Gravitational Wave Signals from Multiple Hidden Sectors

Paul Archer-Smith, Dylan Linthorne, and Daniel StolarskiOttawa-Carleton Institute for Physics, Carleton University,

1125 Colonel By Drive, Ottawa, Ontario K1S 5B6, Canada ∗

We explore the possibility of detecting gravitational waves generated by first-order phase tran-sitions in multiple dark sectors. Nnaturalness is taken as a sample model that features multipleadditional sectors, many of which undergo phase transitions that produce gravitational waves. Weexamine the cosmological history of this framework and determine the gravitational wave profilesgenerated. These profiles are checked against projections of next-generation gravitational waveexperiments, demonstrating that multiple hidden sectors can indeed produce unique gravitationalwave signatures that will be probed by these future experiments.

I. INTRODUCTION

The recent experimental detection of gravitationalwaves [1] gives humanity a new way to observe the uni-verse. Future experiments [2–11] will greatly expandthe frequency range observable. Thus far, experimentshave only observed recent events such as black holemergers, but phase transitions in the early universe canleave an imprint as a stochastic gravitational wave back-ground [12–17]. Thus, searches for this background ofgravitational waves can give direct information of thehistory of the universe before big bang nucleosynthesis.Because gravity is universal, gravitational waves can al-low us to probe hidden sectors that couple very weakly,or not at all, to the Standard Model as long they are re-heated after inflation. This was first explored in [18], andthere has been significant work on this idea since [19–36].

In this work, we explore the possibility of having mul-tiple decoupled hidden sectors. Large numbers of hid-den sectors can solve the hierarchy problem as in theDvali Redi model [37], in the more recently exploredNnaturalness [38] framework, or in orbifold Higgs mod-els [39, 40]. They can also be motivated by dark matterconsiderations [22, 41–43]. Motivated by solutions to thehierarchy problem, we consider hidden sectors with thesame particle content as the Standard Model that haveall dimensionless couplings (defined at some high scale)equal to those of the Standard Model. The only parame-ter that varies across sectors is the dimension-two Higgsmass squared parameter, m2

H . This simple ansatz canlead to very rich phenomenology and interesting grav-itational wave spectra, but we stress that it is only astarting point for exploring the space of theories withmultiple hidden sectors.

In this setup, there are two qualitatively different kindsof sectors:

• Standard Sectors: Those with m2H < 0 where

electroweak symmetry is broken by the vacuum ex-pectation value (vev) of a fundamental scalar. As

∗ Paul.Smith3@carleton.caDylan.linthorne@carleton.castolar@physics.carleton.ca

in [38], we assume that the standard sector withthe smallest absolute value of m2

H is the StandardModel.

• Exotic Sectors: Those with m2H > 0. In this

case, electroweak symmetry is preserved below themass of the Higgs, and broken by the confinementof QCD [44].

Cosmological observations, particularly limits on extrarelativistic degrees of freedom at the time of Big BangNucleosynthesis and the time of the formation of thecosmic microwave background (CMB) [45], require thatmost of the energy in the universe is in the StandardModel sector as we will quantify. Therefore, the hiddensectors cannot be in thermal equilibrium at any time,and the physics of reheating must dump energy pref-erentially in the Standard Model sector. This can beaccomplished with primordial axionlike particle (ALP)models [46, 47] and with the reheaton method [38]. Wewill also explore alternative parameterizations of reheat-ing that satisfy this condition.

In all the above models, there is some energy in the hid-den sectors, and these sectors undergo thermal evolutionindependent of the SM sector. If their initial reheatingtemperature is above their weak scale, the standard sec-tors will undergo phase transitions associated with thebreaking of electroweak symmetry and with confinementof QCD. The exotic sectors will also undergo a phasetransition when QCD confines and electroweak symme-try is broken simultaneously. The condition for thesetransitions to leave imprints on the stochastic gravita-tional wave spectrum is that they strongly first-orderphase transitions (SFOPT) [12–15]. This does not oc-cur at either the electroweak or QCD phase transition inthe SM, but as we will show, it does happen for the QCDphase transition in some standard sectors and in all ex-otic sectors that reheat above the QCD phase transition.

This work is organized as follows: section II introducesthe particle content of the model, section III discusses thephase transition behaviour of both the standard and ex-otic sectors present, section IV lays out hidden sector re-heating, section V applies constraints from cosmologicalobservables allowing for the calculation of gravitationalwave signatures in section VI, and, finally, section VII

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ties everything up.

II. PARTICLE SETUP

We consider the following Lagrangian as in [38]:

L =

N/2∑i=−N/2

Li, (1)

with L0 = LSM being the Standard Model Lagrangian,and Li being a copy of the SM Lagrangian with differentfields, but with all dimensionless parameters the same.Each of the Lagrangians does contain a dimensionful op-erator:

Li ⊂ −(m2H

)iH†iHi (2)

where Hi is a Higgs field in each sector, and the massterm is parametrically given by

(m2H

)i∼ −Λ2

H

N(2i+ r), (3)

where Λ is some high-scale cutoff, N is the number ofsectors, and r is the mass parameter in the SM in unitsof Λ2

H/N . We view the parameterization of Eq. (3) asa random distribution in theory space up to the cut-off Λ: therefore, this setup solves the hierarchy problemif r ∼ O(1) [38]1 and our sector is the one that thathas the smallest absolute value of the Higgs mass pa-rameter. We have taken for simplicity that there areequal numbers of sectors with positive and negative m2

H ,but this assumption does not affect our analysis. ThisNnaturalness framework can be generalized: the varioussectors can possess a wide range of particle content thatcan be freely selected by the model builder. The one ex-ception to this is that “our” sector must consist of theStandard Model.

From the above Lagrangians, the Higgs in sectors withi ≥ 0 will get a VEV given by

vi =√−(m2

H)i/λi ∼ ΛH

√2i+ r

λN, (4)

λi is the quartic coefficient of the scalar potential andis the same across all sectors, λi = λ. This is anotherway to see how this framework can solve the hierarchyproblem: the Higgs VEV is parametrically smaller thanthe cutoff for N 1. The “standard sectors” with i > 0feature electroweak symmetry breaking just like in theSM; however, the VEVs scale with the changing massparameter: vi ∼ vSM

√i. This means that the masses

of the fermions and the W and Z will also increase pro-portional to

√i. The consequences of this scaling on the

1 Constraints require r to be somewhat smaller than 1.

confinement scale of QCD in the i ≥ 1 sectors is furtherdiscussed in Sec. III.

The “exotic sectors” with i < 0 provide a radical de-parture from our own. m2

H > 0 leads to no VEV for theHiggs, and electroweak symmetry is only broken at verylow scales due to the phase transition from free quarksto confinement at the QCD scale ΛQCD [44], and themasses of the W and Z are comparable to those of QCDresonances. The masses of fundamental fermions are pro-duced via four-fermion interactions generated after inte-grating out the SU(2) Higgs multiplet. This leads to verylight fermions:

mf ∼ yfytΛ3QCD/(m

2H)i ≤ 100 eV, (5)

with yf representing the Yukawa coupling to fermion f .As we will see, the extremely light quarks that appearin these sectors dramatically change the nature of theQCD phase transition — unlike the SM, the transition isstrongly first order. Again, this is further developed inSec. III. Crucially, this results in the production of gravi-tational waves. This is the physical signature we explorein this paper; the calculation and results are presentedin Sec. VI.

III. QCD PHASE TRANSITION

We now study the nature of the QCD phase transi-tion across the different sectors. Due to the confiningnature of QCD, the exact nature of the phase transitionis often difficult to ascertain analytically and requires thestudy of lattice simulations. In the SM, it is known thatthe phase transition is a crossover and does not lead togravitational wave signals [48, 49]. In the general casewith three or more colours, the phase transition can bestrongly first order in two regimes [50–52]:

• three or more light flavours and

• no light flavours.

Light indicates a mass small compared to the confine-ment scale ΛQCD, but what that means quantitatively isnot precisely determined. In the SM, the up and downquarks are light, but the strange is not sufficiently lightfor an SFOPT. For the standard sectors in our setup,the quark masses increase with increasing VEV, so forsufficiently large i, all the quarks will be heavier thanΛQCD,2 and those large i sectors will undergo an SFOPTif they are reheated above the the confinement scale.Conversely, exotic sectors with zero VEV feature six verylight quarks, so all the exotic sectors undergo SFOPT atthe temperature of QCD confinement.

2 ΛQCD does vary with i, but the sensitivity is very weak as wewill see below.

3

We now calculate the QCD confinement scale for eachsector following the same procedure as [53]. First, due tothe parameters of each sector being taken to be identicalsave for the Higgs mass squared (thus v 6= vi, where vis the SM VEV), we assume that the strong coupling ofevery sector is identical at some high scale. Using theone-loop running, the β function can be solved:

αis(µ) =2π

11− 2nif3

1

lnµ/Λi, (6)

where nif is the number of quark flavours with mass less

than µ/2 and Λi is the scale where it would confine ifall quarks remain massless. In the SM defined at scaleswell above all the quark masses, we have ΛQCD = 89± 5

MeV in MS [54]. Because we have set the strong cou-plings equal at high scales, Λ = Λi for all i at high scalesfor all sectors. However, since the masses of the quarksin each sector are different, we end up with a uniquerunning of the coupling for each sector. At every quarkmass threshold for a given sector, we match the couplingstrengths above and below the threshold and determinethe new Λi for the lower scale. For example, at the massof the top quark, we match a five-flavour coupling withthe six-flavour one:

αi(5)s (2mi

t) = αi(6)s (2mi

t) (7)

and thus

Λi(5) = (mit)

2/23(Λi(6))21/23. (8)

Suppressing the i’s for notational cleanliness, we can ar-rive at similar relations at the bottom and charm thresh-olds

Λ(4) = (mb)2/25(Λ(5))

23/25,

Λ(3) = (mc)2/23(Λ(4))

25/27.(9)

These can be combined to show that

Λ(3) = (mtmbmc)2/27(Λ(6))

21/27. (10)

This type of matching procedure can be done as manytimes as necessary for a given sector. The process termi-nates when Λi for a given scale is larger than the nextquark mass threshold (i.e running the scale down arrivesat the ΛQCD phase transition before reaching the nextquark mass scale). In cosmological terms, we can en-vision a sector’s thermal history unfolding, whereas theplasma cools below each quark mass threshold and saidquarks are frozen out. At a certain point, the sectorarrives at the QCD phase transition and confinement oc-curs — if this occurs when ≥ 3 quarks are at a muchlower scale or all quarks have already frozen out, we getthe desired phase transition.

A. Standard Sectors

As shown in Eq. (4), for standard sectors with increas-

ing index i, the VEVs of said sectors increase vi ∝√i.

This leads to increasingly heavy particle spectra forhigher sectors — eventually leading to sectors that areessentially pure Yang-Mills that feature strong first-orderphase transitions. This, of course, prompts the question:at what index i do said phase transitions begin? Usingthe methods outlined in the prior section we determineΛQCD to have a relevant value of

Λi(2) = (mism

icm

ibm

it)

2/29(Λi(6))21/29 (11)

at the energy scale we’re interested in. Λi(6) is identical

for all sectors and is taken to have a Standard Modelvalue of Λ

(6)MS = (89± 6) MeV [54]. Rewriting Eq. (11) in

terms of Standard Model variables,

Λi(2) = (msmcmbmti2)2/29(Λ(6))

21/29. (12)

where mq without a superscript is the mass of q in theSM. We take the sector with SFOPT to be the ones whenthe mass of the up quark, down quark, and QCD phasetransition scale are all comparable:

miu ∼ mu

√i ∼ (msmcmbmti

2)2/29(Λ(6))21/29. (13)

This can be solved for i:

ic ∼(msmcmbmt)

4/21(Λ(6))2

(mu)58/21∼ 106. (14)

As we will see in Sec. IV, in the original Nnaturalnesssetup [38], the energy dumped into the ith sector scalesas i−1, so there will not be enough energy in the sec-tors with i > ic to see a signature of these phase tran-sitions. However, if we move away from the originalNnaturalness reheating mechanism and begin exploringmirror sectors with large VEVs and with relative energydensities ρi/ρSM ∼ 10%, a possibility allowed by currentconstraints, we can have sectors with relatively high darkQCD scales that produce detectable gravitational waves.From Eq. (11) we can determine the confinement scaleof an arbitrary mirror sector. If we take Higgs VEVsas high as the GUT scale ∼ 1016 GeV, then we can useEq. (12) to get confinement scales as high as ∼ 38 GeV .The signals of this sector and other test cases like it areexplored in Sec. VI.

B. Exotic Sectors

In every exotic sector the fermion masses are excep-tionally light: their masses are generated by dimensionsix operators with the Higgs integrated out as shownin Eq. (5), and are therefore all below the confine-ment scale. The exotic sectors all have identical one-loop running of the QCD gauge coupling, and thus allhave approximately the same confinement scale given byΛex ∼ 90 MeV. These sectors all have six light fermions,so a strong first order phase transition occurs for all ex-otic sectors at this temperature. The confinement of

4

these sectors directly leads to the production of bothbaryons and mesons as we have the spontaneous break-ing of SU(6)× SU(6) → SU(6) and thus 35 pseudo-Goldstone bosons (pions). The masses obtained throughthe phase transition can be approximated through theuse of a generalization of the Gell-Mann–Oakes–Rennerrelation [55, 56],

m2π =

V 3

F 2π

(mu +md), (15)

where V ∼ ΛQCD, Fπ is the pion decay constant. Oneexpects that within a given sector Fπ ∼ V ∼ ΛQCD [56]and as exotic sectors have Λex ∼ 90 MeV while the SMfeatures ΛQCD = (332± 17) MeV [54] we expect at most

O(1) difference in the√

V 3

F 2π

coefficient relative to the SM

value. So, for pions in exotic sector i:

miπ ∼

√mia +mi

b

mu +mdmπ. (16)

Here, a and b denote the component quark flavours.

IV. REHEATING N SECTORS

A key issue within Nnaturalness is how to predom-inantly gift energy density to our own sector so as tonot be immediately excluded by cosmological constraints,particularly those from number of effective neutrinos(Neff ). Here we review the results of [38]. Reheatingoccurs through the introduction a “reheaton” field. Af-ter inflation, the reheaton field possesses the majority ofthe energy density of the Universe. Although this fieldcan generically be either bosonic or fermionic, we reduceour scope to a scalar reheaton φ. Our focus is primarilythe production of gravitational waves from multiple sec-tors and a fermion reheaton does not change the scalingof the energy density of the exotic sectors and thus doesnot affect expected gravitational wave profiles.

In order to maintain the naturalness of our SM sector,the reheaton coupling is taken to be universal to everysector’s Higgs. However, a large amount of the Universe’senergy density must ultimately be deposited in our ownsector for Nnaturalness to avoid instant exclusion. In or-der to accomplish this, the decay width of the reheatoninto each sector must drop as |mH | grows. If we insistthat the reheaton is a gauge singlet that is both the domi-nant coupling to every sector’s Higgs and lighter than thenaturalness cutoff ΛH/

√N , then we construct a model

that behaves as desired. The appropriate Lagrangian fora scalar reheaton φ is:

Lφ ⊃ −aφ∑i

|Hi|2 −1

2m2φφ

2. (17)

Note that cross-quartic couplings of the form κ|Hi|2|Hj |2that could potentially ruin the spectrum of Nnaturalness

are absent, taken to be suppressed by a very small cou-pling. Effective Lagrangians for the two different typesof sectors present in this theory can be obtained by inte-grating out of the Higgs bosons in every sector:

Lv 6=0φ ⊃ C1ayq

v

m2h

φqqc,

Lv=0φ ⊃ C2a

g2

16π2m2H

φWµνWµν ,

(18)

with Ci representing numerical coefficients, g the weakcoupling constant, and Wµν the SU(2) field strength ten-sor. Immediately from Eq. (18), we can see that the ma-trix element for decays into standard sectors is inverselyproportional to that sectors Higgs mass, Mm2

H<0 ∼1/mhi (since v ∼ mH). The loop decay of φ→ γγ is al-ways subleading and can be neglected. It should be notedthat as one goes to sectors with larger and larger VEVs,the increasing mass of the fermions (mf ∼ vi ∼ vSM

√i)

eventually leads to situations where the decay to two on-shell bottom or charm quarks is kinematically forbidden,mφ < 2mq. For sectors where this kinematic threshold ispassed for charm quarks, the amount of energy in thesesectors becomes so small that contributions to cosmolog-ical observables can be safely ignored. All in all, we endup with a decay width that scales as Γm2

H<0 ∼ 1/m2h.

Since we can expect energy density to be proportional tothe decay width, ρi

ρSM≈ Γi

ΓSM, this indicates that energy

density of standard sectors falls:

ρi ∼ rsρSMi

(19)

with rs being the ratio of the energy density of the firstadditional standard sector over the energy density of oursector. For the exotic sectors, Eq. (18) indicates a ma-trix element scaling Mm2

H>0 ∼ 1/m2Hi

and is also loopsuppressed. This leads to a significantly lower energydensity than the standard sectors. Both the decay widthand energy density for these sectors scale as

Γm2H>0 ∼ ρi ∼ 1/m4

H ∼ 1/i2. (20)

As a final note, in this setup the reheating temperatureof the SM, TRH , has an upper bound on the order of theweak scale. If this bound is not observed, the SM Higgsmass would have large thermal corrections — leading tothe branching ratios into other sectors being problemat-ically large [38]. Thus we only consider relatively lowreheating temperatures . 100 GeV.

Ultimately, after examining the gravitational wave caseproduced by standard Nnaturalness, we also consider amore generic parameterization where the reheating tem-perature of each sector is a free parameter and is in gen-eral uncorrelated with the Higgs mass parameter. Thisallows us to explore a broader model space with multipledark sectors at a huge range of scales. For these models,the reheating mechanism remains unspecified.

5

V. CONSTRAINTS

In general, the multi-hidden-sector models exploredfeature a huge number of (nearly) massless degrees offreedom. Dark photons and dark neutrinos abound inthese sectors and, assuming a relatively high reheat tem-perature, the leptons, quarks, and heavy bosons of thesesectors can also be relativistic. In Nnaturalness this fea-ture is realized quite dramatically: each of the N sectorspossess relativistic degrees of freedom. The presence ofthese particles can have two main effects: extra relativis-tic particles can alter the expansion history of the uni-verse through changes to the energy density or hiddensectors can feature annihilations that reheat the photonsor neutrinos of our sector near Big Bang Nucleosynthe-sis (BBN) and affect the light element abundances. Theeffective number of neutrino species, Neff , is impactedby these contributions and, as such, is the strictest con-straint that must be dealt with when studying these typeof multi-phase-transition models. The SM predicts thatNSMeff = 3.046 [57]. This is in good agreement with the

2σ bounds from studies of the Cosmic Microwave Back-ground (CMB) by Planck combined with baryon acousticoscillations (BAO) measurements [45]:

Neff = 2.99+0.34−0.33. (21)

Various different assumptions about the history of theuniverse can be made and different data sets can be cho-sen to obtain slightly different results [32] — for the pur-poses of this exploratory work, wading through this land-scape is unnecessary. Additionally,

(∆N ieff )CMB

(∆N ieff )BBN

≥ 1 (22)

for any decoupled hidden sector [38]. Because the con-straints on Neff are stronger at photon decoupling thanat BBN, we can focus purely on the constraints providedby the former. Future CMB experiments [58] will im-prove the bound from Eq. (21) by about an order ofmagnitude. This could significantly reduce the allowedtemperature ratio of any hidden sector or, alternatively,could provide evidence for such sectors in a way that iscomplementary to the gravitational wave signatures de-scribed below. For fully decoupled sectors that neverenter (or reenter) thermal equilibrium with our sector,we obtain additional contributions to NSM

eff [32]

∆Neff =4

7

(11

4

)4/3

ghξ4h. (23)

Here, gh represents the effective number of relativisticdegrees of freedom for the hidden sector3, and we pa-rameterize the hidden sector temperature by [32]

ξh ≡ThTγ, (24)

3 gh = Nboson + 7Nfermion/8.

and these should be evaluated at the time of photon de-coupling. We take this approach and generalize it toinclude many additional sectors:

∆Neff =∑i

4

7

(11

4

)4/3

giξ4i . (25)

For a dark sector with one relativistic degree of freedom,its temperature must be TDS ∼ 0.6TSM to not be ex-cluded. Applying the energy density formula [59],

ρi =π2

30giT

4i , (26)

to both said dark sector and the SM and then taking theratio indicates that the dark sector would have an energydensity ρ ∼ 0.038 ρSM.

A. Exotic Sector Contributions

We begin by computing the constraints on exotic sec-tors; these are significantly weaker than those for stan-dard sectors [38]. At the time of photon decoupling,Tγ ∼ 0.39 eV while the temperature of the exotic sec-tors is lower. This means that for sectors with small andmoderate i, we can use Eqs. (5) and (16) to see that thepions will be nonrelativistic leaving at most 7.25 effectivedegrees of freedom per sector from photons and neutri-nos. For very large i, the pions can be much lighter, butthose sectors also have very little energy in them in thestandard reheating scenario. Coupling the number of ef-fective degrees of freedom per sector with the energy den-sity scaling of ∼ 1/m4

H as in Eq. (20) means that the zeroVEV sectors have small temperature ratios. Assuming areheating temperature of 100 GeV and a completely uni-form distribution of sectors, the temperature of the firstexotic sector is slightly more than 6% of our sector atreheating. Applying Eq. (25) to this particular situationgives us:

∆Neff =∑i

4

7

(11

4

)4/3

gi

((TRHE1

/TRH)

i1/2

)4

∼ 10−4,

(27)with TRHE1

/TRH being the ratio of the reheat tempera-tures of the first exotic sector and our own sector (0.06in standard Nnaturalness with r = 1). This sum is dom-inated by i = 1; the sector with the lowest Higgs mass(and thus the most energy density) gives us a contribu-tion of O(10−4) to ∆Neff . Evolving the sector thermalhistories forward in time to the recombination era givesus a slightly larger value, but still of order O(10−4), wellbelow current CMB bounds. It should be noted thatmodifying the exotic sectors’ structure (e.g. adjustingthe exotic sectors to have a lower Higgs mass squared orclustering multiple hidden sectors close to the first ex-otic one) leads to a ∆Neff contribution that is largerthan the base Nnaturalness case. This increase is typi-cally not excluded by current bounds, indicating a large

6

degree of liberty in the structure and number of exotichidden sectors.

B. Standard Sector Contributions

Within the context of vanilla Nnaturalness, the major-ity of contributions arise from standard sectors. This isexplored in detail in [38]; here we briefly summarize thesearguments. All additional standard sectors are very sim-ilar to our own: they have the same particle content andcouplings and differ only by the Higgs mass. As our sec-tor is taken to be the lightest so as to be preferentiallyreheated, every other standard sector features an earlierfreeze-out of their respective particles. This ultimatelyleads to each sector having at most the same number ofrelativistic degrees of freedom as the SM.

In [38], the standard sector contributions are expressedas:

∆Neff =1

ρusν

∑i 6=us

ρi. (28)

In the case that the reheaton is lighter than the lightestHiggs (ours), this can be expressed as

∆Neff ∼Nb∑i=1

1

2i+ 1+y2c

y2b

Nc∑i=Nb+1

1

2i+ 1

' 1

2

(log 2Nb +

y2c

y2b

logNcNb

) (29)

with yc,b representing the charm and bottom Yukawacouplings, respectively, and

Nb,c =

(m2φ

8m2b,c

− 1

2

)(30)

with mφ being the mass of the reheaton.Application of these results indicates that for a major-

ity of the parameter space, vanilla Nnaturalness requiresmild fine-tuning (r in Eq. (4) set to a value . 1). Nu-merical results for the fine-tuning required for variousreheaton masses were presented in [38].

C. Generalized Reheating Scenarios

The generalization of possible reheating mechanismsmentioned in section IV — where the reheating mech-anism no longer depends on the Higgs’ mass parameterof a given sector — opens up a wide range of hiddensectors for study. Specifically, this allows mirror sectorswith large Higgs VEVs to be reheated to significant en-ergy densities and thus produce gravitational waves withenough power to be detected. Crucially, despite thisanalysis being limited to mirror sectors with large Higgsmasses, this analysis pertains to any strong, confining

phase transition at high scales. SinceNeff constraints re-main our strongest cosmological bounds for massive stan-dard sectors, our starting point for exploring the limitsof high transition temperatures is Eq. (25). Assumingheavy, standard sectors (with the only relativistic par-ticles being photons and neutrinos) we can saturate thebounds of Eq. (21) and solve for the maximum tempera-ture allowed for any number of sectors:

Ti ∼ 0.38TSM 1 hidden sector,

Ti ∼ 0.25TSM 5 hidden sectors,

Ti ∼ 0.21TSM 10 hidden sectors,

Ti ∼ 0.12TSM 100 hidden sectors,

(31)

where all the hidden sectors have the same temperatureas one another.

Using these restrictions, we can examine the behaviourof standard sectors with a much larger VEV than ourown. In terms of the Nnaturalness framework, thismeans we can get an SFOPT for QCD if we look at sec-tors with i greater than the critical index of Eq. (14)where all the quark masses are above the QCD confine-ment scale, as long as their temperatures are below thebounds presented here.

VI. GRAVITATIONAL WAVE SIGNALS

We now turn to the gravitational wave signatures ofour setup. At high temperatures, each of the hidden sec-tors has QCD in the quark/gluon phase, but at temper-atures around ΛQCD,i, the ith sector undergoes a phasetransition into the hadronic phase that we computed forthe different sectors in Sec. III. As discussed in that sec-tion, this phase transition will be strongly first order(SFOPT) for certain numbers of light quarks, which willgenerate gravitational waves. This differs from QCD inthe SM sector, as the PT is a crossover and not firstorder [60]. A SFOPT proceeds through bubble nucle-ation, where bubbles of the hadronic phase form in thevacuum of the quark phase. These bubbles will expand,eventually colliding and merging until the entire sectoris within the new phase. These bubbles are described bythe following Euclidean action [61]:

SE(T ) =1

T

∫d3x

[1

2(∇φ)2 + V (φ, T )

], (32)

where the time component has been integrated out dueto nucleation occurring not in vacuum but in a finitetemperature plasma. φ is the symmetry-breaking scalarfield with a nonzero VEV. In the case of the chiralphase transition, the scalar field breaking the SU(Nf )R× SU(Nf )L chiral symmetry is the effective quark con-densate φi ∼ 〈qq〉i of the respective sector. We leavethe thermalized potential V (φ, T ) general. As previouslystated, an exact QCD potential at the time of the chiralphase transition is not well understood outside of lat-tice results. In [31] chiral effective Lagrangian was used

7

to calculate a low-energy thermalized potential for con-fining SU(N). The amount of energy density dumpedinto the individual sectors dictates the energy budget forthe PT and hence for the gravitational waves. Assumingthat the SM sector is radiation dominated, a quantitythat characterizes the strength of the PT is the ratio ofthe latent heat of the phase transition, ε, to the energydensity of radiation, at the time of nucleation [62],

α ≡ ε

g∗π2(Tnucγ )4/30, (33)

with ε being calculable from the scalar potential. As-suming that there is a negligible amount of energy beingdumped back into the SM, which would cause significantreheating of ργ , the latent heat ε should correspond tothe energy density of the hidden sector going through thePT. The parameter g∗ in the denominator of Eq. (33) isthe number of relativistic degrees of freedom at the timeof the phase transition, with contributions from speciesin both the visible and dark sectors. It has weak tem-perature dependence in a single sector, but when dealingwith multiple hidden sectors, g∗ gains contributions fromall N sector’s relativistic degrees of freedom, weighted bytheir respective energy densities

g∗ = g∗,γ +∑i

g∗,i(ξi)4, (34)

with ξ being the temperature ratio defined in Eq. (24).The bounds from effective number of neutrinos [45] meanthat ξi . 1 for all i, so g∗ ≈ g∗,γ . In the case of darkQCD-like chiral phase transitions, the temperature of thephase transition is on the order of the symmetry-breakingscale of the respective sector, T ih ∼ O(ΛQCD,i). The workof [31] calculated α with an effective chiral Lagrangianhas found various upper bounds. We take the optimisticscenario where the numerator is bounded above by thesymmetry breaking scale

αi ≈ ξ4i ≈

(ΛQCD,iTnucγ

)4

, (35)

where Tnucγ is the temperature of the SM photon bath atthe time of the phase transition. Another important pa-rameter to characterize the phase transition is its inversetimescale β [16]. The inverse timescale can be calculatedusing the action in Eq. (32):

β ≡ dSE(T )

dt

∣∣∣∣t=tnuc

. (36)

The ratio of β and the Hubble constant, at the time ofnucleation, H controls the strength of the gravitationalwave (GW) signal,

β

H= Tnuch

dSE(T )

dT

∣∣∣∣T=Tnuch

. (37)

Due to the lack of a general analytic QCD potential,it is not possible to use Eq. (37) to calculate β/H.

There are dimensional arguments [13, 14] that predictβ/H ∼ 4Log(Mp/ΛQCD,i), although these argumentsmake specific assumptions about the potential. In morerecent work, some authors [31, 34] have attempted to es-timate it using first-order chiral effective theories and thePolyakov-Nambu-Jona-Lasinio models which motivates aβ/H of O(104). These studies claim a large range of val-ues with no consensus reached on the precise order of thescaled inverse timescale. Under these circumstances, oursignal projections will consider both extremes of the pa-rameter space where, β/H ∼ 10 − 104. A more realisticscenario may exist in between both cases.

A. Production of Gravitational Waves

Gravitational waves are produced with contributionsfrom different components of the SFOPT’s evolution. Itis commonplace to parameterize the spectral energy den-sity in gravitational waves by [63]

ΩGW(f) ≡ 1

ρc

dρGW(f)

d log(f), (38)

where ρc = 3H2/(8πG) is the critical energy density. Thetotal gravitational wave signal is a linear combination ofthree leading contributions:

h2ΩGW ≈ h2Ωφ + h2Ωv + h2Ωturb. (39)

Each components is scaled by its own unique efficiencyfactor, κ. The three leading-order contributions to theGW power spectrum are as follows:

• Scalar field contributions Ωφ: Caused by colli-sions of the bubble walls, the solutions being com-pletely dependent on the scalar field configuration,with efficiency factor κφ = 1− α∞/α [64, 65].

• Sound wave contributions Ωv: Sound waveswithin the plasma after bubble collision will pro-duce β/H enhanced gravitational waves, with effi-ciency factor κv ∝ α∞/α [66].

• Magnetohydrodynamical contributions ΩB :Turbulence within the plasma, left over from thesound wave propagation, will produce gravitationalwaves with efficiency factor κturb ≈ 0.1κv [67].

The parameter α∞ denotes the dividing line between therunaway regime (α > α∞) and the nonrunaway regime(α < α∞). Explicitly [16, 32, 62],

α∞ =(Tnuch )2

ρR

∑bosons

ni∆m2

i

24+

∑fermions

ni∆m2

i

48

,(40)

for particles with ni degrees of freedom that obtain massthrough the phase transition.

The exotic sectors have essentially massless degrees offreedom pre phase transition and pions with negligible

8

masses post phase transition. Other composite particles,such as baryons, do gain a mass of the order of Λex;this is, however, still much smaller than the order of ρRleading to small α∞ according to Eq. (40).4 Heavy stan-dard sectors that undergo SFOPT for QCD feature nobaryons due to all quarks being above their respectiveQCD scales. They do, however, feature glueballs thatobtain a mass of the order of the SFOPT and, just as inthe case for exotic sectors above, feature small α∞.

Each component of the spectral energy density inEq. (38) is proportional to a power of their respective ef-ficiency factors κ. The relative strength of the efficiencyfactors is dependent on the ratio α∞

α — since both α∞and α are parameterically small [31, 34], a range of pos-sible scenarios can occur. Here, we discuss the two endsof this spectrum: pure runaway walls and pure nonrun-away walls. The former scenario with runaway bubblewalls leads to the efficiency factors for the sound waveand magnetohydrodynamics (MHD) contributions beingsmall and ensures GWs are dominantly produced frombubble collisions, h2ΩGW ≈ h2Ωφ: this is what we as-sume for the remainder of this section. In the latter case,bubbles are nonrunaway (but vw ∼ 1 still [32, 68]) suchthat nonbubble collision contributions being important— ultimately leading to significant changes to the GWprofile. This case and the gravitational waves it producesare examined in the Appendix. Intermediate results areof course possible and would feature profiles somewherein between the two extremes.

The form of the GW energy density at the time ofnucleation is given by [32]

h2Ω∗GW = 7.7× 10−2

(κφα

1 + α

)2(H

β

)2

S(f) (41)

where we use v = 1 for runaway bubbles. Quantitiessuch as Ω∗GW that are calculated at the time of nucle-ation are denoted with an asterisk, and they must thenbe evolved to relate to their values at the time of obser-vation. S(f) is the spectral shape function for the signaland a parametric from has been found through numericalsimulations [65] of bubble wall collisions:

S(f) =3.8 (f/fp)

2.8

1 + 2.8 (f/fp)3.8. (42)

The peak frequency fp is a function of the temperatureof the SM at the time of nucleation. The various hiddensectors can phase transition at different scales and there-fore temperatures, causing a shift in the GW spectrum’speak frequency given by [65]

fp = 3.8× 10−8 Hz

(β

H

)(Tγ

100 GeV

)(g∗

100

) 16

, (43)

4 It should also be noted that although free quarks cease to existpost phase transition in these exotic sectors, their masses are solight that they do not contribute relevant amounts to α∞.

where g∗ is calculated using Eq. (34), although, due tothe lack of substantial reheating into the hidden sectors,the SM contribution is dominant.

Now that the framework has been laid out for the cre-ation of GW from a single SFOPT, we generalize to mul-tiple sectors going under independent, coherent SFOPT.In the models presented in this paper, we consider a sub-set of N hidden sectors that undergo a phase transition ata SM temperature of T iγ . As the GWs propagate in freespace, the energy density and frequency spectrum, at thetime of production Ω∗GW(f), will redshift to today’s valueΩ0

GW(f) = AΩ∗GW((a0/a)f). The redshifting factor Aaccounts for the redshifting of both ρGW and ρc [32, 69],

A ≡(a

a0

)4(H

H0

)2

(44)

where a (a0) and H (H0) are the scale factor and Hubbleconstant at the time of nucleation (observation), respec-tively. Assuming that the sectors are completely decou-pled before and after their respective SFOPT, the totalGW signal that would be measured today is given by thecoherent sum

ΩGW =

N∑i

Ai Ωi,∗GW((a0/a)if). (45)

We assume that the parameters of the SFOPT do not dif-fer between sectors: the relativistic degrees of freedom,phase transition rate, and the dark QCD scale, are allsimilar. This makes the redshifting factor Ai indepen-dent of sector number. Applying this to the standardreheating scenario of Nnaturalness, introduced in IV,we get GW signals as seen in Fig. 1. Plotted are theindividual contributions to the signal from each phasetransitioned sector, as well as the coherent sum of allsectors. Future GW interferometers and pulsar timingarray sensitivity curves are shown in comparison to thesignal. The sensitivity curves are interpreted as the re-gion of possible detection if intersected with the GW sig-nal, and the construction of these curves is detailed inSection VI B. Notice that the total signal is dominatedby the first sector’s contribution. This is caused by thequartic temperature ratio suppression in Eq. (33) and thelarge temperature gaps between adjacent sectors. Such asuppression leads to standard Nnaturalness evading fu-ture detector thresholds by a few orders of magnitude inunits of energy density.

This is not the case if we consider more generalized re-heating scenarios. Once the restriction that sectors withsmall Higgs masses are preferentially reheated has beenlifted, we can explore a much more vast landscape of hid-den sectors than are allowed in the reheaton case. Here,we construct several different scenarios that are both de-tectable and demonstrate a variety of gravitational waveprofiles. Specifically, we explore benchmarks that lead toa deviation in the peak behaviour of the total GW sig-nal (the superposition of stochastic GW from individualSFOPT) from a standard power law signal.

9

FIG. 1. Gravitational wave spectral energy density (solidcurves) for standard Nnaturalness using the scalar reheatonmodel of section IV. The curve corresponding to the sum ofthe sectors is approximately equal to the i = 1 curve. Allcontributions are assumed to be purely from bubble colli-sions Ωφ. The colored solid lines use β/H = 10 whereasthe dashed gray line is the total contribution of all sectorsfor β/H = 104 (the sum of all sectors is roughly equal to thei = 1 curve and sectors beyond the first are below the rangeof this plot). The shaded dashed curves are the power lawnoise curves [70] calculated from expected sensitivity as de-scribed in Section VI B. The ones on the right are space-basedinterferometers: LISA [8] (blue), DECIGO [11] (light blue),BBO [5] (red). The ones on the left are for the pulsar timingarray SKA [7] for exposure time of 5-years (purple), 10-years(orange), and 20-years (green).

It should be noted that the key phenomenological con-straint on all of these models is ∆Neff , giving us a max-imum allowed temperature ratio (when compared to theSM) for each reheated hidden sector: Eq. (31) shows themaximum temperature ratios for specific numbers of ad-ditional hidden sectors. Due to the rather harsh scalingof the GW strength α with the temperature ratio shownin Eq. (35), we take the optimistic approach of keepingthe temperature ratio as high as allowed by CMB datafor all of the hidden sectors.

In the following, we focus on heavy standard sec-tors — pure Yang-Mills sectors with much heavier parti-cles (specifically quarks) and, as shown in Sec. III, theSFOPT these entail. The reason for this arises fromEq. (43): every exotic sector features a phase transitionthat occurs at Λex ∼ 90 MeV. If we maximize the al-lowed temperature ratio, this gives us a (SM) photontemperature Tγ that places our signal directly in the fre-quency void between the detection region of pulsar timingarrays and space-based interferometers (see Sec. VI B).The location of the peak can be changed by droppingthe temperature ratio, but the adjustment required toend up with a signal with an appropriate peak frequencymakes the overall signal too weak to detect. As shownin Sec. III, standard sectors can have much higher tem-perature phase transitions. As such, maintaining themaximum allowed temperature ratio between the hid-

den sector(s) and the SM gives a much larger photontemperature and a proportionally larger peak frequency;ultimately allowing for detection by space-based interfer-ometers.

There are four scenarios that we examine, with keyparameters presented in Table I.

• Maximized signal: A single additional heavy hid-den sector reheated to a temperature that satu-rates current experimental bounds. The SM photonbath temperature at the time of the hidden sectorPT is 87 GeV. In the Nnaturalness framework thisis equivalent to reheating a standard sector withi ∼ 1016 up to the maximum allowed temperatureratio.

• Large split scenario: A scenario where two ad-ditional hidden sectors have been reheated — thesesectors have Higgs VEVs that are split by a factorof

vh1

vh2=√

103. (46)

This results in a difference in the scale of theSFOPTs leading to the SM photon bath temper-ature changing a large amount during the time be-tween the PTs. This, in turn, leads to a large sep-aration in the peak frequency of their gravitationalwave signals. In the Nnaturalness framework thisis equivalent to reheating two standard sectors, onewith i ∼ 1012 and another with i ∼ 1015 up to themaximum allowed temperature ratio.

• Medium split scenario: Similar to the previouscase: these sectors have Higgs VEVs that are splitby a factor of

vh1

vh2=√

10, (47)

resulting in a much smaller difference in the peakfrequency of their gravitational wave signals. Inthe Nnaturalness framework this is equivalent toreheating two standard sectors, one with i ∼ 1012

and another with i ∼ 1013 up to the maximumallowed temperature ratio.

• Five sector scenario: Five sectors are reheatedto the maximum allowed temperature ratio, eachwith VEVs that are

(vhi)/(vh(i+1)) ∼√

3 (48)

larger than the previous sector.

In all cases where multiple sectors are reheated, we as-sume for simplicity that all the hidden sectors are re-heated to the same temperature.

The GW results of these cases are presented in Fig. 2.In all cases, the summed GW signal is detectable by oneor more proposed interferometers. When changing the

10

Parameters for multi-hidden sector benchmarks

Maximized signal

Sector Higgs VEV (GeV) ΛASQCD (GeV) Tγ (GeV) Index

1 24.6 × 109 38.6 87.7 1016

Large split

Sector Higgs VEV (GeV) ΛASQCD (GeV) Tγ (GeV) Index

1 246 × 106 10.8 30.3 1012

2 7.8 × 109 28.1 78.6 1015

Medium split

Sector Higgs VEV (GeV) ΛASQCD (GeV) Tγ (GeV) Index

1 246 × 106 10.8 30.3 1012

2 778 × 106 14.9 41.6 1013

Five sector

Sector Higgs VEV (GeV) ΛASQCD (GeV) Tγ (GeV) Index

1 246 × 106 10.8 38.8 1012

2 426 × 106 12.6 45.2 3 × 1012

3 778 × 106 14.9 53.3 1013

4 1.3 × 109 17.3 62.1 3 × 1013

5 2.5 × 109 20.5 73.3 1014

TABLE I. Outline of parameters used for the various multi-hidden-sector scenarios. The Higgs VEV is the VEV for the givenadditional sector, ΛASQCD is the QCD phase transition in the additional sector, and Tγ is the temperature of the SM photonbath when the SFOPT occurs in the additional sector. The index indicates the equivalent sector from the Nnaturalness model(Eq. (4)). It should be noted that although the various sectors undergo phase transitions at different temperatures, they areall assumed to be reheated to the same initial temperature.

assumptions on β/H, the scenarios in Fig. 2 are still de-tectable for values ranging between O(1) and O(100). Asβ/H increases (decreases) the peak frequency moves tohigher (lower) frequencies, dictated by Eq. (43), whereasthe amplitude decreases (increases) shown in Eq. (41).

The frequency dependence in Eq. (42) takes the form off/fp, this causes a cancellation between the redshiftingfactors. As multiple sectors phase transition at differenttimes, and therefore different SM photon temperatures,the peaks will shift relative to each other, purely fromthe linear temperature dependence of the peak frequencyfp ∼ Tγ given in Eq. (43). This is seen in Fig. 2, wherethe spectrum peaks are shifted causing a peak broadeningof the summed spectrum. The broadening can be sub-stantial if the hidden sectors transition between a largegap of time (temperature). Eventually, a temperaturelimit will be reached where two (or multiple) distinctpeaks will be visible, provided that the amplitudes arecomparable.

B. Detection of Stochastic Graviational Waves

A stochastic gravitational wave background could bedetectable if the signal-to-noise ratio (SNR) is abovesome threshold value, ρ > ρth, dictated by the capabil-ities of future interferometers and pulsar timing arrays(PTAs). These interferometers or PTAs quote their ex-

perimental sensitivies in terms of spectral noise curves,Seff(f), which can be translated into units of energy

density through h2Ωeff(f) = 2π2

3H2 f3Seff(f). If the ex-

periment uses a single (multiple) detector, the autocor-related (cross-correlated) SNR is used in comparing tothe threshold value ρth. The autocorrelated and cross-correlated SNR are explictly given as [71],

ρ2 = T∫ fmax

fmin

df

(h2ΩGW(f)

h2Ωeff(f)

)2

(autocorrelated)

ρ2 = 2T∫ fmax

fmin

df

(h2ΩGW(f)

h2Ωeff(f)

)2

(cross-correlated),

(49)

where T is the exposure time of the experiment. Theintegration covers the entire broadband range of fre-quencies (fmin, fmax). LISA [8] and B-DECIGO [11] areproposed to be single-detector interferometers, whereasBBO [5] and DEICIGO [10] would be built from an arrayof multiple interferometers. GW signals produced froman early cosmological phase transition would be seen asa stochastic background. Assuming that the GW fol-lows a power law background in frequency, it is common-place to quote the power law integrated (PLI) sensitivitycurves [70]. The PLI curves are constructed using infor-mation from the power law form of the signal,

h2ΩGW(f) = h2Ωγ

(f

fref

)γ(50)

11

FIG. 2. Gravitational wave spectral energy density for the various scenarios found in Table I. All contributions are assumedto be purely from runaway bubble collisions Ωφ. The colored solid lines use β/H = 10 where as the dashed gray line is thetotal contribution of all sectors for β/H = 104. The inset is a closer look at the region around the peaks for the β/H = 10case. The shaded curves are the same as Fig. 1.

where γ is the spectral index of the power law, and fref

is an arbitrary reference frequency which has no effect onthe PLI sensitivities. h2Ωγ is the energy density calcu-lated using Eq. (49) with spectral index γ and referencefrequency fref. The method of calculating the PLI curvesinvolves plotting h2ΩGW(f), using Eq. (50), for variousspectral indices γ and for some fixed threshold value ofρth. Each curve will lay tangent to the PLI curve, moreformally,

h2ΩPLI = maxγ

[h2Ωγ

(f

fref

)γ]. (51)

The spectral noise curves used to create the PLI curvesshown in Figs. 1 and 2 were taken from [11, 32, 72–74]for the interferometers and [7, 32] for the Square Kilome-ter Array (SKA) pulsar timing array. We have assumedan observation time of T = 4 years for the interferom-eters and T = 5, 10, 20 years for the various stages ofSKA. In the case of the PTA experiments, the sensitiv-ity curves are dependent on how frequently the pulsar’stiming residuals, δt, are measured. When using Eq. (49)to construct the PLI curves for SKA, the upper integra-tion bound is inversely proportional to pulsar’s timingresidual, fmin = 1/δt. In this work, it is assumed thatδt = 14 days, but this may underestimate the capabili-

ties of SKA as well as the cadences of the pulsar popula-tions. If the timing residuals are lowered the maximumfrequency reach of SKA increases, and the correspondingPLI curves in Figs. 1 & 2 are shifted to the right, possi-bly giving the PTAs sensitivity to some of the scenariosconsidered here.

VII. CONCLUSION

As detection capabilities increase, gravitational wavesignals continue to grow in importance as phenomenolog-ical signatures that can offer us a unique glimpse into theuniverse as it was in the early epochs. The space-basedinterferometers planned for the next generation of GWexperiments will be sensitive enough to begin searchingfor signals of the cataclysmic disruption of space-time dueto SFOPT. As we inch closer to these measurements be-coming available, it becomes important to develop waysto analyze and understand this data.

Here, we examined scenarios, including Nnaturalness,that involve multiple hidden sectors and calculated theGW profiles present. Our GW projections demonstratethat although Nnaturalness with the reheaton scenariopresented in [38] is not projected to be detectable in

12

the near future, more generalized scenarios with multi-ple hidden sector SFOPTs are in an observable regionand will begin to be probed by next-generation space ex-periments. Both cases feature important parts of theirGW signals in the void between frequencies detectableby pulsar timing arrays and space-based interferometers— providing theoretical impetus for new experiments ca-pable of probing this region of frequency space.

Further, our results provide a framework for under-standing and using GW signals in two different ways:first as a unique signal for specific theories featuring mul-tiple SFOPTs and also as a challenge to broaden the un-derstanding of GW detector sensitivity.

In the former case, this demonstrates the power of GWsignals to probe deep into the unknown arena of complexhidden sectors. Individual SFOPTs are understood tocreate GWs that are assumed to follow an approximatepower law. If a model predicts the presence of two, five,or more additional sectors, or features a single extra sec-tor with multiple PTs, deviations from a standard powerlaw can occur. The multiple transitions that occur inthe models outlined here create signals that follow thistrend: although the individual GWs do obey approxi-mate power laws, their sum does not — leading to aunique signal indicating so-called dark complexity. Ex-plicitly, a broadening or distortion of the signal aroundthe peak frequency, precisely where the signal has themost energy, could point to a multi-SFOPT scenario andgently guide us in the direction of multiple hidden sec-tors.

Shifting to the other part of our framework, our resultsleads to the question “how well can experiments probenon-power-law signals?” For frequency ranges away fromthe peak of the total, GW signals the quoted detectionthresholds should hold: the signals fall off as a power lawto a very good approximation. However, for areas aroundthe peak frequency the answer is less clear; the PLI curvesare built under the assumption of a power law. Work hasbeen done [75] in examining GW signals using peak am-plitudes and peak frequencies as the defining observables:this is rooted in the assumption that GW signals have amodel-independent spectral shape around peak frequen-cies. However, our results indicate that this assumptionof model independence cannot hold for all cases: sec-tors with similar (but different) transition temperaturescan create either peak broadening or multihump featuresthat differ significantly from a standard power law shape.This points to the need for future work to better under-stand where the power law approximation breaks downand how this affects detection prospects for the next gen-eration of GW detectors.

ACKNOWLEDGEMENTS

We thank Yang Bai, Djuna Croon, Kevin Earl, andYuhsin Tsai for helpful discussions. P.A.S. and D.L.both thank the 2018 Theoretical Advanced Study In-

stitute in Elementary Particle Physics Program (TASI)for setting up a platform for discussion which seededthe genesis of this project. This work was supportedin part by the Natural Sciences and Engineering Re-search Council of Canada (NSERC). D.L. acknowledgessupport from the NSERC Postgraduate Scholarships-Doctoral Program (PGS D).

APPENDIX: NONRUNAWAY PHASETRANSITIONS

If the phase transition occurs in the nonrunawayregime (α∞ > α), the dominant contributions to the GWenergy densities are given by the sound wave h2Ωv andMHD h2Ωturb components [32],

h2ΩGW ≈ h2Ωv + h2Ωturb. (52)

The new contributions to the GW energy density takeon a different form from Eq. (41) [16],

h2Ω∗v = 1.6× 10−1 v

(κv α

1 + α

)2(H

β

)1

Sv(f),

h2Ω∗turb = 2.01× 101 v

(κturb α

1 + α

)3/2(H

β

)1

Sturb(f).

(53)

Unlike the runaway case, we do not assume v = 1 dueto the bubbles reaching a terminal velocity. The MHDefficiency factor is a fraction of the sound waves, κturb =εκv. Current simulations have motivated a range of ε ∼0.05− 0.10 [16], where we take the optimistic case of ε =0.10. Both contributions have unique spectral shapes,given to be [32],

Sv(f) = (f/fp,v)3

(7

4 + 3(f/fp,v)2

)7/2

,

Sturb(f) =(f/fp,turb)

3

(1 + f/fp,turb)11/3(1 + 8π(f/H)).

(54)

The MHD energy density has a spectral shape dependenton the Hubble rate at the time of nucleation, H. Simi-lar to the scalar spectral shape in Eq. (42), the frequen-cies are scaled by their respective temperature-dependentpeak frequency:

fp,v = 1.9× 10−5 Hz1

v

(β

H

)(Tγ

100 GeV

)(g∗

100

) 16

,

fp,turb = 2.7× 10−5 Hz1

v

(β

H

)(Tγ

100 GeV

)(g∗

100

) 16

.

(55)

We evolve the frequencies and energy densities with thesame redshift factors, Eq. (44) from Sec VI. Fig. 3 shows

13

FIG. 3. Gravitational wave spectral energy density for the various scenarios found in Table I. All contributions are assumedto be from a nonrunaway phase transition with terminal velocity v = 0.95. The coloured solid lines use β/H = 10 whereas thedashed grey line is the total contribution of all sectors for β/H = 104 . The top left figure shows the individual sound waveand MHD contributions. The shaded curves are the same as Fig. 1. In contrast to the runaway case, most scenarios evade theprojected sensitivities.

the sum of the GW energy density for the scenarios inTable I, but instead for the nonrunaway case, with a ter-minal velocity of v = 0.95. For this case we see that thespectra tend to be shifted to lower frequencies and aremore likely to fall in the gap between the interferome-ters and the pulsar-based detectors. On the other hand,larger values of β/H increase the typical frequency, sothis case becomes more sensitive in some scenarios tovalues of β/H on the larger end of the considered range.

In the sound wave case, these parameterizations areextracted from simulations with β/H < 100 correspond-ing to a long-lasting sound wave component. For highβ/H the transition timescale from sound wave to MHDturbulence is much shorter than the Hubble time. When

estimating the model expectations for β/H = 104, weenter a regime at which Eq. (53) may be overestimat-ing the sound wave contribution. Investigations of thisregime have been done in [76–78]. This effect, however,only affects the amplitude of the signal, and our workfocuses on the unique spectral shapes that are formed inthese models. Therefore we project our results for highβ/H in Fig. 3 to motivate the novel spectral profiles.

Finally, we note that because the sound wave andMHD contributions have different spectral shapes, theoverall spectrum has a kink at a frequency above thepeak. In the top left panel of Fig. 3, we show the twocontributions separately in addition to their sum to high-light this effect.

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