initial conditions for |φ| [40].
On top of that, there are also further constraints which must be taken into account in a realistic model. First, thesuperpotential Eq. (1) will lead to the production of cosmic strings at the end of inflation, due to the spontaneous
breaking of U (1)B−L . We will come back to this point in Sec. (3.2). Moreover, the abundance of nonthermally
produced gravitinos, controlled by the symmetry-breaking scale, the common mass of the inflaton and B−L Higgs
in the true vacuum, and the reheating temperature [43, 44],
Y3/2 ∝

2
vB−L
m2S
,
TRH

(14)

must be sufficiently low, so that the sum of nonthermal and thermal (cf. section 4) gravitino abundance does
not produce a gravitino problem [24–28, 45–47]. Note, however, that the nonthermal gravitino abundance canbe suppressed compared to the estimate in Eq. (14), if the massive particle governing the universe during thereheating phase decays sufficiently fast.
Taking all of this together, we find that F-term hybrid inflation is indeed still viable in light of the Planck data,
however some tuning is required. Accepting a non-minimal Kähler potential with some tuning in its coefficients
as well as in the initial conditions for |φ|, accordance with the Planck data can be achieved for θf ∼ π. Staying
with a minimal Kähler potential one has two options to reproduce the experimental data. Either, one tunes theamplitude of the linear term (13) against the CW term (7) in the potential, leading to the situation shown in
Fig. 2. In this case small values for ns can be achieved for θf ∼ 0, but again tuning of the initial condition for
the radial degree |φ| is necessary to prevent the inflaton from being trapped on the wrong side of the hill-top

8

W. Buchmüller, V. Domcke, K. Kamada, K. Schmitz

potential.4 The second possibility is to allow the linear term to dominate over the CW term, however then theinitial phase of φ must be tuned because otherwise one lands on a trajectory where inflation does not end, because
it ‘misses’ the minimum generated by the CW term at small |φ|, such that |φ| always remains larger than thecritical value. For a more detailed analysis of the full two-field inflation model, see Ref. [48].
In summary, successful inflation can be achieved but it imposes constraints on the B−L breaking scale and on
the coupling λ. In the context of the Froggatt-Nielsen flavour model used to parametrize the Yukawa couplings
in Ref. [10], this then constrains the mass of the heaviest of the right-handed neutrinos M1 . In the following, we
will thus consider the restricted parameter spacevB−L = 5 × 1015 GeV ,109 GeV ≤ M1 ≤ 3 × 1012 GeV ,(15)

10−5 eV ≤ m
e 1 ≤ 1 eV ,where the variation of the effective light neutrino mass parameter m
e 1 accounts for the uncertainties of the FroggattNielsen model. The values of vB−L and λ quoted here correspond to the option of choosing θf = π and using
a non-minimal Kähler potential, cf. Ref. [40]. For a discussion of cosmological B−L breaking involving smaller
values for vB−L , cf. Ref. [12].

3.

Tachyonic Preheating and Cosmic Strings

The end of hybrid inflation induces a negative squared mass term for the B−L Higgs field σ 0 in the false vacuum, triggering the U (1)B−L breaking phase transition. The cosmological realization of this phase transition isaccompanied by two important nonperturbative processes, tachyonic preheating [33] and the formation of cosmic
strings [49].

3.1.

Tachyonic preheating

The phase transition
Tachyonic preheating is a fast and nonperturbative process triggered by the tachyonic instability in the scalarpotential in the direction of the Higgs field. As the inflaton field passes a critical point φc , the Higgs field σ 0√
acquires a negative effective mass squared −m2σ , with m2σ = 2λvB−L |φ̇c |t in the linearized equation of motion
for σ 0 close to the instability point φc . This causes a faster than exponential growth of the quantum fluctuationsof the Higgs field σk0 with wave numbers |~k| < mσ [50], while the average value of the Higgs field remains zero.
1/2

2
Once the amplitude of these fluctuations, v(t) = √12 hσ 02 i = √12 hσ 02 (t, x)ix ,5 reaches hσ 02 (t∗ )i = O(vB−L), the45Note that nevertheless, an arbitrary amount of e-folds of inflation can be realized in this setup.
Here, bold letters indicate 3-vectors.

9

A Minimal Supersymmetric Model ofParticle Physics and the Early Universe

curvature of the potential for the homogeneous background field σ 0 becomes positive and the usual oscillating
behaviour of the modes is re-established [33] while v(t) approaches vB−L . A direct consequence of the early phaseof exponential growth are high occupation numbers in the low-momentum Higgs modes and hence a semi-classicalsituation with a large abundance of non-relativistic B−L Higgs bosons.
A further result of this nonperturbative process is the formation of ‘bubble’-like inhomogeneities which randomly
feature different phases of the complex Higgs field [50, 51]. Their initial size is given by the smallest scale amplified
during tachyonic preheating, referred to as k∗−1 . These bubbles expand at the speed of light, thereby colliding witheach other. This phase of the preheating process is an important source of gravitational waves (GWs), cf. [53], apoint to which we will return in Sec. 5. After this very turbulent phase the true Higgs vev is reached in almost
the entire volume, with the regimes of false vacuum reduced to topologically stable cosmic strings, cf. Sec. 3.2,√
separated by the characteristic length scale k∗−1 ≈ ( 2λvB−L |φ̇c |)−1/3 .
Secondary Particle production
The mode equations for the particles coupled to the B−L Higgs field, i.e. for the gauge, Higgs, inflaton andneutrino supermultiplets, feature masses proportional to v(t). The growth of hσ 02 i during tachyonic preheating
thus induces a rapid change of their effective masses. The resulting particle production was studied in Ref. [52],with the results depicted in Fig. 4. Here, for simplicity, an abrupt transition of the inflaton vev to zero is assumed,
introducing the parameter m = mσ (φ̇c t → φc ).6 The left panel shows the evolution of hσ 02 i normalized to thesymmetry-breaking scale vB−L , calculated using a lattice simulation (green curve). For comparison, the red curvevm(t−t∗ )1+tanh. The pink and blue curves depict the number
shows an analytical approximation, v(t) = B−L22
densities of bosonic particles coupled to the Higgs, again calculated using a lattice calculation and an analytical
approximation, respectively. The right panel examines the momentum distribution of these bosons (and also of
fermions coupled to the Higgs) showing the spectrum of occupation numbers. Again both the numerical andanalytical results are shown. We see that just like the Higgs bosons themselves, the particles coupled to it are
produced with very low momentum, i.e. non-relativistically.
Based on these results, the energy and number densities for bosons and fermions coupled to the Higgs boson after
tachyonic preheating have been estimated as [52]7
ρB /ρ0 ' 2 × 10−3 gσ λ f (x1 , 1.3) ,nB (x1 ) ' 1 × 10−3 gσ m3S f (x1 , 1.3)/x1 ,ρF /ρ0 ' 1.5 × 10−3 gσ λ f (x1 , 0.8) ,nF (x1 ) ' 3.6 × 10−4 gσ m3S f (x1 , 0.8)/x1 ,(16)

with f (x1 , x2 ) = (x21 + x22 )1/2 − x2 and x1 = mi /mS , where mi denotes the mass of the respective particle in thetrue vacuum and gσ counts its spin and internal degrees of freedom.
6

The effect of the inflaton dynamics on this nonperturbative particle production requires further investigation.
Note that particle production can be significantly enhanced by quantum effects [54], which require further investigation.
7

10

W. Buchmüller, V. Domcke, K. Kamada, K. Schmitz

φ(t) (Tanh)<φ2(t)>1/2/v (Lattice)nB(x100) (Tanh)nB(x100) (Lattice)

nB(t)

1

<φ2(t)>1/2/v,

0.1

0.01

0.001
5

10

15
time: mt

10

20

25

30

Bosons: LatticeTanhFermions: LatticeTanh1

Occupation number: nk

0.1
0.01
0.001

0.0001
1e-05
1e-06
1e-07

0.1

1
k/m

Figure 4. Numerical and analytical results for particle production during tachyonic√ preheating, taken from Ref. [52]. Left panel:

Evolution of the quantum fluctuations of the Higgs field φ(t) ≡ σ 0 (t)/ 2, normalized to the symmetry-breaking scalev ≡ vB−L , as well as the number density nB of bosonic particles coupled to it. Right panel: Spectrum of occupation
numbers for bosonic and fermionic particles coupled to the Higgs.

3.2.

Cosmic strings

Due to the non-trivial topology of its vacuum manifold, the Abelian Higgs model underlying the B−L phasetransition gives rise to solitonic field configurations, so-called cosmic strings (for a review, see e.g. [55–57]). These
cosmic strings are formed during the process of tachyonic preheating and are topologically stable. The evolutionof the resulting network is governed by the intersection of the infinite strings, which leads to the formation ofclosed loops separated from the infinite string, as well as by the energy loss due to the emission of GWs, Higgs
and gauge particles. After a relaxation time the network reaches the scaling regime, i.e. the typical length scaleof the cosmic string network remains constant relative to the size of the horizon. This implies that a constant11

A Minimal Supersymmetric Model ofParticle Physics and the Early Universe

fraction of the total energy density is stored in cosmic strings throughout the further evolution of the universe
and that there are O(1) cosmic strings per Hubble volume.
In the scaling regime, the cosmic string network is characterized by the energy per unit length µ. In the Abelian
Higgs model, which is based on a field theory featuring a spontaneously broken local U (1) symmetry, µ is givenby [58]2µ = 2πB(β)vB−L,(17)

where β = (mS /mG )2 = λ/(8g 2 ) is the ratio of the masses of the symmetry-breaking Higgs boson and the gaugeboson in the true vacuum, and B(β) is a slowly varying function parametrizing the deviation from the Bogomol’nyi
bound,
B(β) '


 1.04 β 0.195 ,if 10−2 < β  1

.

(18)


 2.4 (ln 2 )−1 , if β < 10−2βFor the special case of β = 1 the Bogomol’nyi bound is saturated and B(1) = 1 [59].
Further important quantities describing the string network are the cosmic string width, given by m−1
G in theAbelian Higgs model, and the length scale ξ separating two strings. From Sec. 3.1, we know that the characteristic
length separating two strings at the time of their formation is√ξ = k∗−1 = ( 2λvB−L |φ̇c |)−1/3 .

(19)This also determines the relaxation time of the cosmic string network, τstring ∼ ξ [50, 60]. Note that in the
Nambu-Goto model, an alternative to the Abelian Higgs cosmic string model which assumes infinitely thin cosmic
strings, the energy scale µ is an input parameter.
Observational prospects
So far, no experimental evidence for the existence of cosmic strings has been found. However, current andupcoming experiments are starting to seriously probe the cosmologically interesting regions of the parameter
space. First, cosmic strings give rise to anisotropies in the CMB temperature map. They distort the surfaceof last scattering of the CMB photons, leaving an imprint on the spectrum observable today. Since the CMB
photons observable today stem from roughly 105 Hubble patches during recombination, these observations aremainly sensitive to the effect of long (Hubble-sized) strings at recombination and not to small cosmic string loops.
In contrast to the perturbations due to inflation, these anisotropies are not phase correlated across distant Hubble
patches and hence the resulting multipole spectrum of the two-point correlation function is suppressed at large
scales and moreover does not show the oscillations characteristic to inflation. Moreover, whereas the primordial
power spectrum due to inflation is (nearly) scale-invariant, the anisotropies on the last scattering surface due tocosmic strings are governed by a characteristic scale. The resulting spectrum thus features a single broad peakassociated with this scale. Due to the re-scattering of a fraction of the CMB photons at reionisation, the CMB
spectrum is, to a lesser extent, also sensitive to the long cosmic strings present at reionisation. This leads to a12

W. Buchmüller, V. Domcke, K. Kamada, K. Schmitz

second, smaller peak in the spectrum, in particular visible in the power spectrum of the B-mode polarization, seee.g. [61] for a recent analysis. The fraction of the amplitude of the scalar power spectrum due to a possible cosmic
string contribution to the CMB temperature anisotropies is conventionally measured at the multipole l = 10 andis referred to as f10 . The Planck data implies that f10 can at most be a few percent, f10 < 2.8% [38].
Second, the gravitational field of cosmic strings gives rise to weak and strong lensing effects of (CMB) photons on
their way from the surface of last scattering or from an astrophysical source to us. The non-observation of such
effects puts a bound on the string tension µ. Again, this effect is mainly sensitive to long (Hubble-sized) strings.
Third, the energy emitted by cosmic strings in the scaling regime is at least partly emitted in form of GWs.
Due to their extremely weak coupling, these can then propagate freely through the universe and are therefore inprinciple detectable today. We will come back to the resulting GW background and the discovery potential of
current and upcoming GW experiments in detail in Sec. 5.
Finally, the Abelian Higgs cosmic string model entails the emission of massive radiation from cosmic strings, i.e.
the emission of the Higgs and gauge particles whose field configurations form the string. If this mechanism is still
active at late times it could yield ultra-high-energetic cosmic rays and GeV-scale γ-rays, which have not been
observed. This too, can be translated into a (model-dependent) bound on µ [62–66].
Currently the most stringent and model-independent bound on the cosmic string tension comes from CMB
observations, Gµ < 3.2 × 10−7 [38], and we shall mainly employ this in the following.
Numerical simulations and theoretical uncertainties
A quantitative understanding of the formation of cosmic strings, the dynamics of the cosmic string network andthe energy loss mechanism during the scaling regime requires lattice simulations. Performing these is extremely
challenging due to the huge range of scales involved in the problem [56]: the width of the string remains constant
while the scales of the network are blown up as the universe expands. Or, in comoving coordinates, the comoving
width of the string shrinks, until it becomes comparable with the lattice spacing and the simulation loses its
validity. There have been different approaches to tackle this problem. Simulations based on solving the field
equations for the Abelian Higgs (AH) model set the comoving width to a finite constant before it comes too
close to the lattice spacing [60, 67, 68]. Simulations based on the Nambu-Goto (NG) string model assume cosmic
strings to be infinitely thin, i.e. strictly one-dimensional objects, throughout the simulation [69–72]. The outcomeof simulations based on these two models is dramatically different. The AH simulations show the formation oflarge, Hubble-sized structures which lose their energy predominantly by emitting massive radiation, i.e. particlesof the Higgs and gauge fields forming the string configuration. The NG simulations on the other hand display theformation of small loops, which lose their energy into GWs. The size of these loops is thought to be controlled
by gravitational backreaction, but is as yet undetermined [56]. Concerning the network of long strings, both
simulations, however, yield a similar result [56]. Which of these two simulations methods is closer to reality iscurrently an open question.
In the following, we will adopt the following hypothesis: For early times, while the comoving cosmic string width13

A Minimal Supersymmetric Model ofParticle Physics and the Early Universeis large compared to the lattice spacing, the AH simulation describes the U (1) phase transition very well. We willthus use the results from these simulations when discussing the formation and early evolution of cosmic strings.
For late times, the AH simulations become questionable and the NG approximations of infinitely thin strings
appears reasonable. Hence for late times, in particular when discussing possible GW signatures from cosmic
strings, cf. Sec. 5, we shall discuss both the AH as well as the NG results.

4.

ReheatingTachyonic preheating nonperturbatively generates a large abundance of non-relativistic B−L Higgs bosons as wellas, to a much lesser extent, non-relativistic abundances of the particles coupled to the Higgs boson, cf. Sec. 3.1.
Among these are the particles of the B−L gauge supermultiplet, which decay quickly due to their comparatively
strong gauge interactions. This sets the initial conditions for the following slow, perturbative reheating process,depicted by the solid arrows in the left panel of Fig. 5: The particles from the symmetry-breaking sector decay
into particles of the N1 supermultiplet. These (s)neutrinos, just as the (s)neutrinos produced through gauge
particle decays and tachyonic preheating as well as thermally produced (s)neutrinos, decay into MSSM particles,
thereby generating the entropy of the thermal bath as well as a lepton asymmetry [73]. Finally, the thermal bathproduces a thermal gravitino abundance, which will turn out to be in the right ball-park to yield the observed
dark matter abundance.
The main tool to obtain a time-resolved description of this reheating process are Boltzmann equations, which
describe the evolution of the phase space densities of the various particles species due to decay and scattering
processes in an expanding universe. After briefly introducing the formalism of Boltzmann equations in Sec. 4.1,
we will turn to the implications for leptogenesis and dark matter production in Sec. 4.2. The results presented
here are based on the analyses of Refs. [10–12].

4.1.

Boltzmann equations

The evolution of the phase space density fX (t, p) of a particle species X is determined by a coupled set ofBoltzmann equations,

E

∂
∂
− Hp
∂t
∂p


fX (t, p) =XX

CX (Xi0 j 0 .. ↔ ij..) ,(20)

i0 j 0 .. ij..

augmented by the Friedmann equation, which governs the evolution of the scale factor. The left-hand side ofEq. (20) describes the evolution of the phase space density in an expanding Friedman-Robertson-Walker (FRW)
universe whereas the collision operators CX on the right-hand side account for all relevant scattering, decay
and inverse decay processes involving the particle X. The set of Boltzmann equations we have to solve here isdetermined by the allowed interactions of the underlying particle physics model, cf. solid blue arrows in the left
panel of Fig. 5.
From the phase space density fX (t, p) one directly obtains the comoving number density NX (t), i.e. the number

14

W. Buchmüller, V. Domcke, K. Kamada, K. Schmitz

Inverse temperature M1  T
Schematic view

Comoving number density abs NHaL

Gauge

Higgs +
InﬂatonRight-handed
Neutrinos
Radiation+
B-L asymmetry
Gravitinos
tachyonic preheating

100

10-1

1050

101

102

103
R

1045

S
N1nt

1040

B-L

G

1035
N1th

10

30

10

25

aiRH

fast processslow process

100

101

102

f

aRH

103

104

aRH

105

106

107

108

Scale factor a

Figure 5. Evolution of the comoving number densities during the reheating process. Left panel: schematic overview, distinguishing production via tachyonic preheating, fast decay processes of the B−L gauge sector and slow processes describedby Boltzmann equations. Right panel: Comoving number densities of the particles of the B−L Higgs sector (S), thee and the B−L asymthermal and nonthermal (s)neutrinos (N1th , N1nt ), the MSSM radiation (R), the gravitinos (G)metry (B−L). Obtained by solving the Boltzmann equations for vB−L = 5 × 1015 GeV, M1 = 5.4 × 1010 GeV,m
f1 = 4.0 × 10−2 eV, mG
f = 100 GeV and mg̃ = 1 TeV. From Ref. [10].

of X particles in a volume (a/GeV)3 , and the energy density ρX (t) by integrating over momentum space,
NX (t) =

a(t)
GeV
Z

ρX (t) = gX

3


nX =

a(t)
GeV

3

Z
gX

d3 p
EX (p) fX (t, p) ,
(2π)3

d3 p
fX (t, p) ,
(2π)3

(21)

with a denoting the scale factor. A rescaling of a leaves the physical number density nX invariant. For convenience,
we will thus set aPH ≡ 1 at the end of preheating. In the following, decay rates Γ, comoving number densities N
and energy densities ρ will sometimes appear with upper and lower indices. In this case, the lower index refers to
the particle species under consideration, while the upper index refers to its origin, e.g. its parent particle or ‘PH’for preheating.

4.2.

Outcome of the reheating process

Solving the Boltzmann equations with the initial conditions given by tachyonic preheating and the successive
decay of the B−L gauge bosons yields a time-resolved picture of the evolution of all particle species. In the rightpanel of Fig. 5, we show an overview of the resulting comoving number densities for a representative parameter
point.
A two-stage reheating process
After the end of preheating, the lion’s share of the energy is stored in non-relativistic B−L Higgs bosons. Assuming15

A Minimal Supersymmetric Model ofParticle Physics and the Early Universe

Inverse temperature M1  T
10-1

100

101

102

101

102

103

1012

THaL @GeVD

1011
1010

100

103

104

f

108

aRH

aiRH

aHtRH L

109

105

106

107

108

Scale factor a
Figure 6.

Temperature of the thermal bath for the same parameter values as in Fig. 5. From Ref. [10].

a hierarchical spectrum of heavy Majorana neutrinos, these decay exclusively into heavy, typically relativistic(s)neutrinos of the first generation, thereby forming the main part of the right-handed (s)neutrino population.
The decay of these (s)neutrinos then generates a thermal bath of MSSM particles. The process of reheating ishence governed by the interplay of two time-scales, the vacuum decay rate of the non-relativistic Higgs bosons
Γ0S and the effective decay rate of the neutrinos produced in the Higgs boson decays ΓS
N1 . The latter differs fromthe zero-temperature decay rate Γ0N1 due to the time-dilation of the relativistic neutrinos,

Γ0S =

1
32π



M1
vB−L

2


1/2
M2
mS 1 − 4 21
,
mS

S
−1
ΓS
(aRH ) Γ0N1
N1 := ΓN1 (aRH ) = γ

with

γ

−1


(a) =

M1
EN1

(S)
,
a

Γ0N1 =1 m
e 1 M12.
24π vEW(22)0
In most of the viable parameter space, we find Γ0S < ΓS
N1 . In this case, ΓS determines the overall time-scale of

the reheating process. On the contrary aRH , defined by H(aRH ) = ΓS
N1 (aRH ), marks a characteristic point in the
middle of the reheating process, which will be in particular relevant for determining the reheating temperature.
Once the Higgs bosons decay into neutrinos, these decay nearly instantaneously into MSSM particles, so that theera of Higgs domination is directly followed by the radiation dominated epoch. On the other hand, if Γ0S > ΓS
N1 , theeffective neutrino decay rate governs the time-scale of reheating. The energy density is then successively governedby non-relativistic Higgs bosons, relativistic nonthermal neutrinos and finally relativistic thermal MSSM particles.
Solving the Boltzmann equations allows us to determine the temperature of the thermal bath throughout thereheating process. As a consequence, the ‘reheating temperature’ is no longer a cosmological input parameter,

16

W. Buchmüller, V. Domcke, K. Kamada, K. Schmitz

Inverse temperature M1  T
100

10-1

1045

101

102

103

 nt
N1nt +N 1

abs NHaL

1040

B-L
eq
2N1

1035

Nonthermal

B-L

1025

100

101

102

103

104

105

NLth > 0

l

ma

er
Th

NLth < 0 th
N1th +N 1

NLnt < 01030

NLnt > 0N2,3 +N 2,3

106

107

108

Scale factor a
Figure 7. Comoving number densities for the nonthermal (NLnt ) and thermal (NLth ) contributions to the total lepton asymmetry
as well as all (s)neutrino species (N1nt + Ñ1nt , N1th + Ñ1th , 2N1eq for comparison and N2,3 + Ñ2,3 ) as functions of thescale factor a. The vertical lines mark the changes in the signs of the two components of the lepton asymmetry. From
Ref. [10].

but is rather determined by the parameters of the B−L Higgs and neutrino sector. In Fig. 6 we show theresulting evolution of the temperature. A remarkable feature is the epoch of nearly constant temperature during
the main part of the reheating process, which arises because the entropy production from the neutrino decays justNcompensates the expansion of the universe. A typical value for this plateau is given by TRH≡ T (aRH ), with aRH

as defined above. The dashed vertical lines labeled aiRH and afRH in Figs. 5 and 6 mark the beginning and theend of the reheating process, defined as the period when the effective production rate of MSSM particles exceeds
the Hubble rate.
Thermal and nonthermal leptogenesis
The decays of the thermally and nonthermally produced neutrinos give rise to a thermal and a nonthermal B−L
asymmetry, as depicted in Fig. 7. The nonthermal lepton asymmetry receives a first contribution from the decayof the heavy (s)neutrinos of the second and third generation. To clearly distinguish this contribution from themain contribution arising due to the decay of the first-generation (s)neutrinos, we have assigned opposite signs to
the parameters 2,3 and 1 quantifying the CP asymmetry in decays of the respective neutrino generations. Thisentails the change of sign visible at a ' 4.6 × 103 in Fig. 7, when the decay of N1S neutrinos becomes efficient andthe main part of the nonthermal asymmetry is produced. Wash-out effects are negligibly small throughout thisprocess and hence, once the production of the nonthermal asymmetry becomes inefficient, the asymmetry freezesout.
The production of the thermal asymmetry is driven by the deviation of the thermal (s)neutrino abundance from17

A Minimal Supersymmetric Model ofParticle Physics and the Early Universe

the equilibrium value. This leads to an initially negative asymmetry with a rapidly increasing absolute value.
This increase slows down as the thermal (s)neutrino abundance approaches the equilibrium value. At arounda ' 6.3 × 104 wash-out processes start to play a role, leading to a decrease of the asymmetry. The situation
rapidly changes when the thermal (s)neutrino abundance overshoots the equilibrium abundance towards theend of the reheating process. This generates an asymmetry with an opposite sign, which overcompensates theasymmetry generated so far. Shortly after, both the wash-out rate and production rate drop significantly below
the Hubble rate and the asymmetry freezes out.
The final values of thermal and nonthermal asymmetry as depicted in Fig. 7 allow us to infer the present baryonntthasymmetry ηB as well as its composition in terms of a nonthermal (ηB) and a thermal (ηB) contribution,

ηB =

n0B
nt
th
= ηB
+ ηB
,
n0γ

nt,th
ηB
= Csph0g∗,sNLnt,thRHg∗,s Nγ

.

(23)

af

RH
0
Here, Csph = 8/23 denotes the sphaleron conversion factor, g∗,s
= 915/4 and g∗,s
= 43/11 are the effective

numbers of relativistic degrees of freedom in the MSSM that enter the entropy density s of the thermal bathin the high- and low-temperature regime, respectively, NLnt,th refers to the comoving number density for thenonthermal and thermal contribution to the lepton asymmetry and Nγ = gγ /g∗,n Nr is the comoving number
density of photons. For our parameter example we findηB ' 3.7 × 10−9 ,nt
ηB' 3.7 × 10−9 ,thηB' 1.9 × 10−14 .

(24)Note that to obtain these values, we have set the CP -viola