Published for SISSA by Springer
Received: January 30, 2019
Revised: March 8, 2019
Accepted: March 12, 2019
Published: March 18, 2019
JHEP03(2019)100
The supercooled universe
Pietro Baratella,a Alex Pomarola,b and Fabrizio Rompinevea
a
IFAE and BIST, Universitat Autònoma de Barcelona,
08193 Bellaterra, Barcelona
b
Departamento de Fı́sica, Universitat Autònoma de Barcelona,
08193 Bellaterra, Barcelona
E-mail: pbaratella@ifae.es, pomarol@ifae.es, frompineve@ifae.es
Abstract: Strongly-coupled theories at the TeV can naturally drive a long period of
supercooling in the early universe. Trapped into the deconfined phase, the universe could
inflate and cool down till the temperature reaches the QCD strong scale. We show how
at these low temperatures QCD effects are important and could trigger the exit from the
long supercooling era. We also study the implications on relic abundances. In particular,
the latent heat released at the end of supercooling could be the reason for the similarities
between dark matter and baryon energy densities. The axion abundance could also be
significantly affected, allowing for larger values of the axion decay constant. Finally, we
discuss how a long supercooling epoch could lead to an enhanced gravitational wave signal.
Keywords: Cosmology of Theories beyond the SM, Beyond Standard Model, Technicolor
and Composite Models
ArXiv ePrint: 1812.06996
Open Access, c The Authors.
https://doi.org/10.1007/JHEP03(2019)100
Article funded by SCOAP3 .
Contents
1 Introduction 1
2 A strongly-interacting new sector at the TeV 3
2.1 Deconfined phase 6
3 Confined-deconfined phase transition and supercooling 8
3.1 Impact of QCD at low T 11
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3.1.1 Fate of the new strong sector for µ below Λdec
QCD 13
4 Cosmological implications 13
4.1 Dark matter abundance 15
4.2 Coincidence between baryon and dark matter abundances 16
4.3 The QCD axion 17
4.3.1 Supercooling and the misalignment mechanism 18
4.3.2 Supercooling and axionic topological defects 19
4.4 Gravitational waves 21
5 Conclusions 25
A Holography and supercooling 26
A.1 The deconfined phase 28
A.2 Free energy of the deconfined phase 29
B Axion dark matter from string-wall network 30
1 Introduction
As the LHC continues exploring the TeV-scale territory, searching for hints on the origin of
the electroweak (EW) scale, it is also quite necessary to look for probes of new physics in
other environments, as for example in the early universe. This is of vital importance due
specially to the LHC energy limitation, as these probes could indirectly provide evidences
for new physics just beyond the LHC reach.
New strongly-interacting sectors at the TeV scale are particularly well-motivated ex-
amples for explaining the origin of the EW scale, and are at present exhaustively searched
at the LHC. These models also present dark matter (DM) candidates that are searched for
at present experiments. In spite of this, there is a generic prediction for these TeV strong
sectors that has not been fully explored and can have a strong impact in the physics of the
early universe: a long epoch of supercooling arising as a result of the transition from the de-
confined to the confined phase. The main purpose of this article is to address under which
–1–
conditions this long epoch of supercooling can happen, and explore the phenomenological
consequences in the early universe.
The basic dynamics which gives birth to such a supercooled universe is the following.
Strongly-interacting theories are expected to be in a deconfined phase at temperatures
above the TeV, in which its constituents behave as a strongly-interacting hot gas. When the
temperature goes down below the TeV, the confined phase becomes energetically favorable.
Nevertheless, when the rank of the strong gauge group N is large, the phase transition is
of first-order and can only be completed through tunneling. Since the tunneling rate is
2
quite suppressed Γ ∝ e−SB ∼ e−N , the transition cannot occur, and the universe enters a
JHEP03(2019)100
potentially eternal epoch of inflation trapped in the deconfined phase. As the temperature
decreases, however, the QCD critical temperature is attained. Since these TeV strong
sectors must be coupled to the SM quarks (to give them masses after the EW breaking),
strong QCD effects become important at low temperatures and can be responsible for
triggering the phase transition and ending the supercooling era. Therefore, a natural
consequence of TeV-scale strong sectors is a long epoch of supercooling, which can last for
several efolds.
The idea of a supercooled universe in which the phase transition is driven by QCD
condensation was first considered in [1], in the context of the Coleman-Weinberg model,
and has been also recently discussed in [2] in strongly-coupled models similar to the ones
discussed here.1 Our first main goal in this article is to address the general conditions
under which this long epoch occurs. We will define the essential ingredients of the strongly-
coupled models needed for supercooling and exiting at temperatures around or below the
QCD strong scale. As we will see, it will be crucial to distinguish between models in which
the EW symmetry is restored in the deconfined phase from those in which it is broken. This
latter possibility has not been considered before, and, as we will see, can lead to important
implications for axion physics. We will also present holographic examples of these type of
models that allow to calculate tunneling rates and determine the exit from supercooling.
Next, we will explore the consequences for cosmology. We will see that the release of en-
tropy at the end of the phase transition leads to a strong dilution of DM relics. This implies
a different range of couplings and masses for WIMP-like DM candidates and wimpzillas.
Also for DM candidates that get their masses after the phase transition (e.g., resonances of
the strong sector), we will see that their relic abundances are mainly determined by their
initial thermal abundance and the dilution from supercooling. Interestingly, we will show
that this can also be the case for the baryon asymmetry, and therefore the supercooled
universe could give an explanation for the observed similarity between the baryon and DM
energy density.
We will also study the implications for the QCD axion. We will see that during
the supercooling epoch the axion could oscillate and quickly relax to the minimum of its
potential. Therefore, its relic abundance could be reduced, meaning that larger values of
the axion decay constant Fa will be favored for axion DM. Whenever axionic topological
1
Supercooling with holographic duals was first considered in [3]. Since then, several studies of the
confinement phase transition have been performed with short periods of supercooling [4–13].
–2–
defects are relevant, their contribution to the axion relic abundance can also be enhanced in
the supercooled universe. This is essentially due to the fact that the inflationary expansion
delays the collapse of the axionic string-wall network compared to the standard case.
Perhaps the most model-independent implication of a confinement phase transition
is the gravitational wave (GW) signal. This has received a lot of attention in recent
literature, but not for the particular case of long supercooling in which the thermal plasma
which surrounds the colliding bubbles of the confined phase is very diluted. We will see
that the GW signal after a long supercooling epoch can be sourced mostly by the collisions
of bubbles of the confined phase, which occurs on a long time scale and thus maximizes
the amplitude of the signal.
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The organization of the paper is as follows. In section 2 we introduce the properties
of the strongly-coupled sectors in the confined phase and deconfined phase (section 2.1).
In section 3 we analyze the deconfinement-confinement phase transition, calculating the
tunneling rates and the temperature for exiting. In section 4 we study the cosmological
consequences of a long period of supercooling for dark matter (section 4.1) and for baryons
(section 4.2). Axion dark matter from misalignment is discussed in section 4.3.1, while the
impact on axionic topological defects is given in section 4.3.2. Gravitational wave signals
are discussed in section 4.4. Section 5 is devoted to conclusions. We add two appendices.
In appendix A we present holographic models with the same properties of the strongly-
coupled sectors that lead to supercooling. In appendix B we review some basics of axion
DM from the string-wall network.
2 A strongly-interacting new sector at the TeV
We are interested in studying the supercooled phase of a strongly-interacting sector, or
its equivalent five-dimensional version based on the AdS/CFT correspondence. We will
identify here the generic properties (at zero T ) of this class of theories relevant for our
analysis:
• We assume that these are strongly-interacting conformal field theories (CFT) with
a large central charge, or, equivalently, a large number of “colors” N . This includes
any gauge theory, e.g. SU(N ), in the large N limit.
• These CFTs will contain a small marginally relevant deformation that will be respon-
sible for generating at the IR the confinement scale Λc ∼ TeV, much smaller than any
UV-scale, ΛUV , as for example, the Planck-scale. This deformation can arise from a
marginal operator added to the CFT, e.g.,
∆L = g Og , (2.1)
where Dim[Og ] = 4 − with 1. The marginal coupling g RG-evolves towards
the IR until a critical value gc is reached at which conformal invariance is lost and a
mass-gap Λc is generated. In this class of theories, like in QCD, we expect a tower
of resonances of different spin, weakly-coupled in the large N limit, and with masses
of order Λc . This corresponds to the confined phase.
–3–
• As these models have usually accidental global symmetries, G, an important param-
eter is the scale of spontaneous symmetry breaking (SSB) G → H. We will refer
to this scale as ΛSSB . This is associated to the nonzero condensate of an operator
of the model OH transforming under G. For example, in QCD this is supposed to
be the quark condensate hq̄qi. We will assume, like in QCD or other gauge theories
studied in the lattice, that ΛSSB ∼ Λc (all condensates are of the same order). This
is needed for phenomenological reasons, as the scale ΛSSB is associated with the elec-
troweak scale that cannot be larger than the masses of the new resonances, bounded
by the LHC to be above the TeV. For example, in composite Higgs models [14] where
the Higgs is a PGB arising from the breaking G → H, the Higgs VEV is given by
JHEP03(2019)100
√
v ' cfSSB where c is an O(1) coefficient and fSSB ' ( N /4π)ΛSSB . In fact, in many
models a small tuning is required (c . 0.2) to fulfill the experimental constraints [15].
• In order to study the phase transitions of the above models, it is crucial to understand
these theories off-shell. This is a very difficult task, even for the holographic versions
discussed in appendix A. Nevertheless, this problem simplifies substantially whenever
the lightest state is the dilaton, µ(x), the Goldstone parametrizing the spontaneous
breaking of the scale invariance. In this case, we can study the effective potential
of the dilaton after integrating out all other heavier states. We can understand
this potential based on symmetries. Since under scale transformations, we have
xµ → ew xµ and µ → e−w µ, the effective potential can be written as
N2
Veff (µ) = λ(µ)µ4 , (2.2)
16π 2
where the dependence of the quartic coupling λ(µ) on µ is dictated by the explicit
breaking of scale invariance. We identify Λc with the minimum of the dilaton po-
tential eq. (2.2), hµi = Λc . Notice that contrary to the Higgs H and other fields,
for the dilaton we will work in the basis in which its kinetic term is non-canonically
normalized:
N2
Lkin (µ) = c1 (∂µ µ(x))2 , (2.3)
16π 2
where c1 is a constant of O(1) and we are assuming that the dilaton is a glueball. This
normalization guarantees hµi ∼ Λc ∼ O(N 0 ) and resonances get masses proportional
to µ with no N factors:
mi ' ri hµi , (2.4)
where ri ∼ O(1). In holographic models the dilaton is the radion and one obtains
c1 = 12 and ri ∼ π (see appendix A). The mass of the dilaton is given by
β0
2 12
m2dil = − λ(hµi) 1 + λ hµi2 , (2.5)
3 4βλ min c1
–4–
where βλ = dλ/d ln µ and βλ0 = dβλ /d ln µ. Notice that the dilaton effective potential
makes sense for mdil . mi ; using eq. (2.4), this gives the rough bound |λ| . ri2 ' π 2 ,
where we have used holography in the last equality.2 We will assume here that
eq. (2.2) describes the vacuum of the theory with the dilaton being the lightest state.
• An important property of this class of models that will be very relevant for the
impact in cosmology is how the condensate OH , that breaks G → H, behaves for
small (off-shell) values of the dilaton µ. We will consider two possibilities:
I. All condensates are proportional to µ. This implies in particular hHi ∝ µ.
JHEP03(2019)100
II. The condensate of OH is determined to be around ΛSSB , independently of µ.
We have then hHi ∝ ΛSSB even when µ ΛSSB .
It is important to understand the behavior of Veff (µ) for the two types of models. In
models of type I we expect λ to evolve logarithmically due to eq. (2.1):
µ
λ(µ) ∼ f (g(µ)) ∼ λ0 + βλ ln + ··· , (2.6)
ΛUV
where βλ ∼ . At around λ(µ) crossing zero, Veff (µ) develops a minimum à la
Coleman-Weinberg. This is illustrated in figure 1. For constant we expect
hµi ∼ e−1/βλ ΛUV , (2.7)
which for ΛUV ∼ MP requires βλ ∼ 1/35 in order to generate Λc ∼ TeV. Holographic
weakly-coupled models with this property have been constructed, and are usually
referred to as Goldberger-Wise models [16]. In these particular models f (g(µ)) can
be computed and is a quadratic function of (µ/ΛUV ) (see e.g. [3]).
On the other hand, models of type II correspond, for example, to models at which the
dimension of the operator OH is not constant but, due to eq. (2.1), RG-evolves till it
becomes imaginary at some scale, signaling the breaking of the conformal symmetry
with hOH i ∼ ΛSSB . They are referred to as walking models (see for example [17]) and
the holographic versions correspond to 5D models with an AdS tachyon as described
in appendix A. For these type of models one finds that for µ ∼ ΛSSB the quartic
goes as [18]
µ
λ(µ) ' λ0 + βλ ln2 , (2.8)
ΛSSB
with βλ ∼ 1, and Veff (µ) can have a minimum at some scale Λc ∼ ΛSSB . For µ < ΛSSB
the quartic becomes almost constant (or at most evolving like eq. (2.6)) as the theory
can be described by another CFT (since part of the original CFT has become massive
at ΛSSB ). This is shown in figure 1.
2
It is instructive to derive the value of λ at the minimum by equating eq. (2.2) to the QCD vacuum
energy with mi ∼ πhµi ∼ GeV and N = 3. One obtains λmin ' 0.9 π 2 as expected since QCD has no light
dilaton.
–5–
μ
>
type I
λ(μ)
type II
μ Λc
JHEP03(2019)100
> |
type I
type II
V(μ)
Figure 1. Behavior of the quartic coupling (upper plot) and effective potential for the dilaton
(lower plot) as a function of µ for models of type I and type II.
• Another relevant ingredient of this class of models is how they couple to the SM
quarks. This coupling must be present, since we need the Higgs (coming from the
new strong sector) to give mass to the SM quarks. The most realistic possibility is
to have a linear coupling of the SM quarks to operators of the strong TeV sector:
∆L = λL q̄L OL + λL q̄R OR + h.c. , (2.9)
that is the portal of the SM quarks to the strong TeV sector, and therefore to the
Higgs. Eq. (2.9) tells us that the composite operators OL,R must transform as a fun-
damental representation under QCD, that implies that there must be states charged
under both the strong TeV sector and QCD. This requirement will have important
implications when, in the supercooling phase, T drops below the QCD confinement
scale ΛQCD .
We can find weakly-coupled models with the properties described above by using the Ad-
S/CFT correspondence (for models of type I, see [14], while for models of type II, see [18]).
The importance of these toy models is that they allow to perform calculations and study
the supercooling phenomena in a more quantitative way (for previous work, see [3–13]).
2.1 Deconfined phase
At high-temperatures a strongly-interacting theory is expected to have a different more
energetically favorable phase in which the constituents are not confined into hadrons. In
QCD this is the quark-gluon plasma phase. For the models described above we will assume
that indeed this phase exists.
–6–
In this deconfined phase models of type I and II will have different properties. Models
of type I are similar to QCD, in which all the condensates are zero and the theory has no
mass gap in the deconfined phase. In particular, we expect that the Higgs does not get
a VEV in this phase. On the other hand, in models of type II we have that hHi ∝ ΛSSB
independently of the scale of condensation µ, and therefore the electroweak symmetry is
also broken in the deconfined phase, hHidec ∼ fSSB , for T < ΛSSB . We notice, as this will be
important later when discussing axion physics, that hHidec is expected to be slightly larger
than the Higgs VEV in the confined phase, hHi ∼ 246 GeV, since to pass experimental
constraints one needs fSSB ∼ TeV [15].
We will be interested later in studying the effects of QCD condensation in the CFT
JHEP03(2019)100
deconfined phase. For this reason, it is important to know what is the QCD scale in the
deconfined phase Λdec
QCD . This scale is generically different from its value in the SM vacuum
ΛQCD ∼ 300 MeV as the CFT, which contains charged states under QCD, can contribute
to the running of gs below the TeV scale. The equation that governs the running of gs at
the one-loop level takes the well known form
g3
dgs 11 2
= β gs = s 2 − Nc + Nf + αN , (2.10)
d ln Q 16π 3 3
where Q is the renormalization scale, while Nf and Nc are respectively the number of
active flavors and colors. The contribution of the CFT is taken into account in the last
term. Even though α is not computable, holography tells unambiguously that it is positive,
since αN corresponds to the inverse of the 5D gauge coupling squared in units of the AdS
length [14]. By use of eq. (2.10) we find that
n
ΛQCD 1−n
Λdec
QCD = ΛQCD , (2.11)
Λc
where we normalized Λdec
QCD to the SM value. The parameter n is given by
βgs (Q < Λc ) − βgs (Q > Λc )
n= . (2.12)
βgs (Q < Λc )
Eq. (2.11) is valid for n < 1. For n ≥ 1 the positive contributions to eq. (2.10) win over the
contributon of the gluons, and QCD is not confining, i.e., Λdec
QCD = 0. Since the CFT always
gives a positive contribution to the beta function, we have βgs (Q > Λc ) > βgs (Q < Λc ),3
and as a consequence Λdec QCD < ΛQCD . Nevertheless, we can allow for the possibility that
the QCD SU(3)c group arises as the low-energy remnant of a larger gauge group (SU(Nc )
for simplicity), spontaneously broken by the strong dynamics of the CFT at the scale Λ c .
In this case n can be negative and then Λdec 4
QCD > ΛQCD . For example, for αN ∼ 1 and
Nc = 5, we get n = −0.5, giving ΛdecQCD ' 5 GeV(Λc /TeV)
1/2 .
3
βgs (Q < Λc ) is negative because it is dominated by the contribution of the gluons.
4
The authors of [19] consider also the possibility to have a larger QCD scale in the early universe by
giving VEV to a scalar coupled to G2µν . This could also be possible here.
–7–
3 Confined-deconfined phase transition and supercooling
If our universe was ever in the deconfined phase described in the previous section, it
must have transitioned to the confined phase at some time during cosmic evolution. The
temperature Tc at which this becomes thermodynamically possible can be estimated as
follows: in the unconfined phase the only relevant scale is temperature, and we then expect
the free energy density to scale as in a hot free gas with O(N 2 ) d.o.f., that is Fdec ∼ −N 2 T 4 .
On the other hand, in the confined phase the CFT acquires a mass gap, Λc , and at T Λc
the heavy resonances are not excited (due to Boltzmann suppression). Therefore the free
energy is dominated by the temperature-independent vacuum energy, Fconf ∼ −N 2 Λ4c . The
JHEP03(2019)100
phase transition can take place as soon as T . Tc ∼ Λc .
In large-N gauge theories it is found that the confinement phase transition is of first
order ; this is also seen in holographic models in which the two phases correspond to two
different solutions of the same Lagrangian (see appendix A), and therefore one can only
go from one to the other by bubble nucleation. It is thus crucial to compute Γ, the rate
at which bubbles are produced: in particular, the transition can be completed only when
Γ & H 4 (see e.g. [20]) where H is the Hubble constant that in the confined phase for T < Tc
is dominated by the vacuum energy:
s
|Veff (hµi)| N mdil Λc
H' 2 ' √ , (3.1)
3MP 4 2π MP
where we have used eq. (2.5) with c1 = 12 and βλ0 . βλ as we will take from now on. In a
semi-classical approximation Γ is given by
Γ = A e−SB , (3.2)
where SB is the action of the critical bubble, or bounce, and A is typically of order R−4 ,
being R the critical bubble radius. When the bounce is dominated by the dynamics of
the large-N sector, we have SB ∝ N 2 and Γ is exponentially suppressed. If the transition
cannot take place the universe inflates in the confined phase and exponentially cools down.
Before going into the details of the tunneling process, let us first compute more precisely
the critical temperature Tc below which confinement becomes favorable; for this we need
to know the free energy in the two phases. As explained above, in the deconfined phase we
can use conformal invariance and N -counting to fix Fdec in the following way:
Fdec = −c2 N 2 T 4 , (3.3)
where c2 is a constant of order 1; using holography we find c2 = π 2 /8, an O(10) larger
than the free particle result (see appendix A for a derivation). In the confined phase, for
temperatures below Λc , we have
Fcon = Veff (hµi) + O(T 4 ) , (3.4)
where thermal corrections mainly come from light degrees of freedom (always present in
realistic models). The critical temperature is given by equating F in the two phases,
–8–
which gives
1/4
Λc mdil 1/2 π 2 /8
Tc ' 300 GeV , (3.5)
TeV2 c2
where we normalize c2 with the value coming from holography.
When the temperature of the universe goes below Tc , bubbles of the confined phase
begin to form. It is in general not an easy task to compute the rate Γ since the bounce can
depend on many fields. To simplify the analysis, one usually looks for a preferred tunneling
direction in field space, in order to reduce it to a single-field problem. In the confined phase
we already argued that the dilaton is the only relevant field if we consider it lighter than
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the other resonances. Nevertheless, it is not at all obvious what are the relevant field and
potential in the deconfined phase, and how to properly connect the two phases. These
points were first discussed by the authors of ref. [3] in the context of the holographic 5D
model. Here, we briefly rephrase their strategy in purely 4D terms. The key observation is
that both phases become indistinguishable in the limit T = µ = 0 at which the conformal
symmetry is recovered in both phases. In figure 2 we plot the potential along the path
that connects in this way the deconfined phase (on the left side) to the confined phase (on
the right side). In the confined phase the field variable is µ, with potential Veff given in
eq. (2.2), while in the deconfined phase the proper variable is the local temperature of the
plasma, Tloc . The free energy density of the CFT plasma as a function of Tloc (see left
of figure 2) can be easily fixed, up to a constant factor, with elementary arguments from
thermodynamics, and has the following expression:
F (Tloc ) = ρ(Tloc ) − T s(Tloc ) = c2 N 2 3Tloc
4 3
− T 4Tloc , (3.6)
where the constant is fixed in such a way that eq. (3.6) matches with eq. (3.3) for Tloc =
T . This result is also derived in appendix A using holography. To completely specify
the system, we finally need to normalize the kinetic term for Tloc (x). Even though it is
not clear how to do it, not even from the holographic perspective, we take this to be
c3 (N 2 /16π 2 )(∂µ Tloc )2 . We will see however that the parameter c3 does not impact sensibly
the calculation of Γ.
At finite temperature bubble nucleation can be induced either thermally or quantum
mechanically. Which process is more likely can be determined by comparing T and R−1 :
when T R 1 (high temperature regime) thermal bubbles are preferred, while for T R 1
bubble nucleation is mainly driven by quantum fluctuations. For T . Tc the two phases
have almost degenerate F and the thin wall regime applies. Moreover, it can be seen that
quantum bubbles are highly disfavoured so that tunneling is driven by thermal fluctuations.
Nevertheless, as was shown in ref. [3], the tunneling rate is too slow for the transition
to be completed in this regime. As the plasma cools to T Tc the properties of the
critical bubble become very different. Firstly, the energy difference between the two phases
increases and thick wall bubbles become favoured. Secondly, especially in the regime of long
supercooling which we are interested in, thermal bubbles production is either disfavoured or
comparable to quantum induced one, as we checked also by numerically solving the bounce
equations for the two cases. Thus in the following we will focus on quantum tunneling in
the thick wall regime.
–9–
µt ⇤c
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AAAB8nicbVDLSsNAFL2pr1pfVZdugkVwVRIRdFkUxGUF+4A2lMl00g6dzISZG6GEfoYbF4q49Wvc+TdO2iy09cDA4Zx7mXNPmAhu0PO+ndLa+sbmVnm7srO7t39QPTxqG5VqylpUCaW7ITFMcMlayFGwbqIZiUPBOuHkNvc7T0wbruQjThMWxGQkecQpQSv1+jHBMSUiu5sNqjWv7s3hrhK/IDUo0BxUv/pDRdOYSaSCGNPzvQSDjGjkVLBZpZ8alhA6ISPWs1SSmJkgm0eeuWdWGbqR0vZJdOfq742MxMZM49BO5hHNspeL/3m9FKPrIOMySZFJuvgoSoWLys3vd4dcM4piagmhmtusLh0TTSjaliq2BH/55FXSvqj7Xt1/uKw1boo6ynACp3AOPlxBA+6hCS2goOAZXuHNQefFeXc+FqMlp9g5hj9wPn8Ad+eRXA==
AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzQOSJcxOZpMhM7PLTK8QQj7BiwdFvPpF3vwbJ8keNLGgoajqprsrSqWw6PvfXmFtfWNzq7hd2tnd2z8oHx41bZIZxhsskYlpR9RyKTRvoEDJ26nhVEWSt6LR7cxvPXFjRaIfcZzyUNGBFrFgFJ300FVZr1zxq/4cZJUEOalAjnqv/NXtJyxTXCOT1NpO4KcYTqhBwSSflrqZ5SllIzrgHUc1VdyGk/mpU3LmlD6JE+NKI5mrvycmVFk7VpHrVBSHdtmbif95nQzj63AidJoh12yxKM4kwYTM/iZ9YThDOXaEMiPcrYQNqaEMXTolF0Kw/PIqaV5UA78a3F9Wajd5HEU4gVM4hwCuoAZ3UIcGMBjAM7zCmye9F+/d+1i0Frx85hj+wPv8AV1ejdY=
AAAB7nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48R8oJkCbOTTjJkdmaZmRXCko/w4kERr36PN//GSbIHTSxoKKq66e6KEsGN9f1vr7CxubW9U9wt7e0fHB6Vj09aRqWaYZMpoXQnogYFl9i03ArsJBppHAlsR5P7ud9+Qm24kg07TTCM6UjyIWfUOqnd6GdCsVm/XPGr/gJknQQ5qUCOer/81RsolsYoLRPUmG7gJzbMqLacCZyVeqnBhLIJHWHXUUljNGG2OHdGLpwyIEOlXUlLFurviYzGxkzjyHXG1I7NqjcX//O6qR3ehhmXSWpRsuWiYSqIVWT+OxlwjcyKqSOUae5uJWxMNWXWJVRyIQSrL6+T1lU18KvB43WldpfHUYQzOIdLCOAGavAAdWgCgwk8wyu8eYn34r17H8vWgpfPnMIfeJ8/f4KPqQ==
AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRi8cW+gVtKJvtpF272YTdjVBCf4EXD4p49Sd589+4bXPQ1gcDj/dmmJkXJIJr47rfTmFjc2t7p7hb2ts/ODwqH5+0dZwqhi0Wi1h1A6pRcIktw43AbqKQRoHATjC5n/udJ1Sax7Jppgn6ER1JHnJGjZUazUG54lbdBcg68XJSgRz1QfmrP4xZGqE0TFCte56bGD+jynAmcFbqpxoTyiZ0hD1LJY1Q+9ni0Bm5sMqQhLGyJQ1ZqL8nMhppPY0C2xlRM9ar3lz8z+ulJrz1My6T1KBky0VhKoiJyfxrMuQKmRFTSyhT3N5K2JgqyozNpmRD8FZfXiftq6rnVr3GdaV2l8dRhDM4h0vw4AZq8AB1aAEDhGd4hTfn0Xlx3p2PZWvByWdO4Q+czx+vqYzY
H 4 gives
c 2 24N 2 T MP
1
T
. 4 ln . (3.10)
12 βλ ln T c Tc2
The temperature at which the above condition is satisfied gives the exit temperature
Te , which can be conveniently traded for the number of efolds of inflation as follows
s !
Tc 1 24N 2 (c1 /12)2 MP
Ne = ln ≈ 1− 1− ln . 17 . (3.11)
JHEP03(2019)100
Te 2 2 MP Tc
βλ ln Tc
When the argument of the square root in eq. (3.11) is negative, there is no solution
to eq. (3.10), meaning that the universe gets trapped in the deconfined phase. This
occurs whenever
βλ 1/2
12
N & 1.2 , (3.12)
c1 1/35
where we normalized βλ according to the discussion after eq. (2.7). We see that
exiting supercooling in these type of models is in tension with having a small βλ ,
which is required to generate the hierarchy between Λc and MP . We could imagine
more favorable situations with βλ not being constant, for example being small at
the UV but rapidly growing towards µ . Λc (the range of µ which is important for
discussing supercooling). This would allow to exit supercooling for larger values of
N , as appreciated in figure 3 where we show Te for different values of N and βλ .
Nevertheless, having small Te (a long period of supercooling) seems to require precise
values of N and βλ , which makes this quite unlikely.
• Type II models. These models have a larger βλ ∼ 1 (for µ ∼ TeV), which leads
to faster tunneling rates. Using eq. (2.8) with λ0 . βλ , we find that exit from
1/2
supercooling roughly happens for N . 1 − 2 (12/c1 ) βλ . When the parameters allow
this, inflation ends for Te not far from Tc . It is difficult to achieve small values of Te
since λ(µ) becomes almost constant for small µ, as it is shown in figure 1. In other
words, we find once more that long periods of supercooling are not expected.
The above conclusions can change drastically when T drops close to Λdec
QCD . As we will
see in the next section, at that scale QCD confines and can drive the supercooling era to
an end.
3.1 Impact of QCD at low T
If the universe is trapped in the CFT deconfined phase, the temperature will drop exponen-
tially till reaching the QCD strong scale, where QCD effects could potentially become im-
portant. The relevance for supercooling of QCD condensates was first discussed in ref. [1] in
the context of the Coleman-Weinberg model (for recent studies see [21–24]). Nevertheless,
there is an important difference with the models discussed here. The Coleman-Weinberg
– 11 –
JHEP03(2019)100
Figure 3. Lines of fixed exit temperature (Te = 1, 0.1, 0.01, 0.001 GeV from below to above) for
type I models, and no exit region eq. (3.12).
is a weakly-coupled model where the scalar (which can be considered a dilaton with a po-
tential as eq. (2.2)) is an elementary particle. For this reason a huge fine-tuning is required
to put to zero the mass term that, on general grounds, could be present. The models
discussed here however are strongly-coupled theories in which the dilaton is composite and
therefore the potential eq. (2.2) arises naturally without fine-tuning.
The first analysis of the impact of QCD in models of composite dilatons was done in
ref. [2]. The authors study the contributions of the QCD vacuum energy to the dilaton
potential. This is estimated to be
Nc2 4
∆Veff (µ) = VQCD (µ) ' −c4 Λ (µ) , (3.13)
16π 2 QCD
where c4 ≈ 7 can be extracted from QCD data (taking Nc = 3 and Λc = 330 MeV), and
ΛQCD (µ) is the QCD strong scale as a function of µ. This dependence on the value of µ
arises, as explained in section 2.1, from the CFT states charged under QCD that affect
the running of gs from high-energies down to Q ∼ µ. Indeed, using eq. (2.10), it can be
shown that
ΛQCD (µ) ∼ µn for µ > Λdec
QCD ,
ΛQCD (µ) ∼ Λdec
QCD for µ < Λdec
QCD , (3.14)
where n is given in eq. (2.12). For n < 1, eq. (3.13) can become larger than λ(µ)µ4 at
small µ. For example, at µ ∼ Λdec
QCD , where ∆Veff /Veff is maximized, this happens for
c4 Nc2 > λ(Λdec 2
QCD )N . (3.15)
– 12 –
If this is the case, eq. (3.13) dominates the dilaton potential and the tunneling rate can be
estimated as6
24N 4 (c1 /12)2
SB (T ∼ Λdec
QCD ) ≈ 1 , (3.16)
c4 Nc2 n(1 − n) n −1
and the condition for exiting supercooling at T ∼ Λdec dec
QCD being SB . 4 log MP ΛQCD /Λc .
2
Since eq. (3.16) scales as N 4 /Nc2 , exit from supercooling is only possible for small values
of N . In particular, we find N . 2 − 3 for Nc = 3 and n ∼ 0.1 − 0.5. For a more detailed
exploration see however ref. [2].
3.1.1 Fate of the new strong sector for µ below Λdec
JHEP03(2019)100
QCD
As T goes below Λdec QCD , we expect the impact of QCD in the tunneling rate to be more
dramatic, as the CFT can be drastically affected. Since some of the CFT constituents are
charged under color, we expect them to take mass at Λdec QCD and decouple at around this
scale, analogously to hadrons in the SM below the GeV. We cannot really compute what
is the fate of the CFT at smaller scales, but it is not unconceivable that, after this drastic
departure from near-conformality, the theory flows into another (quasi) CFT. An example
of this behavior is provided by QCD in the conformal window, where the theory can remain
conformal even after integrating out some flavors. If this is the case, we can parametrize
this effect as a drastic change of the quartic coupling λ → λ + ∆λ ≡ λ̃ at Λdec
QCD , as well as
a change in its beta-function, βλ 1 → β̃λ ∼ 1. Taking λ̃ and β̃λ as free parameters, the
tunneling rate will be given by
24N 2 (c1 /12)2
SB ≈ , (3.17)
Λdec
QCD
|λ̃| + β̃λ ln T
valid under the conditions that ∆λ < 0 and β̃λ > 0 (which are also the conditions that
favor tunneling for T < Λdec
QCD ). Notice that, contrary to eq. (3.16), the bounce action is not
2 2
enhanced by N /Nc , which makes the supercooling exit much more probable. In figure 4
we plot the contours of fixed ∆Ne , the efolds of inflation after the QCD phase transition
is crossed, in the parameter space of |λ̃| and β̃λ , where we have taken Λdec −2 T .
QCD = 10 c
We conclude that under reasonable assumption on the strongly-coupled CFT the uni-
verse could be trapped in the unconfined phase till it reaches T ∼ Λdec QCD . At this tempera-
ture, or slightly below (see figure 4), the tunneling rate to the CFT confined phase can be
large enough due to QCD, leading to the end of this long supercooling epoch.
4 Cosmological implications
How would we know if the universe indeed experienced such a long supercooling epoch?
The purpose of this section is to describe observational consequences of the associated long
period of vacuum domination, which would provide evidence for this phenomenon (see
6
This is is obtained by solving eq. (3.8) with eq. (3.13), taking into account that in the deconfined phase
there is also the contribution ∆F ' −c4 (Nc2 /16π 2 )(Λdec 4 dec
QCD ) . The tunneling point µt is of order ΛQCD and
dec
the bubble size ∼ 1/ΛQCD .
– 13 –
5
.5
4 1
˜
βλ 3
ΔNe =2
2 3
4
JHEP03(2019)100
5
1 6
1 2 3 4 5 6
˜
|λ |
Figure 4. Contours of ∆Ne , the number of efolds after the QCD phase transition, in the scenario
described in section 3.1.1. We take N = 7 and Ne = ln(Tc /Λdec
QCD ) ' 4.6.
also [4–13, 25–31] for related analyses but where mainly only short eras of supercooling
were considered).
Let us start with the universe at high-temperatures, reheated after a first epoch of
inflation, responsible for the generation of the CMB anisotropies. As the temperature
cools down and reaches Tc ∼ O(100) GeV, the universe gets trapped in the CFT deconfined
phase, and a second inflationary epoch begins with Hubble rate given by eq. (3.1). Following
the discussion in section 3.1, we will mainly consider that the exit from supercooling occurs
at or slightly after QCD confinement, i.e. after ∼ 7 − ln(Λdec 7
QCD /330 MeV) efolds. At the
end of the supercooling era, there is an injection of entropy due to the first-order phase
transition. While reheating (RH) in this case can be in general a complicated process,
the natural expectation is that most of the energy available in the deconfined phase is
transferred to radiation in the confined phase. This leads to
1/4 √
π2
4 45 N
Fdec (Tc ) ' g∗ TRH ⇒ TRH ' 1/4
Tc . (4.1)
90 4 g∗
where we have used eq. (3.3) with c2 = π 2 /8. Here g∗ & 100 is the number of relativistic
species after reheating. Thus the entropy injection after supercooling is given by
1/4 1/4 3
sRH 4 g Tc
' √∗ , (4.2)
se 45 N Te
where sRH and se are respectively the entropy density at reheating and Te ; the latter
can be read from eq. (3.6) evaluated at Te . This release of entropy strongly dilutes the
7
The duration of supercooling is in principle not strongly constrained by CMB anisotropies, because
density and/or isocurvature fluctuations are not abundantly produced [32].
– 14 –
abundances of constituents which exist in the deconfined phase. This is in particular the
case for particles heavier than the TeV scale as well as for baryons, as we will discuss in
section 4.1 and section 4.2 respectively. The same conclusion applies to quanta of a QCD
axion field, if the latter oscillates during supercooling (see section 4.3.1). However, the
relevance of axionic topological defects is generically enhanced in the supercooled universe,
as we will discuss in section 4.3.2.
At the end of the first order phase transition, reheating occurs via violent processes,
which source gravitational radiation. While this implication has already been thoroughly
discussed in the literature (see refs. above), our long epoch of inflation can lead to a
maximal signal, dominantly due to the collision of bubbles of the confined phase, as we
JHEP03(2019)100
will discuss in section 4.4.
4.1 Dark matter abundance
Let us start with the implications for dark matter (DM) particles. In the strongly-
interacting scenarios considered above, we can consider two classes of dark matter can-
didates (apart from the QCD axion that will be discussed in the next section): those which
are heavy in both CFT phases, or, alternatively, those which are massive in the confined
phase but massless in the deconfined phase, i.e., m ∝ µ. In both cases, supercooling affects
the relic abundance only if the dark matter is not repopulated after reheating. If a generic
DM particle χ has mass mχ and annihilation cross section σann in the confined phase, its
thermal production after supercooling is subdominant whenever TRH . Tfo , where Tfo is
the freeze-out temperature of χ. For σann mχ ≥ 10−16 GeV−1 , one finds Tfo ∼ O(0.1) mχ .
Therefore, according to eq. (4.1), thermal production after supercooling can be neglected
when mχ & TeV. In what follows we will focus on this case.8
Let us consider first the dilution of DM candidates that are massive in both phases,
dec
mχ ' mχ , and whose thermal freeze-out temperature is above Tc . Their relic abundance
today can be computed starting from the number density-to-entropy ratio Yχ ≡ nχ /s.
After freeze-out, Yχ is constant, except at the phase transition where the entropy changes
according to eq. (4.2). Therefore, the relic abundance of χ is given by
√ 3 dec /T
!
m 10−8 GeV−2
2 N T e χ fo
Ωχ h ' 0.1 1/4
g∗ Tc 20 σann
√ 3 !
mdecχ /Tfo 10−18 GeV−2
N Te
' 0.1 1/4 . (4.3)
g∗ 157 MeV 20 σann
In the second line of eq. (4.3) we have assumed Tc ' 300 GeV and normalized Te to the SM
QCD critical temperature. Eq. (4.3) is simply the standard relic abundance of a WIMP-like
8
Non-thermal production after supercooling can also be relevant in principle. Indeed, in a first-order
phase transition with runaway bubbles, the possibility exists to abundantly produce heavy dark matter
particles during the bubble collisions [9, 33, 34]. This occurs only when collisions are approximately elastic.
However, in our case the two minima of the dilaton potential are highly non-degenerate, which implies that
bubble collisions are likely quite inelastic. This remains true even if the field tunnels to a value of the
potential which is close to the false minimum, since it rolls down the true minimum before the bubbles have
time to meet. Therefore non-thermal production of heavy dark matter from bubble collisions is suppressed
in our scenarios.
– 15 –
particle with a very small prefactor due to supercooling. If we assume σann ∼ gχ4 /(mdec
χ )
2
and mdec −3 for T ∼ 100 MeV.9 If we assume instead
χ ∼ TeV, eq. (4.3) implies gχ . 10 e
gχ ∼ O(1), eq. (4.3) matches the observed DM abundance for mdec 9
χ ' 10 GeV. Thus in
the supercooled universe wimpzillas do not need to be very weakly coupled in order not to
overproduce dark matter.
Let us now move to species which are massless in the deconfined phase mdec χ = 0 and
massive in the confined phase, mχ ∼ µ. We only demand these states to be stable. Their
relic abundance can be computed again from Yχ , which in the deconfined phase is given by
Yχ ∼ gχ T 3 /s, with the entropy s given in eq. (3.6). The number density of χ today can be
thus simply obtained by dividing Yχ by eq. (4.2) and multiplying by the entropy density
JHEP03(2019)100
today. The relic abundance of χ is thus given by
1/4 3
2 gχ κDM 100 mχ Te
Ωχ h ' 0.5 , (4.4)
N 3/2 g∗ TeV 157 MeV
where κDM . 1 is an efficiency prefactor which takes into account the possibility that
χ is further diluted as its mass switches on during the phase transition. If the DM is a
composite state in the confined phase, eq. (4.4) should include a further O(1) prefactor
to take into account the decrease of degrees of freedom in DM from the deconfined to the
confined phase. From eq. (4.4) we can conclude that a TeV-scale particle can reproduce the
observed dark matter abundance if supercooling lasts approximately until the temperature
of the SM QCD phase transition (see [37] for similar conclusions).
4.2 Coincidence between baryon and dark matter abundances
Similarly as dark matter, any baryon asymmetry generated at high temperature will be
diluted by the inflationary expansion during supercooling. This opens the possibility to
explain the smallness of (nB − nB̄ )/s by the dilution factor eq. (4.2). Furthermore, if both
dark matter and baryons arise from relativistic particles that were in thermal equilibrium
at some T > Tc , their abundances will be comparable before supercooling and equally
diluted at the end of the phase transition. This could then provide an explanation for
the coincidence of DM and baryon abundances, with their energy density ratio mainly
determined by the ratio of their masses.
As an illustrative example of the above idea, let us consider the case in which a (B-L)
asymmetry is generated at temperatures above Tc , from the out-of-equilibrium decay of a
very heavy state Ψ (generic in UV completions which include gravity). For T > MΨ , Ψ is in
thermal equilibrium, thus its number density is comparable to the rest of the thermal bath,
i.e. nΨ ∝ gΨ T 3 . Once T . MΨ , the inverse decays are Boltzmann-suppressed and a baryon
asymmetry can be generated if the out-of-equilibrium condition ΓΨ→B-L . H(T ∼ MΨ ) is
fulfilled. If this is the case, we expect nB-L ' nB ∼ CP nΨ , where CP is the CP asymmetry
in the decay of Ψ to (B-L)-charged particles. After supercooling, the baryon number is
9
Heavy particles which interact with nuclei can be looked for at DM direct detection experiments.
Interestingly, if σann ' σχ−nucleon , the region of parameter space suggested by eq. (4.3) is currently probed
by XENON1T [35] and will be further investigated by LZ [36].
– 16 –
diluted in the same way as in the case of DM. Following the same steps which lead to
eq. (4.4), we can estimate the baryon-to-entropy ratio to be
1/4 3
nB CP 100 Te
∼ 0.01 3/2 . (4.5)
s N g∗ Tc
If CP ∼ O(1) and N & 5, the observed baryon asymmetry can be obtained for Te ∼ 1 GeV.
As the origin of eq. (4.5) is similar to DM in eq. (4.4), we can easily estimate the ratio of
baryon to DM energy density to be
ΩB CP mp
' , (4.6)
JHEP03(2019)100
Ωχ κDM mχ
where mp is the proton mass and we have also included a possible dilution factor for χ,
as in eq. (4.4). It is quite interesting that the ratio of baryon and DM relic abundances is
predicted to be very simply related to the ratio of their masses and of the efficiencies of
CP violation and DM number changing processes at the end of the phase transition. In
order to obtain the observed value Ωχ ' 5ΩB , eq. (4.6) leaves two options:
1. mχ ∼ 5 GeV with CP ∼ κDM ∼ O(1). However, in this case one needs to make
sure that the DM is not copiously produced at reheating and that its supercooling
abundance is not overcome by later thermal or sub-thermal production. As explained
above, this forces us to consider TRH mχ , which might be challenging to achieve.
2. mχ & TeV with κDM /CP ∼ 10−3 . This implies that the DM abundance has to
be further diluted during the phase transition. This is not inconceivable: in the
case of the electroweak phase transition it is indeed known that at the bubble walls,
particles can undergo transition radiation and lose energy via the emission of massive
particles [38]. While we do expect the same phenomenon to occur in the deconfined-
confined transition, we are not currently aware of quantitative estimates.
In the light of the arguments presented in this subsection, it would be extremely interesting
to understand whether a low reheating temperature and/or small κDM can be obtained in
the supercooling setup. We leave this task for future work.
4.3 The QCD axion
Let us now discuss the implications of supercooling for the QCD axion. In the deconfined
phase SM quarks can be either massless or massive, depending on whether the Higgs VEV
(or that of the equivalent operator OH ) is zero or not in this phase. On one hand, in models
of type I, all VEVs are zero in the deconfined phase and quarks are then massless. In this
case the axion mass is quite small in the deconfined phase10 and supercooling does not alter
the relic abundance from the misalignment mechanism (see [39] for previous studies). On
the other hand, in models of type II, we have hOH i ∼ ΛSSB ∼ TeV and the SM quarks are
instead massive in the deconfined phase. Axion oscillations can then take place during the
supercooling inflationary expansion as well as 3H . mdec dec
a , where ma is the axion mass in
10
Even for hHi = 0, the axion mass is not zero since there is still a very small Higgs-mediated contribution.
– 17 –
the deconfined phase. In this case the axion field can rapidly relax to the minimum and
its relic abundance from the misalignment mechanism can be diluted, as we will explain in
detail below. Axionic topological defects can instead be affected by supercooling in both
type of models, as it will be discussed at the end of this section.
The standard axion cosmology begins when the PQ symmetry is broken at T ∼ Fa ,
where Fa is the axion decay constant. Below this temperature, the axion dynamics is
dictated by the Klein-Gordon equation
ä + 3H ȧ + V 0 (a, T ) = 0 . (4.7)
JHEP03(2019)100
The potential V is periodic and negligible at high temperatures, while it rises very rapidly
around the QCD phase transition:
m2 (T )F 2
NDW a
V (a, T ) ' a 2 a 1 − cos , (4.8)
NDW Fa
with m2a (T ) = m2a (0)(TQCD /T )n and n ≈ 8 (see [40] for an analytic determination and [41]
for lattice confirmation). Below TQCD ' 157 MeV, the axion mass is given by
3
mu md m2π fπ2 mu md ΛQCD
m2a (0) = 2 2
' . (4.9)
(mu + md ) Fa (mu + md ) Fa2
According to eq. (4.7), axion oscillations start at T = Tosc determined by 3H(Tosc ) '
ma (Tosc ), i.e. at Tosc ∼ GeV for standard axion parameters. Below Tosc axion quanta
behave as DM, according to the misalignment mechanism. The QCD axion relic abundance
depends on whether the PQ symmetry is broken during or after cosmological inflation. In
√
the latter case, the initial average value of the axion field is θi ≡ ai /Fa = π/ 3 and axion
oscillations generate the observed dark matter abundance for Fa ' 1011 GeV. If instead
the PQ symmetry is broken during inflation, larger values of Fa are phenomenologically
viable if θi is appropriately tuned.
4.3.1 Supercooling and the misalignment mechanism
Let us now investigate the effects of supercooling on the misalignment mechanism. We
approximate eq. (4.8) with the quadratic expansion around the minimum, since the effect
of anharmonicities is not especially relevant for our estimates. The value of the axion
field at the end of the supercooling era depends on the axion mass eq. (4.9) with v →
v dec , ΛQCD → Λdec
QCD , the Hubble scale eq. (3.1), and the number of efolds of supercooling.
In particular, eq. (4.7) admits three regimes for the amplitude of the axion field:
ma (Tosc ) 1/2 − 3 Ne
Underdamping: 3H 2ma ; θf ∼ θi e 2 , (4.10)
m(Te )
− 32 Ne 3
Critical damping: 3H ' 2ma ; θf ∼ θi e 1 + Ne , (4.11)
2
m2
a
Overdamping: 3H 2ma ; θf ∼ θi e− 3H 2 Ne . (4.12)
– 18 –
The underdamped case corresponds to the well-known dark matter behavior ρ ∼ a−3 , with
a prefactor due to the temperature dependence of ma . The condition 3H . ma (0) provides
an upper bound on Fa :
!3/2
1011 GeV Λdec
1 TeV 1 TeV QCD
v
dec
1/2
Fa . . (4.13)
N mdil Λc 0.3 GeV 1 TeV
The implication of eq. (4.13) is as follows: on one hand, if Fa is smaller than eq. (4.13), then
the axion oscillates during the supercooling epoch and relaxes to the minimum a/Fa = 0
if Te . ΛdecQCD . On the other hand, if Fa is much larger than eq. (4.13), the axion is
JHEP03(2019)100
slowly-rolling along its potential: according to eq. (4.12), it relaxes to the minimum only
for Ne & 3H 2 /m2a .
The observationally allowed value of Fa depend on the duration of supercooling. In
figure 5 we provide two examples of constraints on Fa and Ne from the overproduction of
axion DM. We use v dec ' 1 TeV and fix N = 5, mdil = 100 (300) GeV,11 Λdec QCD = 330 MeV
(10 GeV) in the left (right) plot in figure 5. The dilution of the axion field does not depend
√
on the initial value of the axion but to fix ideas we have taken ai = (π/ 3)Fa as in
the post-inflationary axion scenario. The dashed orange lines in the plots correspond to
exiting at the QCD critical temperature, TQCD dec ∼ 0.5 Λdec . Assuming that the exit from
QCD
supercooling occurs after the QCD phase transition, the upper bound on Fa can be pushed
slightly above 1012 GeV for Λdec dec
QCD = 330 MeV. For ΛQCD = 10 GeV, Fa slightly above
1013 GeV is allowed. The available parameter space is reduced as Λdec QCD decreases and/or
mdil increases, as dictated by eq. (4.13). In particular for Λdec QCD 300 MeV the axion
field is essentially frozen during the supercooling epoch and the standard computation of
the axion relic abundance is not significantly altered. We also show in figure 5 regions of
parameter space where the QCD axion represents only a fraction of the DM. These are
√
relevant for axion DM detection experiments (e.g. ADMX [43]), which are sensitive to Ωa .
A few comments are in order. One can argue that the constraints on Fa can also be
relaxed in the standard case by demanding that the PQ symmetry is broken during infla-
tion, as this allows to tune the initial condition θi to any desired value. The mechanism
presented here, however, provides a dynamical way to implement this initial condition,
without appealing to any anthropic tuning.12 Furthermore, our setup can still preserve
the predictivity of the standard post-inflationary axion scenario. Indeed, the inflation-
ary expansion during supercooling is too short to homogenize the axion field across the
Hubble volume today, thus in our case one still has to average over randomly distributed
axion values.
4.3.2 Supercooling and axionic topological defects
In the previous subsection we have focused on the relic abundance from the misalignment
mechanism. However, further contributions to the axion energy density can arise from
topological defects (see [46] for an excellent introduction). We now study how the latter are
11
See [42] for constraints on the dilaton mass from collider searches.
12
See [44, 45] for an alternative possibility in the framework of low-scale models of eternal inflation.
– 19 –
1013 1014
Overdamped
Overdamped
Critical damping
13
10
it
Underdamped
Early Ex
1012
Fa [GeV]
Fa [GeV]
Critical damping ⌦a = ⌦CDM
1012
AAACB3icbZDLSsNAFIYnXmu9RV0KEiyCq5KIoBuhWBduxAr2Ak0pk+lpO3RyYeZELCE7N76KGxeKuPUV3Pk2TtsI2vrDwMd/zuHM+b1IcIW2/WXMzS8sLi3nVvKra+sbm+bWdk2FsWRQZaEIZcOjCgQPoIocBTQiCdT3BNS9QXlUr9+BVDwMbnEYQcunvYB3OaOorba551770KPthKZnP+gi3GNSvrhK07ZZsIv2WNYsOBkUSKZK2/x0OyGLfQiQCapU07EjbCVUImcC0rwbK4goG9AeNDUG1AfVSsZ3pNaBdjpWN5T6BWiN3d8TCfWVGvqe7vQp9tV0bWT+V2vG2D1tJTyIYoSATRZ1Y2FhaI1CsTpcAkMx1ECZ5PqvFutTSRnq6PI6BGf65FmoHRUdu+jcHBdK51kcObJL9skhccgJKZFLUiFVwsgDeSIv5NV4NJ6NN+N90jpnZDM75I+Mj29X/5ma
it
⌦a = 0.1 ⌦CDM
Underdamped
Early Ex
AAACC3icbVDJSgNBEO2JW4xb1KOXIUHwFGZE0IsQjAcvYgSzQDIMPZ1K0qRnobtGDMN49uKvePGgiFd/wJt/Y2cRNPFBwev3quiq50WCK7SsLyOzsLi0vJJdza2tb2xu5bd36iqMJYMaC0Uomx5VIHgANeQooBlJoL4noOENKiO/cQtS8TC4wWEEjk97Ae9yRlFLbr7QvvKhR92EpqdWyb7/ebYR7jCpnF+mqZsvWiVrDHOe2FNSJFNU3fxnuxOy2IcAmaBKtWwrQiehEjkTkObasYKIsgHtQUvTgPqgnGR8S2rua6VjdkOpK0BzrP6eSKiv1ND3dKdPsa9mvZH4n9eKsXviJDyIYoSATT7qxsLE0BwFY3a4BIZiqAllkutdTdankjLU8eV0CPbsyfOkfliydYrXR8Xy2TSOLNkjBXJAbHJMyuSCVEmNMPJAnsgLeTUejWfjzXiftGaM6cwu+QPj4xuqzZrP
⌦a = 0.01 ⌦CDM
11
10
AAACDHicbVC7SgNBFJ31GeMrammzGASrsCuCNkIwFjZiBPOAZAmzk7vJkNkHM3fFsKy9jb9iY6GIrR9g5984SVbQxAMDZ845l5l73EhwhZb1ZczNLywuLedW8qtr6xubha3tugpjyaDGQhHKpksVCB5ADTkKaEYSqO8KaLiDyshv3IJUPAxucBiB49NewD3OKGqpUyi2r3zo0U5C01OrZNn3P/c2wh0mlfPLNNUpbY1hzhI7I0WSodopfLa7IYt9CJAJqlTLtiJ0EiqRMwFpvh0riCgb0B60NA2oD8pJxsuk5r5WuqYXSn0CNMfq74mE+koNfVcnfYp9Ne2NxP+8VozeiZPwIIoRAjZ5yIuFiaE5asbscgkMxVATyiTXfzVZn0rKUPeX1yXY0yvPkvphybZK9vVRsXyW1ZEju2SPHBCbHJMyuSBVUiOMPJAn8kJejUfj2Xgz3ifROSOb2SF/YHx8AyFpmwk=
⌦a = ⌦CDM
AAAB+nicbVBNS8NAEN34WetXqkcvwSJ4KokIeiwK6rGC/YA2hM120i7dfLA7UUvMT/HiQRGv/hJv/hu3bQ7a+mDg8d4MM/P8RHCFtv1tLC2vrK6tlzbKm1vbO7tmZa+l4lQyaLJYxLLjUwWCR9BEjgI6iQQa+gLa/uhy4rfvQSoeR3c4TsAN6SDiAWcUteSZlSsvo3m3h/CI2TW0ctczq3bNnsJaJE5BqqRAwzO/ev2YpSFEyARVquvYCboZlciZgLzcSxUklI3oALqaRjQE5WbT03PrSCt9K4ilrgitqfp7IqOhUuPQ150hxaGa9ybif143xeDczXiUpAgRmy0KUmFhbE1ysPpcAkMx1oQyyfWtFhtSSRnqtMo6BGf+5UXSOqk5ds25Pa3WL4o4SuSAHJJj4pAzUic3pEGahJEH8kxeyZvxZLwY78bHrHXJKGb2yR8Ynz+DD5Qm
AAAB+nicbVBNS8NAEN34WetXqkcvwSJ4KokIeiwK6rGC/YA2hM120i7dfLA7UUvMT/HiQRGv/hJv/hu3bQ7a+mDg8d4MM/P8RHCFtv1tLC2vrK6tlzbKm1vbO7tmZa+l4lQyaLJYxLLjUwWCR9BEjgI6iQQa+gLa/uhy4rfvQSoeR3c4TsAN6SDiAWcUteSZlSsvo3m3h/CI2TW0ctczq3bNnsJaJE5BqqRAwzO/ev2YpSFEyARVquvYCboZlciZgLzcSxUklI3oALqaRjQE5WbT03PrSCt9K4ilrgitqfp7IqOhUuPQ150hxaGa9ybif143xeDczXiUpAgRmy0KUmFhbE1ysPpcAkMx1oQyyfWtFhtSSRnqtMo6BGf+5UXSOqk5ds25Pa3WL4o4SuSAHJJj4pAzUic3pEGahJEH8kxeyZvxZLwY78bHrHXJKGb2yR8Ynz+DD5Qm
AAACB3icbZDLSsNAFIYnXmu9RV0KEiyCq5KIoBuhWBduxAr2Ak0pk+lpO3RyYeZELCE7N76KGxeKuPUV3Pk2TtsI2vrDwMd/zuHM+b1IcIW2/WXMzS8sLi3nVvKra+sbm+bWdk2FsWRQZaEIZcOjCgQPoIocBTQiCdT3BNS9QXlUr9+BVDwMbnEYQcunvYB3OaOorba551770KPthKZnP+gi3GNSvrhK07ZZsIv2WNYsOBkUSKZK2/x0OyGLfQiQCapU07EjbCVUImcC0rwbK4goG9AeNDUG1AfVSsZ3pNaBdjpWN5T6BWiN3d8TCfWVGvqe7vQp9tV0bWT+V2vG2D1tJTyIYoSATRZ1Y2FhaI1CsTpcAkMx1ECZ5PqvFutTSRnq6PI6BGf65FmoHRUdu+jcHBdK51kcObJL9skhccgJKZFLUiFVwsgDeSIv5NV4NJ6NN+N90jpnZDM75I+Mj29X/5ma
⌦a = 0.1 ⌦CDM 1011
⇤dec
QCD = 10 GeV
AAACC3icbVDJSgNBEO2JW4xb1KOXIUHwFGZE0IsQjAcvYgSzQDIMPZ1K0qRnobtGDMN49uKvePGgiFd/wJt/Y2cRNPFBwev3quiq50WCK7SsLyOzsLi0vJJdza2tb2xu5bd36iqMJYMaC0Uomx5VIHgANeQooBlJoL4noOENKiO/cQtS8TC4wWEEjk97Ae9yRlFLbr7QvvKhR92EpqdWyb7/ebYR7jCpnF+mqZsvWiVrDHOe2FNSJFNU3fxnuxOy2IcAmaBKtWwrQiehEjkTkObasYKIsgHtQUvTgPqgnGR8S2rua6VjdkOpK0BzrP6eSKiv1ND3dKdPsa9mvZH4n9eKsXviJDyIYoSATT7qxsLE0BwFY3a4BIZiqAllkutdTdankjLU8eV0CPbsyfOkfliydYrXR8Xy2TSOLNkjBXJAbHJMyuSCVEmNMPJAnsgLeTUejWfjzXiftGaM6cwu+QPj4xuqzZrP
JHEP03(2019)100
⌦a = 0.01 ⌦CDM
mdil = 300 GeV
AAACDHicbVC7SgNBFJ31GeMrammzGASrsCuCNkIwFjZiBPOAZAmzk7vJkNkHM3fFsKy9jb9iY6GIrR9g5984SVbQxAMDZ845l5l73EhwhZb1ZczNLywuLedW8qtr6xubha3tugpjyaDGQhHKpksVCB5ADTkKaEYSqO8KaLiDyshv3IJUPAxucBiB49NewD3OKGqpUyi2r3zo0U5C01OrZNn3P/c2wh0mlfPLNNUpbY1hzhI7I0WSodopfLa7IYt9CJAJqlTLtiJ0EiqRMwFpvh0riCgb0B60NA2oD8pJxsuk5r5WuqYXSn0CNMfq74mE+koNfVcnfYp9Ne2NxP+8VozeiZPwIIoRAjZ5yIuFiaE5asbscgkMxVATyiTXfzVZn0rKUPeX1yXY0yvPkvphybZK9vVRsXyW1ZEju2SPHBCbHJMyuSBVUiOMPJAn8kJejUfj2Xgz3ifROSOb2SF/YHx8AyFpmwk=
⇤dec
QCD = 330 MeV
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
1010
mdil = 100 GeV
1010
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
0 5 10 15 20 0 5 10 15 20
Ne
AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRiyepYNpCG8pmO2mXbjZhdyOU0N/gxYMiXv1B3vw3btsctPXBwOO9GWbmhang2rjut1NaW9/Y3CpvV3Z29/YPqodHLZ1kiqHPEpGoTkg1Ci7RN9wI7KQKaRwKbIfj25nffkKleSIfzSTFIKZDySPOqLGSf9/Pcdqv1ty6OwdZJV5BalCg2a9+9QYJy2KUhgmqdddzUxPkVBnOBE4rvUxjStmYDrFrqaQx6iCfHzslZ1YZkChRtqQhc/X3RE5jrSdxaDtjakZ62ZuJ/3ndzETXQc5lmhmUbLEoygQxCZl9TgZcITNiYgllittbCRtRRZmx+VRsCN7yy6ukdVH33Lr3cFlr3BRxlOEETuEcPLiCBtxBE3xgwOEZXuHNkc6L8+58LFpLTjFzDH/gfP4A3vqOtg==
Ne
AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRiyepYNpCG8pmO2mXbjZhdyOU0N/gxYMiXv1B3vw3btsctPXBwOO9GWbmhang2rjut1NaW9/Y3CpvV3Z29/YPqodHLZ1kiqHPEpGoTkg1Ci7RN9wI7KQKaRwKbIfj25nffkKleSIfzSTFIKZDySPOqLGSf9/Pcdqv1ty6OwdZJV5BalCg2a9+9QYJy2KUhgmqdddzUxPkVBnOBE4rvUxjStmYDrFrqaQx6iCfHzslZ1YZkChRtqQhc/X3RE5jrSdxaDtjakZ62ZuJ/3ndzETXQc5lmhmUbLEoygQxCZl9TgZcITNiYgllittbCRtRRZmx+VRsCN7yy6ukdVH33Lr3cFlr3BRxlOEETuEcPLiCBtxBE3xgwOEZXuHNkc6L8+58LFpLTjFzDH/gfP4A3vqOtg==
Figure 5. Constraints on Fa and Ne from DM overproduction. We have taken Λdec QCD =
0.33 (10) GeV in the left (right) plot, as well as N = 5. In the blue region we have 3H > 2ma , in the
green region 3H ≤ 2ma , and in the white region 3H ≤ ma . In the orange region the supercooling
era ends before the condition 3H = ma is satisfied. Contours of Ωa = 1, 0.1, 0.01 ΩCDM are respec-
tively shown as thick, dashed and dotted black lines respectively. The vertical orange dashed line
shows the number of efolds corresponding to the critical QCD temperature during supercooling.
affected by supercooling. The standard lore is that inflationary expansion strongly dilutes
pre-existing topological defects. However, we will argue here that if the PQ symmetry
is broken after cosmological inflation, supercooling generically enhances the relevance of
axionic strings and domain walls.
The history of axionic topological defects starts at T ∼ Fa , when the Peccei-Quinn
symmetry is broken and string-like defects form. They evolve until Tc in a radiation dom-
inated background, according to the so-called scaling behavior : at any given cosmological
time there is approximately one horizon-length string per Hubble volume (for a recent
analysis, see [47–49]). This fast dilution is partially due to the emission of massless axion
quanta from T ∼ Fa to Tosc . The relic number density of such axions is dominated by
particles emitted at Tosc .
In the standard cosmological history, the axion potential eq. (4.8) switches on at the
QCD phase transition and domain walls form, which are attached to the strings. Such
walls pull the strings together and thus destroy the network if NDW = 1 (see [47, 50]).
In the supercooling scenario strings are pushed beyond the cosmological horizon by the
inflationary expansion. At the QCD phase transition during supercooling, strings are not
in causal contact and the network cannot annihilate, since two strings cannot feel the
attractive force of the wall stretching between them. Therefore, the network survives the
first QCD phase transition.
After reheating, as T drops in the ordinary expanding universe, the string network
re-enters the comoving Hubble sphere at the temperature T? ' Te . There are now two
– 20 –
100
SN 1987 A
Ne = 6
10-1
Ne = 8
T? [GeV]
M
D
10-2
C
⌦
Ne = 10
k
=
or
tw
JHEP03(2019)100
ne
AAAB/nicbVBNS8NAEN3Ur1q/ouLJS7AInkoigh6LHvRYoV/QhrDZTtqlm03YnYglFPwrXjwo4tXf4c1/4/bjoK0PBh7vzTAzL0wF1+i631ZhZXVtfaO4Wdra3tnds/cPmjrJFIMGS0Si2iHVILiEBnIU0E4V0DgU0AqHNxO/9QBK80TWcZSCH9O+5BFnFI0U2Ef1IO9qpGrc6SI8Yn4LzbEf2GW34k7hLBNvTspkjlpgf3V7CctikMgE1brjuSn6OVXImYBxqZtpSCkb0j50DJU0Bu3n0/PHzqlRek6UKFMSnan6eyKnsdajODSdMcWBXvQm4n9eJ8Poys+5TDMEyWaLokw4mDiTLJweV8BQjAyhTHFzq8MGVFGGJrGSCcFbfHmZNM8rnlvx7i/K1et5HEVyTE7IGfHIJamSO1IjDcJITp7JK3mznqwX6936mLUWrPnMIfkD6/MH4zuWEQ==
10-3 ⌦ AAACFHicbVDLSgNBEJyNrxhfUY9eFoMgCGFXBL0IwXjwIkYwD0hCmJ10kiGzs8tMrxqW/Qgv/ooXD4p49eDNv3HyOJjEgoaiqpvuLi8UXKPj/FiphcWl5ZX0amZtfWNzK7u9U9FBpBiUWSACVfOoBsEllJGjgFqogPqegKrXLw796j0ozQN5h4MQmj7tSt7hjKKRWtmjxo0PXdqKGwiPGEvAh0D1k+R8Wi9eXidJK5tz8s4I9jxxJyRHJii1st+NdsAiHyQyQbWuu06IzZgq5ExAkmlEGkLK+rQLdUMl9UE349FTiX1glLbdCZQpifZI/TsRU1/rge+ZTp9iT896Q/E/rx5h56wZcxlGCJKNF3UiYWNgDxOy21wBQzEwhDLFza0261FFGZocMyYEd/bleVI5zrtO3r09yRUuJnGkyR7ZJ4fEJaekQK5IiZQJI0/khbyRd+vZerU+rM9xa8qazOySKVhfv8AJn94=
Ne = 12
Ne = 14
10-4
108 109 1010 1011 1012
Fa [GeV]
AAAB+nicbVBNS8NAEN34WetXqkcvwSJ4KokIeiwK6rGC/YA2hM120i7dfLA7UUvMT/HiQRGv/hJv/hu3bQ7a+mDg8d4MM/P8RHCFtv1tLC2vrK6tlzbKm1vbO7tmZa+l4lQyaLJYxLLjUwWCR9BEjgI6iQQa+gLa/uhy4rfvQSoeR3c4TsAN6SDiAWcUteSZlSsvo3m3h/CI2TW0ctczq3bNnsJaJE5BqqRAwzO/ev2YpSFEyARVquvYCboZlciZgLzcSxUklI3oALqaRjQE5WbT03PrSCt9K4ilrgitqfp7IqOhUuPQ150hxaGa9ybif143xeDczXiUpAgRmy0KUmFhbE1ysPpcAkMx1oQyyfWtFhtSSRnqtMo6BGf+5UXSOqk5ds25Pa3WL4o4SuSAHJJj4pAzUic3pEGahJEH8kxeyZvxZLwY78bHrHXJKGb2yR8Ynz+DD5Qm
Figure 6. Constraints on Fa and T? from DM overproduction from the late decay of the ax-
ionic string-wall network. The solid blue line is obtained imposing Ω network = ΩCDM , according
to eq. (4.14). The orange region is forbidden by astrophysical observations. The solid green line
represent the critical temperature of the ordinary QCD phase transition after reheating. The dashed
lines show the corresponding number of efolds of supercooling.
possible scenarios depending on T? :
1. T? & GeV: the standard cosmology of topological defects and their contribution to
axion dark matter is not significantly affected.
2. T? . GeV: in this case strings re-enter the horizon when 3H < ma . Therefore, the
string-wall network is immediately formed and rapidly decays. The contribution to
the axion dark matter abundance is given by (see appendix B for a derivation)
61.75 1/4
2 Fa 100 MeV
Ωnetwork h ' 0.03 . (4.14)
g∗ (T? ) 1010 GeV T?
Constraints on Fa and T? from dark matter overproduction are shown in figure 6. We
are neglecting the contribution coming from the misalignment mechanism. This is
justified for the relevant range of Fa and Ne (see figure 5). The effect of supercooling
is essentially to delay the collapse of the string-wall network, which leads to stronger
bounds on Fa . En passant, let us notice that such a delayed annihilation of axionic
topological defects could lead to the formation of Primordial Black Holes [51].
4.4 Gravitational waves
The exit from the supercooling era occurs via a strong first order phase transition and
thus generates gravitational radiation. Differently from other scenarios discussed in the
– 21 –
literature, we focus here on the case in which the phase transition takes place after a long
inflationary phase. As we will discuss, this implies that the gravitational wave signal is
dominantly due to the collision of bubbles of the confined phase. A further peculiarity of
our setup is the possibility to obtain the maximal strength of the signal, because of a long
duration of the phase transition.
The gravitational wave signal depends mainly on two parameters: the duration of the
phase transition, usually denoted with β −1 , and the latent heat α released during the phase
transition. The former can be computed starting from the time variation of the nucleation
rate, i.e. β ≡ 1/Γ(dΓ/dT ) (see e.g. [52]) and for the case of supercooling is given by [53]
JHEP03(2019)100
" #
β Hsc dSB
= Te −4 . (4.15)
H(TRH ) H(TRH ) dT Te
Here we denote with Hsc and H(TRH ) the Hubble rate during supercooling eq. (3.1) and at
the reheating temperature respectively. For fast reheating they are expected to be similar,
Hsc ' H(TRH ). The latent heat is instead given by [53]
Veff (hµi)
α= . (4.16)
ργ (Te )
In our setup α 1, while β/H(TRH ) can vary between O(1) and O(100). As we will
see below, the strength of the gravitational wave signal is maximized for α 1 and
β/H(TRH ) ∼ 1.
Until recently, it was thought that if a first order phase transition occurs during vacuum
domination, bubbles of true vacuum would reach a runaway regime (see e.g. [53]). This
means that bubbles are accelerated until they actually collide, such that most of the energy
available in the phase transition is deposited in the accelerating walls. If this is the case
and bubbles expand at speeds very close to luminal, then a naive estimate of the wall
γ-factor at the time of collision is γmax ∼ β −1 R0−1 , where R0 ∼ Te−1 is the initial bubble
radius at Te . This gives
H(TRH ) MP Te Te
γmax ∼ ∼ (1013 − 1015 ) , (4.17)
β Λc Λc Λc
for the values of β/H(TRH ) which we find in our models. The subsequent collisions of several
bubbles generate a strong gravitational wave signal, whose spectrum is given by [53]
2
κφ α 2 100 1/3 3
2 −5 H(TRH ) 0.11vw
ΩGW,◦ h ' 1.67 · 10 2
S◦ (ω),
β 1+α g∗ 0.42 + vw
3.8(ω/ω◦ )2.8
S◦ (ω) = , (4.18)
1 + 2.8(ω/ω◦ )3.8
where vw ' 1 is the bubble wall velocity, κφ ' 1 is the fraction of the latent heat which
goes into kinetic energy of the bubble walls and ω is the frequency, whose value at the peak
of the signal is given by
−2 0.62 β TRH g∗ 1/6
ω◦ = 1.65 · 10 mHz 2
. (4.19)
1.8 − 0.1vw + vw H(TRH ) 100, GeV 100
– 22 –
This simple picture has been recently challenged in [38], where it is shown that in a first-
order EWPT bubbles surrounded by a radiation bath tend instead to reach a limit γ-factor,
even when the plasma is significantly diluted. This is due to transition radiation of particles
in the bath: as they cross the wall, they can emit vector bosons whose mass changes across
the wall. This phenomenon translates into the following friction pressure
2
∆PNLO ∼ γgEW ∆mTe3 , (4.20)
where gEW is the electroweak coupling and ∆m is the typical difference in masses across
the wall. In a first-order EWPT this is ∆m ∼ mW . As the bubble expand, γ increases till
JHEP03(2019)100
it reaches a critical γ value, γc , at which ∆PNLO balances the energy difference between the
true and false vacua, and the bubble velocity becomes constant. For a first-order EWPT
with Te ∼ mW , one finds γc & 1/(gEW 2 ).
Now, whenever γc < γmax , the bubble can reach γ-factors of order γc before they
collide. If this is the case (as for example for the first-order EWPT), most of the energy
available in the phase transition will be released to the surrounding plasma. Since this
latter also sources gravitational radiation, either from sound waves or turbulence effects,
the gravitational wave signal will be significantly changed (see e.g. [53]).
The confinement phase transition studied in this paper is different from a first-order
EWPT studied in [38]. In particular, the plasma surrounding the bubbles is strongly
interacting in our case. Nevertheless, we can expect that transition radiation could also
occur here. If we assume that eq. (4.20) approximately holds in our case, we can estimate
3
TeV
γc ∼ , (4.21)
Te
which is larger than γmax for Te . 100 MeV, indicating that γc cannot be achieved for
these small values of Te . As we have seen in the previous section, in the long supercooling
scenarios considered here, we can easily achieve Te . 100 MeV (see for example figure 4).
If this is the case, bubbles would be effectively in the runaway regime until collision. We
can then envision the following options for the gravitational wave signal in our scenarios:
1. Te 100 MeV: bubbles expand and reach γc , giving a gravitational wave signal
dominantly sourced by the plasma which surrounds the bubbles. The amplitude and
spectrum of this type of signals were analyzed in [53], but this may not straightfor-
wardly extend to our case where α is very large. Nevertheless, in the cases which
are currently understood, the slope of the spectrum is larger than in eq. (4.18), the
amplitude is linear in H(TRH )/β and for H(TRH )/β ∼ 1 it is of the same order of
magnitude as eq. (4.18).
2. Te . 100 MeV: in this case it is reasonable to assume that bubbles effectively run
away and the effects of the plasma can be neglected. The gravitational wave signal is
given by eq. (4.18) and depends only on H(TRH )/β for large α. As a concrete example,
let us consider the exit induced by QCD with SB given by eq. (3.17). In this case,
we have β/H(TRH ) ' 100/ ln(Λdec QCD /Te ) − 4, according to eq. (4.15). Thus, having
– 23 –
⇤dec
QCD = 330 MeV, mdil = 1 TeV, N = 5
10.0.
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
10.-1. Te = 1 keV AAAB+3icbVBNS8NAEN34WetXrEcvwSJ4KokIehGKXjxW6Be0IWy2k3bp5oPdibSE+FO8eFDEq3/Em//GbZuDtj4YeLw3w8w8PxFcoW1/G2vrG5tb26Wd8u7e/sGheVRpqziVDFosFrHs+lSB4BG0kKOAbiKBhr6Ajj++m/mdR5CKx1ETpwm4IR1GPOCMopY8s9L0MshvnKc+wgSzMbRzz6zaNXsOa5U4BamSAg3P/OoPYpaGECETVKmeYyfoZlQiZwLycj9VkFA2pkPoaRrREJSbzW/PrTOtDKwglroitObq74mMhkpNQ193hhRHatmbif95vRSDazfjUZIiRGyxKEiFhbE1C8IacAkMxVQTyiTXt1psRCVlqOMq6xCc5ZdXSfui5tg15+GyWr8t4iiRE3JKzolDrkid3JMGaRFGJuSZvJI3IzdejHfjY9G6ZhQzx+QPjM8fTliUmg==
10.2-2.
10 -3. Te = 10 MeV AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBiyUpgh6LXjxWsB/QxrLZTtqlm03Y3Qgl9Ed48aCIV3+PN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFS23Mfs4vatF+uuFV3DrJKvJxUIEejX/7qDWKWRigNE1Trrucmxs+oMpwJnJZ6qcaEsjEdYtdSSSPUfjY/d0rOrDIgYaxsSUPm6u+JjEZaT6LAdkbUjPSyNxP/87qpCa/9jMskNSjZYlGYCmJiMvudDLhCZsTEEsoUt7cSNqKKMmMTKtkQvOWXV0mrVvXcqnd/Wanf5HEU4QRO4Rw8uII63EEDmsBgDM/wCm9O4rw4787HorXg5DPH8AfO5w887o7W
10.
AAAB/HicbVDJSgNBEO2JW4zbaI5eGoPgKcyIoBch6MWLECEbJCH0dGqSJj0L3TViGCa/4sWDIl79EG/+jZ3loIkPCh7vVVFVz4ul0Og431ZubX1jcyu/XdjZ3ds/sA+PGjpKFIc6j2SkWh7TIEUIdRQooRUrYIEnoemNbqd+8xGUFlFYw3EM3YANQuELztBIPbtY66WQXbvOpIPwhOk9NLKeXXLKzgx0lbgLUiILVHv2V6cf8SSAELlkWrddJ8ZuyhQKLiErdBINMeMjNoC2oSELQHfT2fEZPTVKn/qRMhUinam/J1IWaD0OPNMZMBzqZW8q/ue1E/SvuqkI4wQh5PNFfiIpRnSaBO0LBRzl2BDGlTC3Uj5kinE0eRVMCO7yy6ukcV52nbL7cFGq3CziyJNjckLOiEsuSYXckSqpE07G5Jm8kjdrYr1Y79bHvDVnLWaK5A+szx+S+ZS2
aLIGO
10.-4.
10.-5. LISA C4
-6.
1010.6-7. ET
10.
AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBiyURUY9FLx4r2A9oY9lsN+3SzSbsToQS+iO8eFDEq7/Hm//GbZuDtj4YeLw3w8y8IJHCoOt+O4WV1bX1jeJmaWt7Z3evvH/QNHGqGW+wWMa6HVDDpVC8gQIlbyea0yiQvBWMbqd+64lrI2L1gOOE+xEdKBEKRtFKLc99zM4uJ71yxa26M5Bl4uWkAjnqvfJXtx+zNOIKmaTGdDw3QT+jGgWTfFLqpoYnlI3ogHcsVTTixs9m507IiVX6JIy1LYVkpv6eyGhkzDgKbGdEcWgWvan4n9dJMbz2M6GSFLli80VhKgnGZPo76QvNGcqxJZRpYW8lbEg1ZWgTKtkQvMWXl0nzvOq5Ve/+olK7yeMowhEcwyl4cAU1uIM6NIDBCJ7hFd6cxHlx3p2PeWvByWcO4Q+czx9DAo7a
aLIGO O5
LISA C3
2 -8.
⌦GW h10.
AAAB+XicbVBNS8NAEJ3Ur1q/oh69LBbBU0mKoMeiB71ZwX5AG8Nmu2mXbjZhd1MoIf/EiwdFvPpPvPlv3LY5aOuDgcd7M8zMCxLOlHacb6u0tr6xuVXeruzs7u0f2IdHbRWnktAWiXksuwFWlDNBW5ppTruJpDgKOO0E45uZ35lQqVgsHvU0oV6Eh4KFjGBtJN+2+/cRHWI/u+3ko6esnvt21ak5c6BV4hakCgWavv3VH8QkjajQhGOleq6TaC/DUjPCaV7pp4ommIzxkPYMFTiiysvml+fozCgDFMbSlNBorv6eyHCk1DQKTGeE9UgtezPxP6+X6vDKy5hIUk0FWSwKU450jGYxoAGTlGg+NQQTycytiIywxESbsComBHf55VXSrtdcp+Y+XFQb10UcZTiBUzgHFy6hAXfQhBYQmMAzvMKblVkv1rv1sWgtWcXMMfyB9fkDWGaTdQ==
LISA C2
10.-9.
JHEP03(2019)100
-10.
1010.10 DECIGO
10.-11.
AAAB73icbVBNSwMxEJ2tX7V+VT16CRbBiyURQY9FLx4r2A9o15JNs21oNrsmWaEs/RNePCji1b/jzX9j2u5BWx8MPN6bYWZekEhhLMbfXmFldW19o7hZ2tre2d0r7x80TZxqxhsslrFuB9RwKRRvWGElbyea0yiQvBWMbqZ+64lrI2J1b8cJ9yM6UCIUjFontQl+yM4InvTKFVzFM6BlQnJSgRz1Xvmr249ZGnFlmaTGdAhOrJ9RbQWTfFLqpoYnlI3ogHccVTTixs9m907QiVP6KIy1K2XRTP09kdHImHEUuM6I2qFZ9Kbif14nteGVnwmVpJYrNl8UphLZGE2fR32hObNy7AhlWrhbERtSTZl1EZVcCGTx5WXSPK8SXCV3F5XadR5HEY7gGE6BwCXU4Bbq0AAGEp7hFd68R+/Fe/c+5q0FL585hD/wPn8AquGPDw==
10.-12.
LISA C1
10.-13.
-14.
1010.14 BBO
10.-15. -8.8 -7. -6. -5. -4. -3. -2. -1.
AAAB73icbVBNSwMxEJ31s9avqkcvwSJ4sSRS0GPRi8cK9gPatWTTbBuaza5JVihL/4QXD4p49e9489+YtnvQ1gcDj/dmmJkXJFIYi/G3t7K6tr6xWdgqbu/s7u2XDg6bJk414w0Wy1i3A2q4FIo3rLCStxPNaRRI3gpGN1O/9cS1EbG6t+OE+xEdKBEKRq2T2gQ/ZOekOumVyriCZ0DLhOSkDDnqvdJXtx+zNOLKMkmN6RCcWD+j2gom+aTYTQ1PKBvRAe84qmjEjZ/N7p2gU6f0URhrV8qimfp7IqORMeMocJ0RtUOz6E3F/7xOasMrPxMqSS1XbL4oTCWyMZo+j/pCc2bl2BHKtHC3IjakmjLrIiq6EMjiy8ukeVEhuELuquXadR5HAY7hBM6AwCXU4Bbq0AAGEp7hFd68R+/Fe/c+5q0rXj5zBH/gff4AsPWPEw==
10.
10 10. 10. 10. 10. 10. 10. 10. 10. 10. 10.1. 10.2. 10.3. 10.4. 10.
105.5 AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBiyURwR6LXjxWsB/QxrLZTtqlm03Y3Qgl9Ed48aCIV3+PN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFS23Mfs4vatF+uuFV3DrJKvJxUIEejX/7qDWKWRigNE1Trrucmxs+oMpwJnJZ6qcaEsjEdYtdSSSPUfjY/d0rOrDIgYaxsSUPm6u+JjEZaT6LAdkbUjPSyNxP/87qpCWt+xmWSGpRssShMBTExmf1OBlwhM2JiCWWK21sJG1FFmbEJlWwI3vLLq6R1WfXcqnd/Vanf5HEU4QRO4Rw8uIY63EEDmsBgDM/wCm9O4rw4787HorXg5DPH8AfO5w9GDI7c
AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1Fip6Q3KFbfqLkDWiZeTCuRoDMpf/WHM0gilYYJq3fPcxPgZVYYzgbNSP9WYUDahI+xZKmmE2s8Wh87IhVWGJIyVLWnIQv09kdFI62kU2M6ImrFe9ebif14vNeGNn3GZpAYlWy4KU0FMTOZfkyFXyIyYWkKZ4vZWwsZUUWZsNiUbgrf68jppX1U9t+o1ryv12zyOIpzBOVyCBzWowz00oAUMEJ7hFd6cR+fFeXc+lq0FJ585hT9wPn8Aep2MtQ==
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AAAB+nicbVBNS8NAEN34WetXqkcvwSJ4KokIeix66bGC/YAklM120i7dfLA7UWvsT/HiQRGv/hJv/hu3bQ7a+mDg8d4MM/OCVHCFtv1trKyurW9slrbK2zu7e/tm5aCtkkwyaLFEJLIbUAWCx9BCjgK6qQQaBQI6weh66nfuQCqexLc4TsGP6CDmIWcUtdQzK14SwYC6HsID5o3Hid8zq3bNnsFaJk5BqqRAs2d+ef2EZRHEyARVynXsFP2cSuRMwKTsZQpSykZ0AK6mMY1A+fns9Il1opW+FSZSV4zWTP09kdNIqXEU6M6I4lAtelPxP8/NMLz0cx6nGULM5ovCTFiYWNMcrD6XwFCMNaFMcn2rxYZUUoY6rbIOwVl8eZm0z2qOXXNuzqv1qyKOEjkix+SUOOSC1EmDNEmLMHJPnskreTOejBfj3fiYt64Yxcwh+QPj8we+7JRN
![Hz]
Figure 7. Relic abundance of gravitational waves from the collision of cosmic bubbles in the
supercooling scenario (thick lines, as in the legend), according to eq. (4.18). The sensitivity curves
of several future ground- and space-based interferometers are also shown (see [53] for the sensitivity
of different configurations of LISA).
Te ΛdecQCD leads to β/H(TRH ) ∼ 1 − 10. Very interestingly, we therefore find that
whenever the friction of the plasma can be neglected by exiting from supercooling at
sufficiently low temperatures, the duration of the phase transition is very long and
the GW signal is maximized. This conclusion is not significantly affected by the exact
value of mdil and N . This is true also for Λdec dec
QCD , as long as ΛQCD & 100 MeV.
13
In figure 7 we show two representative gravitational wave spectra, obtained by fixing
mdil = 1 TeV, Λdec
QCD = 330 MeV, N = 5 and |λ̃| = 0 for simplicity. We vary instead
the exit temperature Te . We see that a very strong signal can indeed be obtained
if exit occurs much below Λdec QCD (blue curve): the peak amplitude is close to the
upper bound from dark radiation at BBN [54] and the signal is detectable at the
least sensitive configuration of the LISA interferometer, possibly even at the ET. A
weaker signal (orange line), which is nevertheless detectable at more sensitive config-
urations of LISA, is instead obtained if the exit occurs closer to Λdec
QCD . Both signals
are significantly larger than the more commonly considered case β/H(TRH ) & 100.
However, we stress that a detailed analysis of the thermal friction in the confinement
phase transition is required to more confidently assess the shape and size of the GW
spectrum in our case.
13
If Λdec
QCD 100 MeV and exit occurs due to QCD, then friction can always be neglected, independently
from the value of β/H(TRH ).
– 24 –
5 Conclusions
We have studied the conditions for having a long period of supercooling driven by strongly-
coupled theories at the TeV, which arises due to the small tunneling rate of the confinement
phase transition. We have calculated these rates for several examples, and showed that
quite generically (for reasonable values of N and the dilaton mass) the universe gets trapped
in the deconfined phase, starting a new period of inflation. As the supercooled universe
expands and cools down, the temperature reaches the QCD strong scale, Λdec QCD , where QCD
effects become relevant. We have shown that these effects can be calculated for T & Λdec QCD
but are suppressed by Nc2 /N 2 . For T < Λdec the CFT drastically changes as QCD-color
JHEP03(2019)100
QCD
states get massive and decouple from the rest of the CFT, affecting the tunneling rate by
O(1) effects. Under some reasonable assumptions we have shown that these can lead to an
end of supercooling after Ne ∼ 5 − 10 efoldings.
We have also studied the cosmological consequences of a long period of supercooling.
Very generically, at the end of supercooling any DM candidate is diluted by eq. (4.2). Under
the assumption that DM is not reheated again after the phase transition, we have seen that
WIMP-like DM candidates need to have very different properties. In particular, see 4.3,
their masses can be as large as 109 − 103 GeV for generic DM couplings of order 1 − 10−3
for Te ∼ 100 MeV. For DM candidates that get masses from the TeV strong scale mχ ∼ Λc ,
and are massless in the deconfined phase, their relic abundances are mainly determined
by their thermal number density at T ∼ Tc and the dilution factor eq. (4.2), as shown in
eq. (4.4). This can also be the case for baryons, as we have seen in the particular case
in which the baryon asymmetry is generated from decaying particles that were in thermal
equilibrium at high temperatures. This provides an interesting explanation for having a
ratio of baryon to DM energy densities roughly proportional to the ratio of their masses,
even though the mechanisms that originate them were very different.
One of the main impact of a long supercooling epoch is for the QCD axion abundance.
We find that during supercooling we can have H ∼ ma (not possible for inflation at higher
energies), implying that the axion can oscillate towards its minimum. This is possible in
models in which the EW symmetry is broken in the deconfined phase. This leads to an
extra dilution of axions as compared to the standard case, which changes the predictions
for being DM. We find that larger values of Fa are required (see figure 5), weakening the
signal in experiments searching for DM axions such as ADMX [43]. On the other hand, the
contributions from axionic topological defects can be enhanced in a supercooled universe.
This is because, due to the inflationary expansion, the axionic string-wall network can
collapse later than in the standard cosmology. Therefore, the network can provide a larger
contribution to the axion relic abundance, which would limit the allowed values of Fa , see
figure 6.
Finally, we have also explored the GW signal arising from the end of the supercooling
epoch. We identified two peculiarities of our setup, which can simplify the estimates of
the GW spectrum, as well as maximize the signal. Firstly, due to the long inflationary
expansion, the plasma which surrounds the bubble can be severely diluted: this implies
α 1 and that the friction on the bubble motion can be likely neglected. Therefore,
– 25 –
bubble collisions can be effectively considered as the main source of gravitational radiation.
Secondly, due to the very same reason, one can have β/H ∼ O(1) − O(10), which leads to
a stronger GW signal than in the usual case, see figure 7.
There are many things left for future work. It would be very useful to better understand
how the CFT behaves at T < Λdec QCD in order to get a better prediction for Te . For DM
candidates we have seen that the reheating process at the end of the phase transition
deserves more attention as it is crucial to determine their abundances. Also it is important
to understand transition radiation effects in the confinement phase transition as this plays
a crucial role in the GW signal.
JHEP03(2019)100
Acknowledgments
We thank J. Garriga., B. von Harling, A. Katz, O. Pujolàs and G. Servant for discussions.
A.P. has been supported by the Catalan ICREA Academia Program. This work has also
been partly supported by the grants FPA2014-55613-P, FPA2017-88915-P, 2017-SGR-1069
and SEV-2016-0588.
A Holography and supercooling
Using the AdS/CFT dictionary [55, 56], we can identify weakly-coupled five-dimensional
models with the same properties of the strongly-coupled theories described in section 2.
This provides us with simple calculable models to address the supercooling phenomena in
a more quantitative way. Many examples were considered before in the literature [3–13].
Here we briefly recall the entries of the dictionary that are useful for studying our problem.
The “dual” five-dimensional models are characterized by an AdS5 geometry:
L2
ds2 = ηµν dxµ dxν + dz 2 ,
2
(A.1)
z
solution to the 5D Einstein equations for metric GM N with negative cosmological constant
√
Z
S5 = 2M5 d5 X −G(R5 (G) + 12L−2 ) ,
3
(A.2)
where L is the AdS curvature radius, M5 the 5D Planck mass and R5 the 5D Ricci scalar.
This five-dimensional space is assumed to be ending by a hard-wall, often called IR-brane, at
some position in the extra dimension, z = zIR . For eq. (A.1) to actually be a solution ending
with a hard-wall at zIR , the hard-wall must have a negative tension τIR = −24M53 L−1 .
When this condition is met the value of zIR is not dynamically fixed but free to take
any value. In fact zIR spans a moduli space of equivalently solutions to eq. (A.2). The
AdS/CFT dictionary tells that the radion corresponds to the dilaton of the CFT model
via the identification µ = 1/zIR . The tuning of the IR-brane tension τIR to the value above
corresponds to tune the dilaton potential to zero.
The moduli space parametrized by zIR is actually part of a larger manifold of solu-
tions, parametrized by zIR itself and a four dimensional constant matrix gµν , generalizing
– 26 –
the matrix ηµν in eq. (A.1). This moduli space is associated to massless fields in 4D, cor-
responding to the spacetime dependent fluctuations of the moduli “coordinates” zIR (or µ)
and gµν . When rewriting the Einstein 5D action eq. (A.2) in terms of the moduli fields,
we get the 4D action
√
Z
3
d4 x −g 2(L−2 − µ2 )R(g) − 12(∂µ µ(x))2 ,
S = (M5 L) (A.3)
where R is the 4D Ricci scalar constructed from gµν (x). The AdS/CFT correspondence
relates the 5D Planck mass in units of 1/L to the rank of the 4D gauge group N , according
to (M5 L)3 ≈ N 2 /16π 2 . The dilaton µ is not canonically normalized, and eq. (A.3) fixes
JHEP03(2019)100
the coefficient of eq. (2.3) to c1 = 12. The massive Kaluza-Klein modes are interpreted in
the 4D as resonances of the strongly coupled CFT, and their masses are given by
1
mn ≈ n + πµ , n = 1, 2, . . . , (A.4)
4
corresponding to fix ri ∼ π in eq. (2.4).
In the 5D models discussed here we also need the equivalent of the 4D marginally
relevant operator eq. (2.1) with Dim[Og ] = 4 − . By use of the AdS/CFT dictionary,
we know that operators OΦ in the CFT are mapped to 5D fields in AdS with the same
quantum numbers. Also the dimensions of the 4D scalar operators are related to the masses
of the 5D fields according to
q
Dim[OΦ ] = 2 + 4 + MΦ2 L2 , (A.5)
where MΦ is the mass of Φ. Therefore, the marginally relevant operator eq. (2.1) is asso-
ciated to an almost massless scalar in the five-dimensional theory, MΦ2 ∝ . Furthermore,
we have the mapping g = Φ|zUV 6= 0, that triggers a nonzero profile for Φ in the bulk.
Let us now move to the symmetry breaking pattern. The AdS/CFT dictionary tells
that having the global symmetries G in the CFT corresponds to gauging a group G in the
5D bulk, and the SSB G → H by a scalar condensate translates to having an extra 5D
scalar in the bulk, H5 , transforming under G, that gets an nonzero VEV on the IR-brane.
The SM Higgs H is associated to be one of the zero-modes of H5 after KK-reduction.
In realistic models the potential of the radion/dilaton should not be fixed to zero but
be generated from the 5D model in order to naturally make hµi ∼ TeV. Let us qualitatively
discuss how this can arise in the 5D versions of models of type I and II studied in the paper:
I. In this case the radion/dilaton potential arises from the almost massless field Φ, as in
Goldberger-Wise models [16]. H5 has a potential on the IR-brane that gives a non-
zero VEV for H5 at the IR-brane. We have in this case that the VEV profile of H5
is peaked towards the IR-brane and therefore hHi ∝ µ. This possibility is illustrated
by the left figure of figure 8.
2 ∼ Φ(z). At some z, we
II. H5 is coupled to Φ that gives him a z-dependent mass, MH 5
have that MH2 becomes smaller than the BF-bound (M 2 < −4/L2 ), and H becomes
5 H5 5
– 27 –
I)
AAAB7XicbVBNSwMxEJ3Ur1q/qh69BIugl7Irgh6LXvRWwX5Au5Rsmm1js8mSZIWy9D948aCIV/+PN/+NabsHbX0w8Hhvhpl5YSK4sZ73jQorq2vrG8XN0tb2zu5eef+gaVSqKWtQJZRuh8QwwSVrWG4FayeakTgUrBWObqZ+64lpw5V8sOOEBTEZSB5xSqyTmt0wwndnvXLFq3oz4GXi56QCOeq98le3r2gaM2mpIMZ0fC+xQUa05VSwSambGpYQOiID1nFUkpiZIJtdO8EnTunjSGlX0uKZ+nsiI7Ex4zh0nTGxQ7PoTcX/vE5qo6sg4zJJLZN0vihKBbYKT1/Hfa4ZtWLsCKGau1sxHRJNqHUBlVwI/uLLy6R5XvW9qn9/Uald53EU4QiO4RR8uIQa3EIdGkDhEZ7hFd6QQi/oHX3MWwsonzmEP0CfP4X1jmw=
H5
AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0nEoseilx4r2A9oQ9lsJ+3SzSbsboQS+iO8eFDEq7/Hm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ/dzvPKHSPJaPZpqgH9GR5CFn1Fip0w9C0hjUBuWKW3UXIOvEy0kFcjQH5a/+MGZphNIwQbXueW5i/Iwqw5nAWamfakwom9AR9iyVNELtZ4tzZ+TCKkMSxsqWNGSh/p7IaKT1NApsZ0TNWK96c/E/r5ea8NbPuExSg5ItF4WpICYm89/JkCtkRkwtoUxxeythY6ooMzahkg3BW315nbSvqp5b9R6uK/W7PI4inME5XIIHN1CHBjShBQwm8Ayv8OYkzovz7nwsWwtOPnMKf+B8/gBNhI7g
zIR ⌘ 1/µ
AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXNVEBF0W3eiuin1AE8JkOmmHziRxHoUau/BX3LhQxK2/4c6/cdpmoa0HLhzOuZd77wlTRqVynG+rsLC4tLxSXC2trW9sbtnbOw2ZaIFJHScsEa0QScJoTOqKKkZaqSCIh4w0w/7l2G8OiJA0ie/UMCU+R92YRhQjZaTA3nsIMk9weH078si9pgPoHntcB3bZqTgTwHni5qQMctQC+8vrJFhzEivMkJRt10mVnyGhKGZkVPK0JCnCfdQlbUNjxIn0s8n9I3holA6MEmEqVnCi/p7IEJdyyEPTyZHqyVlvLP7ntbWKzv2MxqlWJMbTRZFmUCVwHAbsUEGwYkNDEBbU3ApxDwmElYmsZEJwZ1+eJ42TiutU3JvTcvUij6MI9sEBOAIuOANVcAVqoA4weATP4BW8WU/Wi/VufUxbC1Y+swv+wPr8ATpolZI=
II) AdS5
AAAB8HicbVBNSwMxEJ2tX7V+VT16CRbBU9kVix6rXjxWtB/SLiWbzbahSXZJskJZ+iu8eFDEqz/Hm//GtN2Dtj4YeLw3w8y8IOFMG9f9dgorq2vrG8XN0tb2zu5eef+gpeNUEdokMY9VJ8CaciZp0zDDaSdRFIuA03Ywupn67SeqNIvlgxkn1Bd4IFnECDZWeuwFEboK7/u1frniVt0Z0DLxclKBHI1++asXxiQVVBrCsdZdz02Mn2FlGOF0UuqlmiaYjPCAdi2VWFDtZ7ODJ+jEKiGKYmVLGjRTf09kWGg9FoHtFNgM9aI3Ff/zuqmJLv2MySQ1VJL5oijlyMRo+j0KmaLE8LElmChmb0VkiBUmxmZUsiF4iy8vk9ZZ1XOr3t15pX6dx1GEIziGU/DgAupwCw1oAgEBz/AKb45yXpx352PeWnDymUP4A+fzB6cDj6Q=
AdS05
AAAB8XicbVBNSwMxEJ2tX7V+VT16CRbRU9kVRY9VLx4r2g9sl5LNZtvQJLskWaEs/RdePCji1X/jzX9j2u5BWx8MPN6bYWZekHCmjet+O4Wl5ZXVteJ6aWNza3unvLvX1HGqCG2QmMeqHWBNOZO0YZjhtJ0oikXAaSsY3kz81hNVmsXywYwS6gvclyxiBBsrPXaDCF2F973z41654lbdKdAi8XJSgRz1XvmrG8YkFVQawrHWHc9NjJ9hZRjhdFzqppommAxxn3YslVhQ7WfTi8foyCohimJlSxo0VX9PZFhoPRKB7RTYDPS8NxH/8zqpiS79jMkkNVSS2aIo5cjEaPI+CpmixPCRJZgoZm9FZIAVJsaGVLIhePMvL5LmadVzq97dWaV2ncdRhAM4hBPw4AJqcAt1aAABCc/wCm+Odl6cd+dj1lpw8pl9+APn8wcJ8o/V
AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBZBLyURQY9FL/ZWwX5AG8pmO2mXbjZhdyOU0B/hxYMiXv093vw3btsctPXBwOO9GWbmBYng2rjut1NYW9/Y3Cpul3Z29/YPyodHLR2nimGTxSJWnYBqFFxi03AjsJMopFEgsB2M72Z++wmV5rF8NJME/YgOJQ85o8ZK7V4Qknr9ol+uuFV3DrJKvJxUIEejX/7qDWKWRigNE1Trrucmxs+oMpwJnJZ6qcaEsjEdYtdSSSPUfjY/d0rOrDIgYaxsSUPm6u+JjEZaT6LAdkbUjPSyNxP/87qpCW/8jMskNSjZYlGYCmJiMvudDLhCZsTEEsoUt7cSNqKKMmMTKtkQvOWXV0nrsuq5Ve/hqlK7zeMowgmcwjl4cA01uIcGNIHBGJ7hFd6cxHlx3p2PRWvByWeO4Q+czx8bbI6/
I) H5
AAAB7XicbVBNSwMxEJ3Ur1q/qh69BIugl7Irgh6LXvRWwX5Au5Rsmm1js8mSZIWy9D948aCIV/+PN/+NabsHbX0w8Hhvhpl5YSK4sZ73jQorq2vrG8XN0tb2zu5eef+gaVSqKWtQJZRuh8QwwSVrWG4FayeakTgUrBWObqZ+64lpw5V8sOOEBTEZSB5xSqyTmt0wwndnvXLFq3oz4GXi56QCOeq98le3r2gaM2mpIMZ0fC+xQUa05VSwSambGpYQOiID1nFUkpiZIJtdO8EnTunjSGlX0uKZ+nsiI7Ex4zh0nTGxQ7PoTcX/vE5qo6sg4zJJLZN0vihKBbYKT1/Hfa4ZtWLsCKGau1sxHRJNqHUBlVwI/uLLy6R5XvW9qn9/Uald53EU4QiO4RR8uIQa3EIdGkDhEZ7hFd6QQi/oHX3MWwsonzmEP0CfP4X1jmw=
H5
AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0nEoseilx4r2A9oQ9lsJ+3SzSbsboQS+iO8eFDEq7/Hm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ/dzvPKHSPJaPZpqgH9GR5CFn1Fip0w9C0hjUBuWKW3UXIOvEy0kFcjQH5a/+MGZphNIwQbXueW5i/Iwqw5nAWamfakwom9AR9iyVNELtZ4tzZ+TCKkMSxsqWNGSh/p7IaKT1NApsZ0TNWK96c/E/r5ea8NbPuExSg5ItF4WpICYm89/JkCtkRkwtoUxxeythY6ooMzahkg3BW315nbSvqp5b9R6uK/W7PI4inME5XIIHN1CHBjShBQwm8Ayv8OYkzovz7nwsWwtOPnMKf+B8/gBNhI7g
AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0nEoseilx4r2A9oQ9lsJ+3SzSbsboQS+iO8eFDEq7/Hm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ/dzvPKHSPJaPZpqgH9GR5CFn1Fip0w9C0hjUBuWKW3UXIOvEy0kFcjQH5a/+MGZphNIwQbXueW5i/Iwqw5nAWamfakwom9AR9iyVNELtZ4tzZ+TCKkMSxsqWNGSh/p7IaKT1NApsZ0TNWK96c/E/r5ea8NbPuExSg5ItF4WpICYm89/JkCtkRkwtoUxxeythY6ooMzahkg3BW315nbSvqp5b9R6uK/W7PI4inME5XIIHN1CHBjShBQwm8Ayv8OYkzovz7nwsWwtOPnMKf+B8/gBNhI7g
zIR ⌘ 1/µ
AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXNVEBF0W3eiuin1AE8JkOmmHziRxHoUau/BX3LhQxK2/4c6/cdpmoa0HLhzOuZd77wlTRqVynG+rsLC4tLxSXC2trW9sbtnbOw2ZaIFJHScsEa0QScJoTOqKKkZaqSCIh4w0w/7l2G8OiJA0ie/UMCU+R92YRhQjZaTA3nsIMk9weH078si9pgPoHntcB3bZqTgTwHni5qQMctQC+8vrJFhzEivMkJRt10mVnyGhKGZkVPK0JCnCfdQlbUNjxIn0s8n9I3holA6MEmEqVnCi/p7IEJdyyEPTyZHqyVlvLP7ntbWKzv2MxqlWJMbTRZFmUCVwHAbsUEGwYkNDEBbU3ApxDwmElYmsZEJwZ1+eJ42TiutU3JvTcvUij6MI9sEBOAIuOANVcAVqoA4weATP4BW8WU/Wi/VufUxbC1Y+swv+wPr8ATpolZI=
zSSB ⇠ 1/⇤SSB
AAACDHicbVDLSgMxFL1TX7W+qi7dBIvgqs6IoMtSNy5cVGof0BlKJpO2ocnMkGSEOvQD3Pgrblwo4tYPcOffmLYDauuBwOGcc7m5x485U9q2v6zc0vLK6lp+vbCxubW9U9zda6ookYQ2SMQj2faxopyFtKGZ5rQdS4qFz2nLH15O/NYdlYpF4a0exdQTuB+yHiNYG6lbLN13U1cKVK9Xx65iAjkn7rUZD/CPblJ22Z4CLRInIyXIUOsWP90gIomgoSYcK9Vx7Fh7KZaaEU7HBTdRNMZkiPu0Y2iIBVVeOj1mjI6MEqBeJM0LNZqqvydSLJQaCd8kBdYDNe9NxP+8TqJ7F17KwjjRNCSzRb2EIx2hSTMoYJISzUeGYCKZ+SsiAywx0aa/ginBmT95kTRPy45ddm7OSpVqVkceDuAQjsGBc6jAFdSgAQQe4Ale4NV6tJ6tN+t9Fs1Z2cw+/IH18Q1Gspp5
zIR ⌘ 1/µ
AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXNVEBF0W3eiuin1AE8JkOmmHziRxHoUau/BX3LhQxK2/4c6/cdpmoa0HLhzOuZd77wlTRqVynG+rsLC4tLxSXC2trW9sbtnbOw2ZaIFJHScsEa0QScJoTOqKKkZaqSCIh4w0w/7l2G8OiJA0ie/UMCU+R92YRhQjZaTA3nsIMk9weH078si9pgPoHntcB3bZqTgTwHni5qQMctQC+8vrJFhzEivMkJRt10mVnyGhKGZkVPK0JCnCfdQlbUNjxIn0s8n9I3holA6MEmEqVnCi/p7IEJdyyEPTyZHqyVlvLP7ntbWKzv2MxqlWJMbTRZFmUCVwHAbsUEGwYkNDEBbU3ApxDwmElYmsZEJwZ1+eJ42TiutU3JvTcvUij6MI9sEBOAIuOANVcAVqoA4weATP4BW8WU/Wi/VufUxbC1Y+swv+wPr8ATpolZI=
Figure 8. Models of type I (left) and type II (right).
JHEP03(2019)100
0
II) AdS AdSIn
an5AdS5 tachyon. 5 this case H5 gets a nonzero profile in the 5D bulk that grows as
AAAB8HicbVBNSwMxEJ2tX7V+VT16CRbBU9kVix6rXjxWtB/SLiWbzbahSXZJskJZ+iu8eFDEqz/Hm//GtN2Dtj4YeLw3w8y8IOFMG9f9dgorq2vrG8XN0tb2zu5eef+gpeNUEdokMY9VJ8CaciZp0zDDaSdRFIuA03Ywupn67SeqNIvlgxkn1Bd4IFnECDZWeuwFEboK7/u1frniVt0Z0DLxclKBHI1++asXxiQVVBrCsdZdz02Mn2FlGOF0UuqlmiaYjPCAdi2VWFDtZ7ODJ+jEKiGKYmVLGjRTf09kWGg9FoHtFNgM9aI3Ff/zuqmJLv2MySQ1VJL5oijlyMRo+j0KmaLE8LElmChmb0VkiBUmxmZUsiF4iy8vk9ZZ1XOr3t15pX6dx1GEIziGU/DgAupwCw1oAgEBz/AKb45yXpx352PeWnDymUP4A+fzB6cDj6Q=
AAAB8XicbVBNSwMxEJ2tX7V+VT16CRbRU9kVRY9VLx4r2g9sl5LNZtvQJLskWaEs/RdePCji1X/jzX9j2u5BWx8MPN6bYWZekHCmjet+O4Wl5ZXVteJ6aWNza3unvLvX1HGqCG2QmMeqHWBNOZO0YZjhtJ0oikXAaSsY3kz81hNVmsXywYwS6gvclyxiBBsrPXaDCF2F973z41654lbdKdAi8XJSgRz1XvmrG8YkFVQawrHWHc9NjJ9hZRjhdFzqppommAxxn3YslVhQ7WfTi8foyCohimJlSxo0VX9PZFhoPRKB7RTYDPS8NxH/8zqpiS79jMkkNVSS2aIo5cjEaPI+CpmixPCRJZgoZm9FZIAVJsaGVLIhePMvL5LmadVzq97dWaV2ncdRhAM4hBPw4AJqcAt1aAABCc/wCm+Odl6cd+dj1lpw8pl9+APn8wcJ8o/V
z2, till H5 reaches the minimum of its 5D potential V (H5 ) and becomes constant in z.
AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBZBLyURQY9FL/ZWwX5AG8pmO2mXbjZhdyOU0B/hxYMiXv093vw3btsctPXBwOO9GWbmBYng2rjut1NYW9/Y3Cpul3Z29/YPyodHLR2nimGTxSJWnYBqFFxi03AjsJMopFEgsB2M72Z++wmV5rF8NJME/YgOJQ85o8ZK7V4Qknr9ol+uuFV3DrJKvJxUIEejX/7qDWKWRigNE1Trrucmxs+oMpwJnJZ6qcaEsjEdYtdSSSPUfjY/d0rOrDIgYaxsSUPm6u+JjEZaT6LAdkbUjPSyNxP/87qpCW/8jMskNSjZYlGYCmJiMvudDLhCZsTEEsoUt7cSNqKKMmMTKtkQvOWXV0nrsuq5Ve/hqlK7zeMowgmcwjl4cA01uIcGNIHBGJ7hFd6cxHlx3p2PRWvByWeO4Q+czx8bbI6/
H5 The value of z where the maximum is achieved, zSSB , is associated with the SSB scale
AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0nEoseilx4r2A9oQ9lsJ+3SzSbsboQS+iO8eFDEq7/Hm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ/dzvPKHSPJaPZpqgH9GR5CFn1Fip0w9C0hjUBuWKW3UXIOvEy0kFcjQH5a/+MGZphNIwQbXueW5i/Iwqw5nAWamfakwom9AR9iyVNELtZ4tzZ+TCKkMSxsqWNGSh/p7IaKT1NApsZ0TNWK96c/E/r5ea8NbPuExSg5ItF4WpICYm89/JkCtkRkwtoUxxeythY6ooMzahkg3BW315nbSvqp5b9R6uK/W7PI4inME5XIIHN1CHBjShBQwm8Ayv8OYkzovz7nwsWwtOPnMKf+B8/gBNhI7g
∼ ⇠1/z
ΛSSBzSSB SSB
1/⇤ . When including
SSB
the metric back-reaction, one gets that the AdS5
zIR ⌘ 1/µ
AAACDHicbVDLSgMxFL1TX7W+qi7dBIvgqs6IoMtSNy5cVGof0BlKJpO2ocnMkGSEOvQD3Pgrblwo4tYPcOffmLYDauuBwOGcc7m5x485U9q2v6zc0vLK6lp+vbCxubW9U9zda6ookYQ2SMQj2faxopyFtKGZ5rQdS4qFz2nLH15O/NYdlYpF4a0exdQTuB+yHiNYG6lbLN13U1cKVK9Xx65iAjkn7rUZD/CPblJ22Z4CLRInIyXIUOsWP90gIomgoSYcK9Vx7Fh7KZaaEU7HBTdRNMZkiPu0Y2iIBVVeOj1mjI6MEqBeJM0LNZqqvydSLJQaCd8kBdYDNe9NxP+8TqJ7F17KwjjRNCSzRb2EIx2hSTMoYJISzUeGYCKZ+SsiAywx0aa/ginBmT95kTRPy45ddm7OSpVqVkceDuAQjsGBc6jAFdSgAQQe4Ale4NV6tJ6tN+t9Fs1Z2cw+/IH18Q1Gspp5
geometry evolves at around zSSB into another AdS5 with different curvature. The
AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXNVEBF0W3eiuin1AE8JkOmmHziRxHoUau/BX3LhQxK2/4c6/cdpmoa0HLhzOuZd77wlTRqVynG+rsLC4tLxSXC2trW9sbtnbOw2ZaIFJHScsEa0QScJoTOqKKkZaqSCIh4w0w/7l2G8OiJA0ie/UMCU+R92YRhQjZaTA3nsIMk9weH078si9pgPoHntcB3bZqTgTwHni5qQMctQC+8vrJFhzEivMkJRt10mVnyGhKGZkVPK0JCnCfdQlbUNjxIn0s8n9I3holA6MEmEqVnCi/p7IEJdyyEPTyZHqyVlvLP7ntbWKzv2MxqlWJMbTRZFmUCVwHAbsUEGwYkNDEBbU3ApxDwmElYmsZEJwZ1+eJ42TiutU3JvTcvUij6MI9sEBOAIuOANVcAVqoA4weATP4BW8WU/Wi/VufUxbC1Y+swv+wPr8ATpolZI=
dual 4D interpretation of this is that a CFT RG-evolves into another CFT at Λ SSB .
The tachyon profile gives also a contribution to the radion potential and a minimum
for µ can be achieved, as studied in ref. [18]. One usually finds hµi ∼ ΛSSB . When
moving the IR-brane off-shell towards the horizon z → ∞, the VEV of H5 is not
proportional to µ and then does not go to zero as in type I models. This is illustrated
in the right figure of figure 8.
Let us finally provide the 5D implementation of eq. (2.9). The fermionic operators OL,R
corresponds to 5D fermions with the same quantum numbers as the SM quarks. This
implies that the QCD group must be a gauged symmetry in the 5D bulk. At low energies,
the zero-modes of these 5D fermions corresponds to the SM quarks.
A.1 The deconfined phase
In 5D holographic versions the deconfined phase is characterized by a different solution
to the 5D Einstein equations (arising from eq. (A.2)) at finite temperature T where we
must take the Euclidian version of the theory, with compact time it ∈ [0, 1/T ]. This
solution is called AdS5 -Schwarzschild Black Hole (AdS-BH) geometry, and the metric can
be expressed as
4
L2
2 2 1 z
ds = − 2 −f (z)dt + dx2i + dz 2 , f (z) = 1 − . (A.6)
z f (z) zh
This metric has a BH horizon at zh replacing the infrared brane (z runs from z = 0 to
z = zh ). The BH horizon is associated to a Hawking temperature, given by Th = 1/(πzh ).
To avoid singularities, we must match the BH Hawking temperature with the temperature
of the system: T = 1/(πzh ) (see A.2 for more details).
On the AdS-BH background eq. (A.6) the 5D fields which were discussed above, and in
particular H5 , take in general a different profile. We are especially interested in the value
of ΛSSB in models I and II. In type I models H5 cannot get a nonzero profile in the AdS-BH
– 28 –
solution since, contrary to the IR-brane, there is no potential for H5 in the BH solution.
On the other hand, in models of type II, H5 is an AdS5 tachyon and then also turns on
even in the absence of an IR-brane potential. Indeed, we can find a nonzero profile for H5
in the AdS-BH where at z = zh the regularity of the solution implies
4
∂z H5 + V (H5 ) = 0. (A.7)
z z=zh
Therefore, the EW symmetry is broken in type II models also in the deconfined phase.
JHEP03(2019)100
A.2 Free energy of the deconfined phase
The energy of solution eq. (A.6) is interpreted in the 4D as the free energy of the plasma
phase, with identification zh = 1/(πTloc ). In fact, when zh 6= 1/(πT ) the metric eq. (A.6)
has a conical singularity at z = zh that gives a finite contribution to the free energy density
equal to [3]
4 3 4 T
Fcone = 8π (M5 L) Tloc 1 − . (A.8)
Tloc
We see that it correctly vanishes when Tloc = T . A remaining piece comes from the regular
region z 6= zh . This is UV divergent, but what matters for tunneling is its finite difference
with the energy of full AdS5 . This gives
FBH (z 6= zh ) − FAdS = −2π 4 (M5 L)3 Tloc
4
. (A.9)
When added together, the two contributions eq. (A.8) and eq. (A.9) give
π2 2
FBH = 2π 4 (M5 L)3 (3Tloc
4 3
− 4T Tloc )= 4
N (3Tloc 3
− 4T Tloc ), (A.10)
8
where the last equality comes from the identification (M5 L)3 = N 2 /16π 2 . With this
identification we get c2 = π 2 /8 for eq. (3.3). The minimum of eq. (A.10) is attained when
T = Tloc , as it must be since Einstein equations forbid conical singularities.
This result can also be determined, modulo an overall constant, from elementary ther-
modynamics and scale invariance as follows. If we put a small region of volume V to
temperature Tloc , its free energy density (with respect to a thermal bath at temperature
T ) will be F = (E(Tloc , V ) − T S(Tloc , V ))/V . The total energy and entropy E and S
are extensive variables, and scale invariance tells that the densities E/V and S/V are
4 and T 3 )
appropriate powers of the only available scale Tloc (that is respectively Tloc loc
4 3
F (Tloc ; T ) = aTloc − bT Tloc . (A.11)
We know that F is a minimum at thermal equilibrium, that is when Tloc = T . This fixes
the ratio a/b = 3/4 universally for any CFT.
– 29 –
B Axion dark matter from string-wall network
The aim of this appendix is to review the estimate of the contribution to the total axion
dark matter abundance from the annihilation of the string-wall network (see e.g. [50]). The
energy of the network is dominated by the walls, thus we have
ρnetwork ' σH, (B.1)
where σ ' 8ma (0)Fa2 is the wall tension. This energy is eventually entirely released in
axion quanta which behave like dark matter. The process is expected to occur very quickly
after T? . The relic number density of the radiated particles is thus given by
JHEP03(2019)100
ρnetwork a(t? ) 3
nnetwork (t0 ) = , (B.2)
Ēa a(t0 )
where Ēa is the mean energy of the radiated axions at T? . This has to be extracted
from numerical simulations: here we follow the results of [50], according to which Ēa '
3.23 ma (T? ). The relic energy density of such axions is given by
a(t? ) 3
ma (o)
ρnetwork (t0 ) = ρnetwork (T? ) . (B.3)
3.23ma (T? ) a(t0 )
Finally the relic abundance is
1/4
2 61.75 Fa 100 MeV
Ωnetwork h ' 0.03 10
. (B.4)
g∗ (T? ) 10 GeV T?
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