Stasis in an Expanding Universe:A Recipe for Stable Mixed-Component Cosmological Eras

Keith R. Dienes,1, 2, ∗ Lucien Heurtier,3, † Fei Huang,4, 5, ‡ Doojin Kim,6, §

Tim M.P. Tait,5, ¶ Brooks Thomas7, ∗∗

1Department of Physics, University of Arizona, Tucson, AZ 85721 USA2Department of Physics, University of Maryland, College Park, MD 20742 USA

3IPPP, Durham University, Durham, DH1 3LE, United Kingdom4CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,

Chinese Academy of Sciences, Beijing 100190, China5Department of Physics and Astronomy, University of California, Irvine, CA 92697 USA

6Mitchell Institute for Fundamental Physics and Astronomy,Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843 USA

7Department of Physics, Lafayette College, Easton, PA 18042 USA

One signature of an expanding universe is the time-variation of the cosmological abundances ofits different components. For example, a radiation-dominated universe inevitably gives way to amatter-dominated universe, and critical moments such as matter-radiation equality are fleeting.In this paper, we point out that this lore is not always correct, and that it is possible to obtaina form of “stasis” in which the relative cosmological abundances Ωi of the different componentsremain unchanged over extended cosmological epochs, even as the universe expands. Moreover, wedemonstrate that such situations are not fine-tuned, but are actually global attractors within certaincosmological frameworks, with the universe naturally evolving towards such long-lasting periods ofstasis for a wide variety of initial conditions. The existence of this kind of stasis therefore givesrise to a host of new theoretical possibilities across the entire cosmological timeline, ranging frompotential implications for primordial density perturbations, dark-matter production, and structureformation all the way to early reheating, early matter-dominated eras, and even the age of theuniverse.

CONTENTS

I. Introduction, motivation, and basic idea 1

II. Stasis: General considerations 4A. Minimal condition for the existence of stasis 4B. Conditions for the persistence of stasis 5

III. A model of stasis 6

IV. Stasis as a global attractor 11

V. Stasis in the presence of additional energycomponents 15

VI. Discussion and cosmological implications 17

Acknowledgments 25

A. Stasis as a cosmological coherent state? 26

References 27

∗ Email address: dienes@arizona.edu† Email address: lucien.heurtier@durham.ac.uk‡ Email address: huangf4@uci.edu§ Email address: doojin.kim@tamu.edu¶ Email address: ttait@uci.edu∗∗ Email address: thomasbd@lafayette.edu

I. INTRODUCTION, MOTIVATION, ANDBASIC IDEA

One of the earliest and most profound discoveries ofmodern cosmology is that we live in an expanding uni-verse. With this one discovery, the age-old paradigm ofan everlasting static universe was overthrown, replacedby a universe whose fundamental characteristics are time-dependent. Chief among these characteristics are theabundances of the different components which contributeto its energy density. Indeed, it is traditional to refer tothe different epochs through which the universe evolvesin terms of the abundances which dominate during thoseepochs, with the current paradigm positing that the uni-verse passed from an initial inflationary epoch domi-nated by vacuum energy to a reheating epoch dominatedby the energy of an oscillating inflaton to a radiation-dominated post-reheating epoch to a matter-dominatedpost-reheating epoch — one which is only now giving wayto a second epoch dominated by vacuum energy.

This passage from epoch to epoch is almost inevitable.Indeed, according to the Friedmann equations, cosmo-logical expansion induces a redshifting effect that causesthe abundances of the different energy components ofthe universe to scale with time in different ways. It cantherefore happen that the smallest one now will later bevast (just as the present now will later be past); theseabundances are constantly changing.

As an example, let us focus on the energy densitiesρM and ργ associated with matter and radiation re-

arX

iv:2

111.

0475

3v2

[as

tro-

ph.C

O]

18

Jan

2022

2

spectively, along with their corresponding abundancesΩM ∼ ρM/H2 and Ωγ ∼ ργ/H2, where H(t) = a/a is theHubble parameter and a(t) is the scale factor. In general,ρM and ργ evolve as a−3 and a−4 respectively, but thetime-dependence of the scale factor a(t) in turn depends(through the Friedmann equations) on the instantaneousmix of components in the associated cosmology. We thusobtain an evolving non-linear system in which the valuesof the abundances ΩM and Ωγ at any moment influencetheir own instantaneous rates of change. For example,we might start in a radiation-dominated universe withΩγ ΩM , but in such a universe Ωγ remains approxi-

mately constant while ΩM ∼ t+1/2. As a result we findthat ΩM grows and eventually becomes significant com-pared to Ωγ . Indeed, with ΩM ≈ Ωγ we find ΩM ∼ t+2/7

while Ωγ ∼ t−2/7. Thus ΩM continues to grow even be-yond Ωγ . Eventually we enter a matter-dominated epochwith ΩM Ωγ , whereupon ΩM remains approximately

constant while Ωγ ∼ t−2/3. Our radiation-dominatedepoch has thus become a matter-dominated epoch, sim-ply as a result of cosmological expansion. In a similarway, a matter-dominated epoch generally gives way toa vacuum-energy-dominated epoch. Indeed, special mo-ments such as those exhibiting matter-radiation equalityare fleeting, since even at the instant when ΩM = Ωγwe see that ΩM is growing while Ωγ is shrinking. Thelesson, then, seems clear: In an expanding universe, therelative sizes of the different contributions are continuallyin flux. As a result, epochs containing non-trivial mix-tures of energy components are generally unstable, withcomponent ratios such as ΩM/Ωγ perpetually evolving intime regardless of where such epochs might be situatedalong the cosmological timeline.

In this paper, we shall demonstrate that this generalexpectation need not always hold true. In particular, weshall demonstrate that it is possible to construct scenar-ios in which such mixed-component cosmological eras canbe stable over extended epochs lasting as many e-foldsas desired, with values of ΩM and Ωγ remaining strictlyconstant despite cosmological expansion. In other words,we shall demonstrate that it is generally possible to havelong-lasting epochs which are not matter-dominated orradiation-dominated, and not necessarily dominated byany particular component at all! We shall refer to suchepochs as periods of “stasis”. For example, we shall pro-vide an explicit model that gives rise to an extended sta-sis epoch exhibiting strict matter-radiation equality, withΩM = Ωγ holding throughout. Extended epochs lastingarbitrary numbers of e-folds can also be constructed ex-hibiting other ratios between ΩM and Ωγ .

As we shall demonstrate, these scenarios emerge natu-rally in realistic scenarios of physics beyond the StandardModel. Moreover, we shall demonstrate that these stasisstates are not fine-tuned, and are indeed global dynami-cal attractors within these scenarios. Thus, within thesescenarios, the universe need not begin within a periodof stasis in order for stasis to arise — the universe willnecessarily evolve into a stasis state for a wide variety

of initial conditions. Finally, we shall find in all casesthat the stasis state also has a natural ending after whichnormal cosmological evolution resumes. Thus our stasisepoch has both a beginning and an end — a feature whichpotentially allows it to be “spliced” into various pointsalong the standard cosmological timeline.

It may initially seem impossible to arrange such pe-riods of stasis between matter and radiation. After all,for the reasons discussed above, matter inevitably dom-inates over radiation; this is so intrinsic a prediction ofthe Friedmann equations that this conclusion seems un-avoidable. On the other hand, matter can decay backinto radiation. This then might provide a natural coun-terbalance to the effects of cosmological expansion, caus-ing the matter abundance ΩM to shrink while the radi-ation abundance Ωγ grows. Our idea, then, is a simpleone: can these two effects be balanced against each other?More specifically, can particle decay be balanced againstcosmological expansion in order to induce an extendedtime interval of stasis during which the matter and radi-ation abundances each remain constant?

Of course, particle decay is a relatively short process,localized in time. In order to have an extended periodof stasis we would therefore require an extended periodduring which particle decays are continually occurring.This would be the case if we had a large tower of mat-ter states φ` (` = 0, 1, ..., N − 1), with each state se-quentially decaying directly (or preferentially) into radi-ation. As is well known from the Dynamical Dark Mat-ter framework [1–3], many scenarios for physics beyondthe Standard Model give rise to precisely such towers ofdark-matter states. The question is then whether thesesequential decays down the tower could be exploited inorder to sustain an extended period of cosmological sta-sis.

This is clearly a tall order, and at first glance such abalancing might seem to be impossible. In order to appre-ciate the difficulties involved, let us consider how such anidealized scenario might work. A sketch of such a scenarioappears in Fig. 1, where we have illustrated the behaviorof the individual abundances Ω`(t) of each of the matterfields within the tower (blue). In general, as discussedabove, each abundance Ω`(t) initially grows according toa common power-law as the result of cosmological ex-pansion; for convenience and simplicity this power-lawgrowth is sketched within Fig. 1 as linear. However, oncethe appropriate decay time τ` is reached for each com-ponent (idealized in Fig. 1 as a sharp transition), thebehavior of Ω`(t) changes and now reflects an exponen-tial decay. Of course, our goal is for all of this to occur insuch a way that the sum

∑` Ω`(t) — i.e., the total mat-

ter abundance ΩM also shown in Fig. 1 (red) — remainsconstant.

Given this sketch, we can immediately see the compli-cations involved. First, in order to keep the sum ΩM con-stant, at any moment we need to somehow be cancellingthe power-law growth of the abundances Ω` of the lightercomponents which have not yet decayed against the expo-

3

Γℓ

Ω

ΩM

Γ`

t

ΩM =∑

`Ω` = constant

FIG. 1. A sketch of the basic stasis mechanism: the abundances Ω`(t) of individual matter components (blue) each experiencepower-law growth due to cosmological expansion before eventually experiencing exponential decay. For simplicity in thisidealized sketch, this power-law growth is shown as linear while the transition to exponential decay is sketched as sharp.Nevertheless this process conspires to keep the total matter abundance ΩM ≡

∑` Ω`(t) constant (red), thereby producing an

extended epoch of cosmological stasis. Note that this process is highly non-trivial: the power-law growth for some Ω` mustbe balanced against the exponential decays of the other Ω` while the exponent of the power-law growth is correlated with thevalue of the total ΩM through the Friedmann equations and the decay rate Γ` of each component is correlated with the timeτ` at which Ω` hits a maximum and the decay begins to dominate. As a result this system does not exhibit time-translationinvariance even though the total ΩM remains constant. Despite its complexity, we shall demonstrate that the stasis state notonly occurs naturally in well-motivated physics scenarios but is actually a global dynamical attractor in such situations, withthe system generically evolving towards stasis even if it does not begin in stasis.

nential decays of the abundances Ω` of the heavier stateswhich have. Second, we see that each successive Ω` mustreach a greater maximum value before decaying than didthe previous abundance Ω`+1, since with each decay wehave fewer remaining matter states contributing to ΩM .Third, it follows from the Friedmann equations that theexponent of the common power-law growth experiencedby each Ω` prior to decay must be correlated with the to-tal matter abundance ΩM — an observation which pro-vides a non-linear “feedback” constraint on our system.Finally, as our decays proceed down the tower, the de-cay lifetimes τ` are continually increasing. This impliesthat the corresponding decay widths Γ` are continuallydecreasing, which means that the successive exponentialdecays must occur with slower and slower rates. All ofthese features are illustrated in Fig. 1.

With all of these tight constraints, it is remarkable thatsuch a stasis state with constant ΩM can ever emerge.However, we shall demonstrate that this is exactly whatoccurs. As a result, the existence of this kind of stasisstate gives rise to a host of new theoretical possibilitiesacross the entire cosmological timeline, ranging from po-tential implications for primordial density perturbations,dark-matter production, and structure formation all the

way to early reheating, early matter-dominated eras, andthe age of the universe.

This paper is organized as follows. In Sect. II, we startby studying the stasis phenomenon itself and derive a setof mathematical conditions that must be satisfied withinany period of stasis. At this stage of our analysis, thevery notion of a stasis state implicitly requires that sucha state be truly eternal, without beginning or end. InSect. III, we then present a model of stasis — i.e., a gen-eral model which arises naturally in many extensions ofthe Standard Model and which generally satisfies theseconditions. Thus, our model gives rise to stasis. How-ever, we shall find that our model contains certain “edge”(or “boundary”) effects that cause the system to deviatefrom true stasis at times which are extremely early orlate compared with the time at which our tower of statesis originally produced. Thus, within our model, we shallfind that our stasis epoch is actually a finite one in whichthere exist both a natural entrance into as well as exitfrom stasis. This is ultimately a beneficial feature, imply-ing that our stasis state is ultimately of finite duration,after which normal cosmological evolution resumes. InSect. IV, we then study what happens when such sys-tems are not originally in stasis, and demonstrate that

4

the stasis state is a global dynamical attractor. Thus, re-gardless of the initial conditions, our system will alwayseventually enter into a stasis state and remain in stasisuntil all decays have concluded.

Taken together, these results demonstrate that the sta-sis state is both stable and robust. In Sect. V, we thenconsider various extensions of our results. In particu-lar, we study the behavior that emerges when additionalenergy components beyond radiation and matter are in-troduced into the cosmology. Finally, in Sect. VI, wesummarize our results and consider various possible theo-retical and phenomenological implications of stasis acrossthe cosmological timeline. We also outline ideas for fu-ture research.

We emphasize that our main interest in this paper isthe stasis phenomenon itself — i.e., the theoretical possi-bility that such stable mixed-component eras can be real-ized within an expanding universe. Needless to say, phe-nomenological constraints may make it difficult to intro-duce stasis epochs into certain portions of the standardcosmological timeline. For example, those portions of thetimeline after nucleosynthesis are deeply constrained byobservational data and therefore cannot be significantlymodified. Such stasis epochs nevertheless represent a vi-able phenomenological possibility during earlier periodsalong the cosmological timeline, such as during earlierportions of radiation-domination or even during reheat-ing. We shall therefore study stasis as a general theo-retical phenomenon throughout most of this paper, anddefer our discussion of its phenomenological implicationsto Sect. VI.

II. STASIS: GENERAL CONSIDERATIONS

Throughout this paper, “stasis” will refer to any ex-tended period during which the total matter and radi-ation abundances ΩM and Ωγ remain constant despitecosmological expansion. In this section, we provide ananalytical discussion of stasis, with the goal of obtain-ing mathematical conditions that characterize this stateand must therefore be satisfied therein. In particular, weshall do this in two separate steps:

• we shall first determine a condition that character-izes stasis at any moment in time — i.e., a minimalcondition necessary for stasis to exist ; and

• we shall then determine two additional conditionsthat must hold in order for stasis to persist over anextended period.

We shall now address each of these issues in turn.

A. Minimal condition for the existence of stasis

Let us begin by assuming a flat Friedmann-Robertson-Walker (FRW) universe containing only

• a tower of matter states φ` where the indices ` =0, 1, 2, .... are assigned in order of increasing mass;and

• radiation (collectively denoted γ) into which the φ`can decay.

We shall let ρ` and ργ denote the corresponding energydensities and Ω` and Ωγ the corresponding abundances.We shall also let Γ` denote the decay rates for the φ`.

Recall that for any energy density ρi (where i = `, γ),the corresponding abundance Ωi is given by

Ωi ≡8πG

3H2ρi (2.1)

where H is the Hubble parameter and G is Newton’sconstant. From this it follows that

dΩidt

=8πG

3

(1

H2

dρidt− 2

ρiH3

dH

dt

). (2.2)

We can simplify this expression through the use of theFriedmann “acceleration” equation for dH/dt, which inthis universe takes the form

dH

dt= −H2 − 4πG

3

(∑

i

ρi + 3∑

i

pi

)

= −H2 − 4πG

3

(∑

`

ρ` + 2ργ

)

= − 12H

2 (2 + ΩM + 2Ωγ)

= − 12H

2 (4− ΩM ) . (2.3)

Note that in passing to the second line of Eq. (2.3) wehave recognized that matter and radiation have w = 0and w = 1/3 respectively, where wi ≡ pi/ρi is theequation-of-state parameter for component i. Thus p` =0 and pγ = ργ/3. Likewise, in passing to the third line wehave defined the total matter abundance ΩM ≡

∑` Ω`,

and in passing to the fourth line we have imposed theconstraint ΩM + Ωγ = 1. Substituting Eq. (2.3) intoEq. (2.2) we then obtain

dΩidt

=8πG

3H2

dρidt

+HΩi (4− ΩM ) , (2.4)

yielding

dΩMdt

=8πG

3H2

∑

`

dρ`dt

+HΩM (4− ΩM )

dΩγdt

=8πG

3H2

dργdt

+HΩγ (4− ΩM ) . (2.5)

These are thus general relations for the time-evolution ofΩM and Ωγ in terms of dρ`/dt and dργ/dt. Of course,since ΩM + Ωγ = 1, it follows that dΩM/dt = −dΩγ/dt.From Eq. (2.5) we therefore obtain the self-consistencyconstraint

8πG

3H2

(∑

`

dρ`dt

+dργdt

)= H (ΩM − 4) , (2.6)

5

which is tantamount to asserting that ρ` and ργ are theonly contributions to the total energy density of the uni-verse.

Given the relations in Eq. (2.5), our final step is toinsert appropriate equations of motion for dρ`/dt anddργ/dt. It is here that we introduce the idea that theproduction of radiation γ comes from the decays of theφ`. Since each φ` is assumed to decay into radiation γwith rate Γ`, and given that each decay process conservesenergy, these equations of motion are given by

dρ`dt

= − 3Hρ` − Γ`ρ`

dργdt

= − 4Hργ +∑

`

Γ`ρ` . (2.7)

Note that these equations of motion indeed satisfy theconstraint in Eq. (2.6). The results in Eq. (2.5) thentake the form

dΩMdt

= −∑

`

Γ`Ω` +H(ΩM − Ω2

M

)(2.8)

with dΩγ/dt = −dΩM/dt. Note that ΩM−Ω2M = ΩM (1−

ΩM ) = ΩMΩγ .The differential equation for ΩM in Eq. (2.8) is com-

pletely general, describing the complete time-evolution ofΩM and Ωγ . It is important to realize that these differ-ential equations do not imply that ΩM or Ωγ are mono-tonic functions of time. In general, each Ω` comprisingΩM has a complicated time dependence: as evident fromEq. (2.8), the decay process tends to push Ω` downward,while the existence of an induced Ωγ > 0 in the back-ground cosmology tends to affect the Hubble expansionin such a way as to push Ω` upwards. Thus, each Ω` caneither rise or fall as a function of time, implying that ΩM— and therefore Ωγ — can likewise either rise or fall asa function of time. This is ultimately the result of thecompetition between the two terms on the right side ofEq. (2.8). Of course, for situations in which the decaywidths Γ` are all significantly greater than H, the effectsof the decays will dominate and in that case dΩM/dt willbe strictly negative, resulting in a monotonically fallingΩM .

Given the result in Eq. (2.8), we now seek a steady-state “stasis” solution in which ΩM and Ωγ are constant.Clearly such a solution will arise if the effects of the φ`decays are precisely counterbalanced by the Hubble ex-pansion. While there are many ways of seeking such asolution, we shall do this in two steps. First, we shallimpose the condition that dΩM/dt = 0. This then yieldsthe constraint

∑

`

Γ`Ω` = H(ΩM − Ω2M ) . (2.9)

This is clearly a necessary (but not sufficient) conditionfor stasis. In the next subsection, we shall determine the

additional conditions under which Eq. (2.9), once satis-fied at some time t∗, actually remains satisfied over anextended time interval.

Note that since 0 ≤ ΩM ≤ 1, both sides of Eq. (2.9)are necessarily non-negative. Indeed, the right side ofthis equation can equivalently be written as HΩMΩγ orH(Ωγ − Ω2

γ). It also follows from Eq. (2.9) that no so-lution for dΩM/dt = 0 is even possible at a given timeunless

∑

`

Γ`Ω` ≤H

4. (2.10)

This provides an upper limit on the possible decay widthsΓ`. When this inequality is saturated with dΩM/dt = 0we necessarily have ΩM = Ωγ = 1/2. Otherwise, if theinequality in Eq. (2.10) is satisfied but not saturated,other values of ΩM (both bigger and smaller than 1/2)are in principle possible.

B. Conditions for the persistence of stasis

As we have seen, Eq. (2.9) furnishes us with a minimalcondition for stasis. In general, however, such a condi-tion will be satisfied only at a particular instant of timet∗. By itself, this would clearly not lead to a true sta-sis state in which ΩM is constant. For the purposes ofunderstanding stasis, we are therefore interested in deter-mining the additional condition(s), if any, that will allowEq. (2.9) to remain satisfied over an extended period oftime. Indeed, it is only in this way that we can obtain atrue period of stasis.

Starting from Eq. (2.9), there are two ways in whichwe might demand that ΩM actually remain constant atsome fixed stasis value ΩM . First, at t = t∗, we couldimpose not only dΩM/dt = 0, but also dnΩM/dt

n = 0for all integers n > 1. This would then guarantee theabsence of any time evolution for ΩM . However, ratherthan impose this infinite set of constraints, we can dothis in a much quicker way: assuming that the equalityin Eq. (2.9) has been achieved at some time t∗, we nowsimply need to demand that both sides of this equationevolve with time in the same manner. This would ensurethat Eq. (2.9) remains satisfied under time evolution.

Before proceeding, we note two important implicationsof imposing such an additional requirement. First, bydemanding that both sides of Eq. (2.9) behave identi-cally under time evolution, we are actually demandingan eternal stasis in which ΩM is fixed, without begin-ning or end. Of course, this sort of eternal stasis is onlyan idealized abstraction which cannot be representativeof a realistic cosmology and which requires, in particular,a correspondingly perpetual decay process. Nevertheless,understanding the mathematics of such an idealized sta-sis will ultimately prove useful in allowing us to under-stand how to achieve a more realistic period of stasis inwhich ΩM remains constant at some value ΩM over anextended but finite time interval.

6

The second implication of demanding that both sidesof Eq. (2.9) evolve identically with respect to time isthat t∗, which we have identified as the time at whichEq. (2.9) is instantaneously satisfied, now becomes noth-ing more than a fiducial reference time, i.e., an arbitrarychoice which cannot carry physical significance. Indeed,although we shall find it useful to assume that Eq. (2.9)is satisfied at t = t∗ before working our way towardsa solution for all t, no physical condition for stasis canultimately depend on the choice of t∗.

In order to determine the extra conditions we requirefor stasis, let us now study how each term in Eq. (2.9)evolves with time during a supposed period of stasis.First, we observe that we can solve for the Hubble pa-rameter directly via Eq. (2.3). Since ΩM is presumedconstant with some fixed value ΩM during stasis, we ob-tain the exact solution

H(t) =

(2

4− ΩM

)1

t=⇒ κ =

6

4− ΩM(2.11)

where κ corresponds to the parametrization H(t) =κ/(3t). Indeed, from Eq. (2.11) we verify the standardresults that κ = 2 for ΩM = 1 (i.e., a matter-dominateduniverse), while κ = 3/2 for ΩM = 0 (i.e., a radiation-dominated universe). We emphasize that Eq. (2.11) isan exact result only under the stasis assumption, guar-anteeing that ΩM — and therefore κ — remain strictlyconstant. This solution for H(t) in turn implies that dur-ing stasis, the scale factor grows as

a(t) = a∗

(t

t∗

)κ/3= a∗

(t

t∗

)2/(4−ΩM )

(2.12)

with t∗ representing an arbitrary fiducial time and the ‘∗’subscript indicating that the relevant quantity is evalu-ated at t = t∗. Moreover, from the first equation withinEq. (2.7) we find the solution

ρ`(t) = ρ∗`

[a(t)

a∗

]−3

e−Γ`(t−t∗)

= ρ∗`

(t

t∗

)−6/(4−ΩM )

e−Γ`(t−t∗) . (2.13)

This in turn implies that

Ω`(t) = Ω∗`

(t

t∗

)2−6/(4−ΩM )

e−Γ`(t−t∗) . (2.14)

Given these results, we thus have two conditions whichmust be satisfied simultaneously in order to have an ex-tended period of stasis:

∑

`

Ω`(t) = ΩM

∑

`

Γ`Ω`(t) =2ΩM (1− ΩM )

4− ΩM

1

t(2.15)

where Ω`(t) is given in Eq. (2.14). Indeed, these twoconditions ensure that Eq. (2.9) is satisfied not only in-stantaneously, at one particular time t∗, but eternally,for all times. Note that these results together imply theconstraint

∑` Γ`Ω`∑` Ω`

=2(1− ΩM )

4− ΩM

1

t. (2.16)

Going forward, our goal will be to find systems for whichboth constraints in Eq. (2.15) are satisfied as exactly aspossible.

At first glance, it may seem surprising that the con-straints in Eq. (2.15) can ever be satisfied. Indeed, theforms of these constraints present two immediate chal-lenges. First, while the right side of the second con-straint in Eq. (2.15) drops like 1/t, the individual com-ponent abundances Ω`(t) never drop as 1/t: as indicatedin Eq. (2.14), they either grow as a power-law duringearly times t 1/Γ` (when the exponential decay is notyet dominant), or they fall exponentially as t ∼ 1/Γ`(ultimately overriding the power-law growth). However,what Eq. (2.15) is telling us is that these two effectsmust somehow cancel within the sum

∑` Γ`Ω`(t), leav-

ing behind an overall 1/t dependence. Second, all of thismust happen for each individual Ω`(t) while somehow si-multaneously keeping their sum

∑` Ω`(t) fixed, so that

the rising values of Ω` from what are presumably lightermodes with smaller widths (experiencing later decays)perfectly compensate for the exponential decays of theheavier modes (which experience earlier decays). Notethat this second challenge is indeed different from thefirst: the second challenge concerns the unweighted sumof the individual Ω`, while the first challenge is sensitiveto the sum in which each contribution Ω` is weightedby the corresponding width Γ`. However, as we shallnow demonstrate, very accurate simultaneous solutionsto both constraints in Eq. (2.15) can nevertheless indeedbe found.

III. A MODEL OF STASIS

In Sect. II, we obtained two conditions, as listed withinEq. (2.15), which together yield an eternal stasis existingfor all times. However, in reality, these conditions cannotbe strictly satisfied for all times. For example, regardlessof whether the tower of φ` states is finite or infinite, thereis an early time immediately after these states are pro-duced during which the decay process is just beginning.However, because the universe is expanding even at thisearly time, the required balancing between decay and cos-mological expansion cannot yet have been achieved, andconsequently we do not yet expect to have realized sta-sis. Likewise, there will eventually come a time at whichall of the decays will have essentially concluded. At thispoint we expect our period of stasis to end.

Despite these observations, the critical issue is whetherthere exist solutions for the spectrum of decay widths

7

Γ` and abundances Ω` across our tower of stateswhich will at least lead to a period of stasis during the se-quential decay process. Given the discussion at the endof Sect. II, it might seem that such solutions for Γ`and Ω` must be very carefully arranged. Remarkably,however, we shall now demonstrate that there exist solu-tions for Γ` and Ω` which not only satisfy these con-straints but which are also relatively simple and whichemerge naturally in realistic scenarios for physics beyondthe Standard Model.

Towards this end, let us consider a spectrum of de-cay widths Γ` and abundances Ω` which satisfy thegeneral scaling relations

Γ` = Γ0

(m`

m0

)γ, Ω

(0)` = Ω

(0)0

(m`

m0

)α(3.1)

where α and γ are general scaling exponents, where themass spectrum takes the form

m` = m0 + (∆m)`δ (3.2)

with m0 ≥ 0, ∆m > 0, and δ > 0 treated as generalfree parameters, and where the superscript ‘0’ withinEq. (3.1) denotes the time t = t(0) at which the φ` are ini-tially produced (thereby setting a common clock for thesubsequent φ` decays). Our goal is then to determinethose values — if any — of the eight parameters

α, γ, δ,m0,∆m,Γ0,Ω(0)0 , t(0) (3.3)

for which the constraints in Eq. (2.15) can be satis-fied. Of course, within the context of this model, wehave ΩM = 1 at t = t(0) before the decay process has

begun. This in turn requires that we choose Ω(0)0 =[∑N−1

`=0 (m`/m0)α]−1

.

Before proceeding further, our choice of the scaling re-lations in Eqs. (3.1) and (3.2) deserves comment. It mayinitially seem that we have adopted these relations for thesole purpose of achieving stasis. However, these exactrelations actually have an independent history [1, 2] ascharacterizing the towers of states that naturally emergewithin a variety of actual models of physics beyondthe Standard Model. For example, taking the φ` asthe Kaluza-Klein (KK) excitations of a five-dimensionalscalar field compactified on a circle of radius R (or aZ2 orbifold thereof) results in either m0,∆m, δ =m, 1/R, 1 or m0,∆m, δ = m, 1/(2mR2), 2, de-pending on whether mR 1 or mR 1, respectively,where m denotes the four-dimensional scalar mass [1, 2].Alternatively, taking the φ` as the bound states of astrongly-coupled gauge theory yields δ = 1/2, where∆m and m0 are determined by the Regge slope and in-tercept of the strongly-coupled theory, respectively [4].Thus δ = 1/2, 1, 2 serve as compelling “benchmark”values. Likewise, γ is generally governed by the partic-ular φ` decay mode. For example, if φ` decays to pho-tons through a dimension-d contact operator of the form

O` ∼ c`φ`F/Λd−4 where Λ is an appropriate mass scaleand where F is an operator built from photon fields, wehave γ = 2d − 7. Thus values such as γ = 3, 5, 7 canserve as relevant benchmarks. Finally, α is governed bythe original production mechanism for the φ` fields. Forexample, one typically finds that α < 0 for misalignmentproduction [1, 2], while α can generally be of either signfor thermal freeze-out [5].

Given this general model, we can now evaluate thesums which appear on the left sides of our constraintequations in Eq. (2.15). Our goal, of course, is to avoidassuming stasis and to find the conditions under whichour model nevertheless satisfies these stasis constraints.

We begin by focusing on the behavior of the abun-dance of any individual component Ω`(t). In Eq. (2.14),we derived the time-dependence of Ω`(t), but this deriva-tion assumed that we were already within stasis. Indeed,within the calculation leading to Eq. (2.14), the assump-tion of stasis entered into the form of the gravitational

redshift factor (t/t(0))2−6/(4−ΩM ); without assuming sta-sis, this factor would be much more complicated. Forsimplicity and generality, we shall therefore let h(ti, tf )denote the net gravitational redshift factor that accruesbetween any two times ti and tf . We thus have

Ω`(t) = Ω(0)` h(t(0), t) e−Γ`(t−t(0)) . (3.4)

Note that this h-factor is necessarily `-independent sincethe gravitational redshift affects all components equally.We then find that

∑

`

Ω`(t) = Ω(0)0 h(t(0), t)

∑

`

(m`

m0

)αe−Γ0

(m`m0

)γ(t−t(0))

(3.5)where we have used the scaling relations in Eq. (3.1).

In order to evaluate this sum, we shall make three ap-proximations. First, we shall take the continuum limit

∆m→ 0 , N →∞ (3.6)

such that mmax ≡ m0+(∆m)(N−1)δ is held constant. Inthis limit, the masses m` become a continuous variablem ranging from m0 to mmax, so that for any functionf(m`/m0) we can replace

N−1∑

`=0

f

(m`

m0

)

→ 1

δ

∫ mmax

m0

dm

m−m0

(m−m0

∆m

)1/δ

f

(m

m0

)

=1

δ

∫ mmax−m0

0

dm

m

( m

∆m

)1/δ

f

(m

m0+ 1

)(3.7)

where the additional integrand factor is the Jacobiand`/dm. Simultaneously, we shall also take

m0 → 0 , mmax →∞ (3.8)

in such a way that ∆m/m0 — and therefore all values ofm`/m0 ∼ m/m0 in Eq. (3.1) — are kept constant. This

8

limit extends the range of m-integration in Eq. (3.7) from0 to ∞. Finally, we shall also approximate

f

(m

m0+ 1

)≈ f

(m

m0

)(3.9)

within Eq. (3.7). This represents a “warping” of ourintegrand which is relevant only for small values ofm/m0.

We shall later verify that all three of these approxima-tions are relatively harmless, with effects that can easilybe understood and interpreted. However, the net effectof these three approximations is that we can convert our`-sum in Eq. (3.5) into an m-integral via

N−1∑

`=0

f

(m`

m0

)→ 1

δ

∫ ∞

0

dm

m

( m

∆m

)1/δ

f

(m

m0

),

(3.10)whereupon Eq. (3.5) becomes

∑

`

Ω`(t) =Ω

(0)0

δ(∆m)1/δh(t(0), t)

×∫ ∞

0

dmm1/δ−1

(m

m0

)αe−Γ0

(mm0

)γ(t−t(0))

.

(3.11)

For γ > 0 and α+1/δ > 0, we then find that this integralcan be evaluated in closed form, yielding the result

∑

`

Ω`(t) =Ω

(0)0

γδ

( m0

∆m

)1/δ

Γ

(α+ 1/δ

γ

)

× h(t(0), t)[Γ0(t− t(0))

]−(α+1/δ)/γ

(3.12)

where Γ(x) denotes the Euler gamma-function. Likewise,repeating the same steps for

∑` Γ`Ω`(t), we obtain

∑

`

Γ`Ω`(t) =Γ0Ω

(0)0

γδ

( m0

∆m

)1/δ

Γ

(α+ γ + 1/δ

γ

)

× h(t(0), t)[Γ0(t− t(0))

]−(α+γ+1/δ)/γ

.

(3.13)

Thus, dividing Eq. (3.13) by Eq. (3.12) and recalling thatΓ(x+ 1)/Γ(x) = x, we obtain

∑` Γ`Ω`(t)∑` Ω`(t)

=α+ 1/δ

γ

1

t− t(0). (3.14)

Comparing this result with the constraint equation inEq. (2.16), the first thing we notice is that our modelhas produced a power-law time dependence in the timedifference t − t(0) rather than in t itself. In principle,this therefore does not satisfy the criterion in Eq. (2.16).However, this criterion can be approximately satisfied solong as

t(0) t . (3.15)

In other words, as we originally anticipated, for stasis toemerge within our model we must restrict our attentionto periods of time which are sufficiently far beyond the φ`production time that the initial “edge” effects have diedaway. Indeed, the precision with which this power-lawscaling requirement is satisfied only increases the furtherwe are from these initial edge effects. Eq. (3.15) is thusour first condition for this model, indicating that we donot expect stasis to develop in this model until consider-ably after t(0). This thereby provides a natural beginningto the stasis period.

Let us now assume that Eq. (3.15) is satisfied. Com-paring the overall coefficients in Eqs. (2.16) and (3.14) wethen obtain a constraint on suitable values of (α, γ, δ):

1

γ

(α+

1

δ

)=

2(1− ΩM )

4− ΩM. (3.16)

Equivalently, for any tower of states parametrized by(α, γ, δ), this can be inverted in order to obtain the cor-responding predicted value of ΩM during stasis, yielding

ΩM =2γδ − 4(1 + αδ)

2γδ − (1 + αδ). (3.17)

For example, with (α, γ, δ) = (1, 5, 1), Eq. (3.17) yieldsΩM = 1/4. However, for (α, γ, δ) = (1, 7, 1), Eq. (3.17)yields ΩM = 1/2. This is then a case of stasis withmatter-radiation equality ! In general, from Eq. (3.17) wesee that stasis with matter-radiation equality will occurprovided

1 + αδ

γδ=

2

7. (3.18)

The fact that we must have 0 ≤ ΩM ≤ 1 places boundson the possible values of α, γ, and δ that can give riseto stasis. For example, in accordance with our expec-tation that the heavier φ` will decay more rapidly thanthe lighter φ`, we can restrict our attention to cases withγ > 0. In such cases, Eq. (3.17) in conjunction with0 ≤ ΩM ≤ 1 and the requirement that α + 1/δ > 0[as indicated above Eq. (3.12)] immediately leads to arestricted range for α:

− 1

δ< α ≤ γ

2− 1

δ. (3.19)

The conditions in Eqs. (3.16) and (3.17) emerged fromdemanding that our model satisfy Eq. (2.16). However,Eq. (2.16) only emerged as the quotient of the two morefundamental constraints in Eq. (2.15). We must thereforealso demand that these constraints are each individuallysatisfied. Of course, having already satisfied Eq. (2.16),we need only concentrate on one of these constraints. Letus assume that our model indeed gives rise to stasis fort t(0), and then verify that this assumption leads toa self-consistent result. Assuming stasis for t t(0), we

9

can write

h(t(0), t) = h(t(0), t∗)h(t∗, t)

= h(t(0), t∗)

(t

t∗

)2−6/(4−ΩM )

(3.20)

where t∗ t(0) is some fiducial time beyond which stasishas developed. Inserting this into our result in Eq. (3.12)then yields

∑

`

Ω`(t) =Ω

(0)0

γδ

( m0

∆m

)1/δ

Γ

(α+ 1/δ

γ

)

× h(t(0), t∗)

(t

t∗

)2−6/(4−ΩM )[Γ0(t− t(0))

]−(α+1/δ)/γ

.

(3.21)

However, given the conditions in Eqs. (3.15) and (3.16),we see that the rather complicated time-dependence inEq. (3.21) cancels! This by definition verifies that we areindeed within a period of stasis, with a constant ΩM ≡∑` Ω`(t). We are thus left with only one additional self-

consistency constraint on our model:

ΩM =Ω

(0)0

γδ

( m0

∆m

)1/δ

Γ

(α+ 1/δ

γ

)

× h(t(0), t∗)

(1

Γ0t∗

)(α+1/δ)/γ

. (3.22)

At first glance, this constraint equation is not particularlyilluminating. However, via Eqs. (3.4), (3.16), and (3.20),we see that

Ω(0)0 h(t(0), t∗)

(1

Γ0t∗

)(α+1/δ)/γ

= Ω0(τ0)eΓ0(τ0−t(0))

≈ eΩ0(τ0) (3.23)

where τ0 ≡ 1/Γ0 and where in passing to the second linewe have assumed that τ0 t(0). Substituting Eq. (3.23)into Eq. (3.22) we thus obtain the constraint

ΩM ≈ X Ω0(τ0) (3.24)

where

X ≡ e

γδ

( m0

∆m

)1/δ

Γ

(α+ 1/δ

γ

). (3.25)

Note that this proportionality constant X does not in-clude any of the potentially large ratios of time intervalsthat originally appeared in Eq. (3.22). Thus, as long as∆m ∼ m0, we find that X ∼ O(1).

It is not difficult to interpret this result. Ordinarily,ΩM receives significant contributions Ω`(t) from each ofthe individual components. However, by the time wereach t ≈ τ0, the lightest component is just about tobegin decaying while all of the heavier components havealready decayed to various extents. Thus the dominantcontribution to ΩM at t = τ0 comes from Ω0(τ0), while

the contributions Ω`(τ0) with ` ≥ 1 are exponentiallysuppressed. We then expect that ΩM will be approxi-mately equal to Ω0(τ0), with the proportionality coeffi-cient X in Eq. (3.24) including (and the difference X − 1quantifying) the residual contributions from all of theheavier states as well as the approximations made inpassing from the exact (discrete) sum in Eq. (3.5) to the

integral in Eq. (3.11). Of course, for t >∼ τ0, our decayprocess ends and our system exits from the stasis state.

We therefore conclude that our system satisfies the re-quirements for an extended period of stasis during thedecay process so long as the conditions in Eqs. (3.15),(3.17), and (3.24) are satisfied. Alternatively, comparingwith the parameter list in Eq. (3.3), we see that Eq. (3.15)constrains t(0) (or simply requires that this productiontime be significantly earlier than any ensuing period ofstasis), while Eq. (3.17) gives the resulting value of ΩMin terms of α, γ, δ and Eq. (3.24) involves all of theremaining parameters and can be viewed as constraining

the overall scale Ω0(τ0) [or equivalently Ω(0)0 ]. Although

it might seem that the constraints in Eqs. (3.17) and(3.24) represent fine-tunings that we must impose on theparameters of our model, we shall find in Sect. IV thatno such fine-tuning is required, and that stasis ultimatelyemerges within this model even if these constraints arenot originally satisfied.

As evident from the above derivation, we have madea number of approximations in obtaining these results.In particular, in order to evaluate the sum in Eq. (3.5),we made three approximations: we treated our tower ofstates as a continuum, as in Eq. (3.6); we then tookthe limits in Eq. (3.8), thereby essentially disregarding“edge” effects at the top and bottom of our tower; andfinally we made the approximation in Eq. (3.9). Theseapproximations indicate that the precise power-law time-dependence that is required for stasis according to theconstraints in Eq. (2.15) is at best only approximate fora realistic discrete tower of states φ`. However, it is easyto understand the situations in which these approxima-tions might fail, thereby disturbing the power-law behav-ior and consequently disrupting the resulting stasis. Ingeneral, for γ > 0, the decay widths Γ` increase as afunction of the masses m`, implying that the heavier φ`states tend to decay first while the lighter φ` decay later.For this reason, we expect that the approximations inEqs. (3.8) and (3.9) — approximations which primarilycome into play only at the tops or bottoms of our towers— will primarily affect the behavior of our system onlyat extremely early and/or late times, respectively. Thisis consistent with our further condition in Eq. (3.15). In-deed, as a particularly dramatic example of these edgeeffects, we observe that with γ > 0 and α + 1/δ > 0,our integral results in Eqs. (3.12) and (3.13) actually di-verge as t approaches the initial production time t(0). Ofcourse, this divergence is completely spurious, since ouractual model has a finite number of states within the de-caying tower and thus contains no such divergences. Thisis clear illustration of the fact that our integral approxi-

10

mation is highly inaccurate at such early times.

It is precisely the failure of our approximations at ex-tremely early and/or late times which explains why ourstasis (which would otherwise have been strictly eternal,i.e., time-independent, as in Sect. II) actually has a be-ginning and an end, emerging in full force only after thefirst few decays have already occurred and ending as thesystem approaches the final decays. We therefore regardthese edge effects as beneficial features, indicating thatthere will necessarily exist both an entrance into, as wellas an exit from, our stasis epoch, such as would be re-quired in any realistic cosmological scenario. At othertimes far from these “edge” effects, we shall neverthelessfind numerically that this stasis is quite robust.

In order to illustrate the stasis epoch — together withits beginning and end — we can perform a direct numer-ical study of this system, with the time evolution deter-mined through exact numerical solutions of the relevantBoltzmann equations for our discrete tower of decayingstates without any approximations. In Fig. 2 we illus-trate the behavior of the individual abundances Ω`(t)as well as the resulting behavior for their sum ΩM (t),where for concreteness we have chosen the benchmarkvalues (α, γ, δ) = (1, 7, 1) [for which ΩM = 1/2], with∆m = m0, N = 300, and ΓN−1/H

(0) = 0.01. As an-ticipated, we see that we not only have a robust periodof stasis lasting ≈ 15 e-folds, but we also have a clearentrance into this epoch as well as an exit from it. Dur-ing the stasis epoch, we nevertheless find that ΩM holdssteady at the value ΩM predicted in Eq. (3.17). As weshall shortly see, similar results hold for other values ofour benchmarks as well, leading to other values for ΩM .

Two final comments are in order. First, we observefrom the right panel of Fig. 2 that although the indi-vidual abundance contributions Ω`(t) have fairly compli-cated behaviors, they each attain a maximum value atapproximately t ≈ τ` ≡ 1/Γ` before beginning their ex-ponential decays. [More precisely, the maximum value ineach case occurs at t` = ζτ` where ζ ≡ 2 − 6/(4 − ΩM ),but these extra factors of ζ will cancel below and canthus be ignored.] Moreover, when plotting log Ω versusN ∼ log t (such as in this panel), we see that these max-imum values all lie along a straight line which we mayconsider to be the “envelope” function for the individualΩ`(t). This linear envelope function is a critical ingredi-ent in producing the stasis state.

It is easy to see how this envelope function emergesduring stasis. From Eqs. (3.4) and (3.20) we see that

Ω`(τ`) = Ω(0)` h(t(0), t∗)

(τ`t∗

)(α+1/δ)/γ

e−1 (3.26)

where t∗ is any fiducial time within stasis and where we

have approximated e−Γ`(τ`−t(0)) ∼ e−1 for τ` t(0). We

then find

Ω`(τ`)

Ω0(τ0)=

Ω(0)`

Ω(0)0

(τ`τ0

)(α+1/δ)/γ

=

(m`

m0

)α(τ`τ0

)(α+1/δ)/γ

=

(Γ`Γ0

)α/γ (τ`τ0

)(α+1/δ)/γ

=

(τ`τ0

)1/(γδ)

, (3.27)

indicating that this envelope line has constant positiveslope 1/(γδ).

The existence of this envelope line provides an addedperspective regarding the constraint in Eq. (3.24). It isclear that this rising envelope line must eventually inter-sect the horizontal ΩM = constant line. What Eq. (3.24)tells us is that this intersection point occurs near t ≈ τ0,as illustrated in the right panel of Fig. 2.

Our second comment is that we are now also in a posi-tion to estimate the duration of the stasis state. In gen-eral, if we disregard the “edge effects” at the beginningand end of the decay process, we can roughly identify thestasis state as stretching from the decay of the heavieststate in the tower at t ≈ τN−1 until the decay of the light-est state at t ≈ τ0, where we have treated each of thesequantities as significantly greater than t(0). We then findthat the number of e-folds during stasis is given by

Ns ≡ log

[a(t = τ0)

a(t = tN−1)

]=

2

4− ΩMlog

(ΓN−1

Γ0

)

=2γ

4− ΩMlog

(mN−1

m0

)

=2γ

4− ΩMlog

[1 +

∆m

m0(N − 1)δ

]

≈ 2γδ

4− ΩMlogN (3.28)

where we have used Eq. (2.12) in the first equality andwhere in passing to the final line we have taken N 1and ∆m/m0 ∼ O(1). We thus see that we can adjust thenumber of e-folds associated with the stasis epoch simplyby adjusting the number of states in the tower.

11

0 5 10 15 20 25N

0.0

0.2

0.4

0.6

0.8

1.0Ω

ΩM

ΩM0

100

200

300`

0 5 10 15 20 25N

10−3

10−2

10−1

100

Ω

ΩM

ΩM

0

100

200

300`

FIG. 2. The individual matter abundances Ω` (orange/blue) and the corresponding total matter abundance ΩM (red), plottedas functions of the number N of e-folds since the initial φ` production. These curves were generated through a direct numericalsolution of the relevant Boltzmann equations for our discrete tower of decaying states without invoking any approximations, andcorrespond to the parameter choices (α, γ, δ) = (1, 7, 1) [for which ΩM = 1/2], with ∆m = m0, N = 300, and ΓN−1/H

(0) = 0.01.In the left panel the abundances are plotted on a linear scale, while in the right panel these same abundances are plotted ona logarithmic scale. We see that our system begins with ΩM = 1 at t = t(0), with each individual Ω` component exhibitinga non-trivial behavior, first growing as a power-law due to cosmological redshifting before ultimately decaying exponentially.Despite this complexity, their sum ΩM nevertheless evolves towards a stasis epoch in which ΩM remains essentially constantfor approximately 15 e-folds before exiting stasis. Even longer periods of stasis can be produced if N is increased. This therebyprovides a concrete realization of the basic stasis mechanism sketched in Fig. 1. Eventually the stasis ends as we approach thefinal decays of the lightest modes.

IV. STASIS AS A GLOBAL ATTRACTOR

In previous sections we have studied the properties ofthe stasis state and developed a model in which this statenaturally arises. Indeed, in Fig. 2 we demonstrated thisnumerically for a particular choice of parameters in ourmodel. However, it may seem that this choice was some-how fine-tuned. To address this issue, we shall now re-turn to the basic dynamical equations that underlie thissystem and demonstrate that the stasis state is actuallya global attractor for this system. Thus, regardless of theparticular parameter choices we might make within ourmodel, we are inevitably drawn into a stasis epoch.

We begin our analysis with Eq. (2.8). Indeed, thisequation serves as the fundamental equation of motionfor our system and thus governs its dynamics. Our goal,then, is to demonstrate that all solutions to this equationwithin our model framework inevitably head towards thestasis solution ΩM → ΩM , where ΩM is the value ofΩM during stasis. Unfortunately, Eq. (2.8) contains twoquantities whose general connections to ΩM and ΩM arenot obvious: these are the `-sum

∑` Γ`Ω` and the Hubble

parameter H(t). Our first task will therefore be to derivegeneral expressions for each of these quantities within ourmodel, but without assuming stasis.

Let us first consider∑` Γ`Ω`. We have already eval-

uated this quantity in Sect. III, obtaining the resultin Eq. (3.13). Indeed, like the corresponding result inEq. (3.12), this result is completely general and does not

rely on any assumption of stasis. As a result, the quo-tient of these two results in Eq. (3.14) is also completelygeneral. We therefore have the result

∑

`

Γ`Ω` =

(α+ 1/δ

γ

)ΩM

t− t(0)

=

[2(1− ΩM )

4− ΩM

]ΩM

t− t(0), (4.1)

where in passing to the second line we have used theresult in Eq. (3.16). Note, in particular, that the resultin Eq. (3.16) also holds independently of stasis since it isnothing more than a rewriting of the (α, γ, δ) parametersin terms of the eventual stasis value ΩM . We thus seefrom Eq. (4.1) that in general

∑` Γ`Ω` depends on both

ΩM and ΩM .Let us now turn to the Hubble parameter H(t). We

previously evaluated H(t) in Eq. (2.11), but that deriva-tion assumed stasis. We must therefore now proceedmore generally by integrating Eq. (2.3). This immedi-ately yields the relation

1

H− 1

H(0)= (t− t(0))

[4− 〈ΩM 〉

2

](4.2)

where H(0) is the Hubble value at t = t(0) and where〈ΩM 〉 at any time t is the time-averaged value of ΩMsince t = t(0):

〈ΩM 〉 ≡1

t− t(0)

∫ t

t(0)dt′ ΩM (t′) . (4.3)

12

Eq. (4.2) then immediately yields

H(t) =2

4− 〈ΩM 〉1

t− t(0), (4.4)

where we have assumed H(0)(t− t(0)) 1. As expected,this result reduces to the result in Eq. (2.11) if we arewithin an eternal stasis, but otherwise depends on thecomplete time-history of the Hubble parameter since t =t(0) and thus makes absolutely no assumptions about theactual time-evolution of ΩM .

Inserting our results for∑` Γ`Ω` and H(t) from

Eqs. (4.1) and (4.4) into Eq. (2.8), we find that our equa-tion of motion for this system now takes the form

dΩMdt

=ΩM

t− t(0)

[2(1− ΩM )

4− 〈ΩM 〉− 2(1− ΩM )

4− ΩM

]. (4.5)

Of course, this result immediately allows us to verify thatdΩM/dt = 0 when ΩM = 〈ΩM 〉 = ΩM , consistent withour original (eternal) stasis solution. However, in general,we see that this equation — although first-order in time-derivatives — actually depends on two time-dependentvariables, ΩM and 〈ΩM 〉, which need not have any directrelation to each other.

One way to analyze the dynamics of this system is torecognize that the definition of 〈ΩM 〉 in Eq. (4.3) actuallyprovides us with another first-order differential equation,this one for 〈ΩM 〉:

d〈ΩM 〉dt

=1

t− t(0)[ΩM − 〈ΩM 〉] . (4.6)

Indeed, this equation is nothing but the time-derivativeof the definition in Eq. (4.3). The two coupled first-orderequations (4.5) and (4.6) could then be combined into asingle second-order differential equation for ΩM . How-ever, it will prove simpler (and more conceptually trans-parent) to retain these two first-order equations, treatingΩM and 〈ΩM 〉 as independent variables, and then studythe behavior of the corresponding two-variable dynamicalsystem

dΩMdt

=1

t− t(0)f(ΩM , 〈ΩM 〉)

d〈ΩM 〉dt

=1

t− t(0)g(ΩM , 〈ΩM 〉) ,

(4.7)

where

f(ΩM , 〈ΩM 〉) ≡ ΩM

[2(1− ΩM )

4− 〈ΩM 〉− 2(1− ΩM )

4− ΩM

]

g(ΩM , 〈ΩM 〉) ≡ ΩM − 〈ΩM 〉 .(4.8)

We note in passing that we are always free to shift our in-dependent variable for this system from t to log t, therebyrendering the right sides of Eq. (4.7) independent of time.This system is therefore effectively autonomous.

It is clear from these equations that the stasis solutioncorresponds to ΩM = 〈ΩM 〉 = ΩM . Moreover, this solu-tion will be a local attractor if both of the eigenvalues ofthe corresponding Jacobian matrix J are negative whenJ is evaluated at the stasis point. In our case, the Jaco-

bian matrix is given by J = (t− t(0))−1J where J is thetime-independent Jacobian matrix

J =

(∂ΩM f ∂〈ΩM 〉f∂ΩM g ∂〈ΩM 〉g

). (4.9)

Evaluated at the stasis point, this matrix takes the form

J∣∣s

=

(A B1 −1

)(4.10)

where the symbol |s indicates that the expression is eval-uated within stasis and where

A ≡ − 2ΩM

4− ΩM, B ≡ 2ΩM (1− ΩM )

(4− ΩM )2. (4.11)

The corresponding eigenvalues are therefore given by

λ± =−(4 + ΩM )±

√Ω

2

M − 16ΩM + 16

2(4− ΩM ), (4.12)

whereupon we see that

λ± < 0 for all 0 ≤ ΩM ≤ 1 . (4.13)

We therefore conclude that the stasis state is (at least) alocal attractor, stable against small deviations δΩM andδ〈ΩM 〉. Thus, if perturbed, our system will necessarilyreturn to the stasis state.

Of course, the question remains as to whether oursystem will flow to the stasis state if we are originallyfar from it. In such cases, ΩM and 〈ΩM 〉 need not beclose to ΩM and need not even be close to each other.The best way to answer this question is therefore to as-sume arbitrary initial values for ΩM and 〈ΩM 〉 within therange 0 ≤ ΩM , 〈ΩM 〉 ≤ 1, and then examine how thesetwo variables evolve under the time-evolution specified inEq. (4.7). In other words, we seek to determine the trajec-tories that our system maps out in the (ΩM , 〈ΩM 〉)-planeaccording to Eq. (4.7). In Fig. 3, we illustrate these tra-jectories for the case in which the stasis solution is givenby ΩM = 〈ΩM 〉 = ΩM = 1/2. As we see, all trajectoriesfor this system ultimately flow towards this stasis point.Moreover, similar trajectory maps emerge regardless ofthe chosen stasis point. We therefore conclude that thestasis state is not only a local attractor, but actually aglobal one.

At first glance, given that our initial conditions at theproduction time t = t(0) are always ΩM = 〈ΩM 〉 = 1, itmight seem that most of the trajectories plotted in Fig. 3are irrelevant for our situation. However, we must recallthat in deriving the differential equations in Eq. (4.7)that govern our system — particularly in establishing

13

0

0

0.2

0.2

0.4

0.4

0.6

0.6

0.8

0.8

1

1

ΩM

0 0

0.2 0.2

0.4 0.4

0.6 0.6

0.8 0.8

1 1〈Ω

M〉

FIG. 3. Trajectories (blue curves) within the (ΩM , 〈ΩM 〉)-plane for the system defined in Eq. (4.7). For concretenesswe have shown the case with stasis values ΩM = 〈ΩM 〉 =ΩM = 1/2 (central red dot), but similar results emerge forall stasis values ΩM . Because this system is effectively au-tonomous, any point along a given trajectory can be taken asa starting point without affecting the subsequent trajectory.Given these trajectories, we see that the stasis state serves asa global attractor for this system.

Eq. (4.1) — we made a number of approximations whenpassing from the discrete sum in Eq. (3.5) to the integralform in Eq. (3.11). These approximations were discussedin detail in Sect. III, and are listed in Eqs. (3.6), (3.8),and (3.9). Some of these approximations turn out to beof little consequence for the current situation, such astaking the continuum limit as in Eq. (3.6). However, themmax →∞ approximation within Eq. (3.8) can be signif-icant, since this approximation essentially eliminates thetransient “edge” effects that arise at early times imme-diately after the production time t(0), when the heaviestφ` states are just beginning to decay. Since the integralapproximation ignores these edge effects, the correspond-ing dynamical equations in Eq. (4.7) are valid only afterthese edge effects have died away.

These edge effects can nevertheless have significant im-pacts on the dynamics of our system. For example, ifthe decay widths are relatively large, the most massiveφ` states will decay extremely promptly, thereby induc-ing a significant initial depletion of ΩM and 〈ΩM 〉 thatoccurs before Eq. (4.7) becomes valid. We shall see ex-plicit examples of this phenomenon below. Thus, whilethe differential equations in Eq. (4.7) accurately describethe dynamics of our system after the initial edge effectshave died away, these initial edge effects are capable ofshifting ΩM and 〈ΩM 〉 to new locations Ω′M and 〈ΩM 〉′in the (ΩM , 〈ΩM 〉)-plane which are quite far from their

original t(0) location ΩM = 〈ΩM 〉 = 1. It is then thesenew values Ω′M and 〈ΩM 〉′ which constitute the “initial”point for the subsequent trajectory in Fig. 3. Thus, tothe extent that we regard our dynamical system as gov-erned by the differential equations in Eq. (4.7), we shouldproperly regard the initial conditions for this dynamics tobe those associated with Ω′M and 〈ΩM 〉′. In other words,the initial conditions for the trajectories in Fig. 3 shouldbe taken to be those that exist not at the production timet(0), but rather at a subsequent time after which the ini-tial edge effects have died away. These edge effects canthus can place our system on an entirely different trajec-tory than that beginning at ΩM = 〈ΩM 〉 = 1.

Fortunately, this observation does not affect our con-clusion that the stasis state is a global attractor. Nomatter what behaviors are induced within our systemby the initial edge effects, we know that they must ulti-mately yield values Ω′M and 〈ΩM 〉′ which remain withinthe plane shown in Fig. 3. Indeed, we have seen in Fig. 3that any such trajectory within this plane eventuallyleads to the stasis state. Thus our stasis state remainsa global attractor even when the initial edge effects areincluded.

Changing the parameters of our model can alter notonly the initial edge effects but also the resulting stasisabundance ΩM . However, as discussed above, the emer-gence of a stasis state at ΩM is a robust phenomenonwhich persists even as the values of the parameters ofour model are changed. For example, in Fig. 4, we showthe time-evolution of ΩM for a variety of values of α andγ, taking δ = 1 as a fixed benchmark. In each case,we see that ΩM immediately begins to fall from 1 buteventually enters into a stasis epoch in which ΩM is es-sentially fixed at the corresponding stasis value ΩM pre-dicted by Eq. (3.17). Indeed, as evident within Fig. 4,this stasis epoch can persist for many e-folds (and can beextended indefinitely by increasing N) before dissipatingwhen the lightest φ` states decay. We emphasize thatthese plots were generated as direct numerical solutionsof the relevant Boltzmann equations without invokingany approximations. They thus accurately represent theactual behavior of our system.

Even more dramatically, the emergence of the stasisstate is also robust against changes in the overall de-cay rate for the φ` particles. This feature is illustratedin Fig. 5. In general, for any tower of φ` states whosedecay rates Γ` are connected through the scaling rela-tions in Eq. (3.1), we can parametrize this overall de-cay rate through the dimensionless quantity ΓN−1/H

(0)

where ΓN−1 is the decay rate Γ` for the most massiveparticle (i.e., that with ` = N − 1) and where H(0) de-notes the value of the Hubble parameter at the produc-tion time t = t(0). When ΓN−1/H

(0) 1, the decaysbegin relatively slowly after the production time t(0) andthe initial edge effects are correspondingly mild. As aresult, the approximations leading to Eq. (4.7) becomevalid rather quickly, with Ω′M and 〈ΩM 〉′ still fairly close

to 1. By contrast, when ΓN−1/H(0) 1, a significant

14

0 10 20 30 40 50N

0.0

0.2

0.4

0.6

0.8

1.0

ΩM

(0, 7)

(0, 5)

(1, 7)

(0, 3)

(1, 5)

(2, 7)

FIG. 4. The total matter abundance ΩM , plotted as a function of the number N of e-folds since the initial φ` production timet(0) for a variety of different benchmark values of (α, γ) satisfying the constraints in Eq. (3.19). For each plot we have taken

δ = 1, ∆m = m0, N = 105, and H(0)/ΓN−1 = 0.1 as relevant benchmarks. Each curve begins at ΩM = 1 at the initial time t(0)

and eventually falls to ΩM → 0 as t → ∞. However, we see that in each case there is a prolonged epoch lasting many e-foldsduring which ΩM (t) settles into a stasis state with ΩM (t) = ΩM ; indeed the duration of the stasis state can be increased atwill simply by increasing N . For each value of (α, γ), the corresponding stasis values ΩM are indicated with horizontal dashedlines. Ultimately in each case, the period of stasis ends as we reach the final φ` decays; the resulting oscillations in ΩM reflectthe discretization effects associated with the successive final decays.

0 2 4 6 8 10 12N

0.0

0.2

0.4

0.6

0.8

1.0

ΩM

log10 [Γ

N−

1 /H(0

) ]

−2

−1

0

1

2

3

4

FIG. 5. The total matter abundance ΩM , plotted as a function of the number N of e-folds since the initial φ` productiontime t(0) for a variety of different values of ΓN−1/H

(0), where H(0) is the value of the Hubble parameter at the φ` production

time t(0). For this plot we have chosen the benchmark values (α, γ, δ) = (1, 7, 1) and taken ∆m = m0 and N = 300. We see

that as ΓN−1/H(0) increases, our decays occur more rapidly and ΩM therefore drops more rapidly from its initial value 1 after

t(0). Indeed, for ΓN−1/H(0) >∼ 3, the initial drop in ΩM is so rapid that ΩM initially drops below the stasis value ΩM before

rebounding due to cosmological expansion; ΩM therefore ultimately approaches the stasis value ΩM from below. Such curvesthen correspond to trajectories in Fig. 3 for which stasis is approached from the ΩM < 1/2 region. However, in all cases oursystem is inevitably drawn towards the same stasis configuration, illustrating that the stasis state is indeed a global attractorfor this system.

15

number of the heaviest φ` states decay promptly aftert(0), leading to a rapid depletion in ΩM and 〈ΩM 〉. In-deed, in such cases ΩM may even initially fall below theeventual stasis value ΩM , as illustrated in Fig. 5. Never-theless, the stasis state still serves as an attractor in suchcases; the only difference is that ΩM now approaches itsstasis value ΩM from below rather than above. Indeed,in such cases these edge effects have produced an “ini-tial” value Ω′M which lies below ΩM , so that our systemfollows a trajectory within Fig. 3 along which the valueof ΩM increases rather than decreases.

We conclude, then, that our stasis state is a global at-tractor, emerging regardless of the initial conditions andregardless of the values of the parameters in our model.Our stasis state is thus the Rome of our dynamical sys-tem, and all roads lead to it.

V. STASIS IN THE PRESENCE OFADDITIONAL ENERGY COMPONENTS

Thus far, we have considered the emergence of stasiswithin universes consisting of only matter and radiation.Given this, a natural question is to understand how thispicture is modified if our universe also contains an addi-tional energy component X beyond matter and radiation,with general equation-of-state parameter wX . To studythis, we can repeat our derivations, only now allowing foran initial abundance ΩX in addition to ΩM and Ωγ .

Our derivation proceeds exactly as before. Includingan energy contribution for which pX = wXρX , we findthat Eq. (2.3) now becomes

dH

dt= − 1

2H2 [2 + ΩM + 2Ωγ + (1 + 3wX)ΩX ]

= − 12H

2 [4− ΩM + (3wX − 1)ΩX ] (5.1)

where in passing to the second line we have now identifiedΩγ = 1− ΩM − ΩX . This in turn implies that Eq. (2.4)becomes

dΩidt

=8πG

3H2

dρidt

+HΩi [4− ΩM + (3wX − 1)ΩX ] .

(5.2)While Eq. (2.7) continues to apply without modification,we now additionally have dρX/dt = −3(1 + wX)HρX .This of course assumes that X is uncoupled from mat-ter or radiation. Substituting these results into Eq. (5.2)we then find that the time-evolutions of our three abun-dances ΩM , Ωγ , and ΩX are described by a system ofthree coupled differential equations:

dΩMdt

= −∑

`

Γ`Ω` +HΩM (Ωγ + 3wXΩX)

dΩγdt

=∑

`

Γ`Ω` −HΩγ [ΩM + (1− 3wX)ΩX ]

dΩXdt

= HΩX [Ωγ − 3wX(ΩM + Ωγ)] . (5.3)

Of course, dΩM/dt+ dΩγ/dt+ dΩX/dt = 0, so only twoof these equations are independent.

As expected, these equations reduce to Eq. (2.9) whenΩX = 0 and when we can therefore identify Ωγ = 1 −ΩM . Moreover, we see from the third equation withinEq. (5.3) that if ΩX vanishes at any initial time, then ΩXremains vanishing for all times, thereby reproducing ourprevious results. However, we are now interested in stasisconfigurations in which ΩM , Ωγ , and ΩX are all constant

but non-zero, with values ΩM , Ωγ , and ΩX ≡ 1−ΩM−Ωγrespectively.

Since stasis requires a non-zero constant ΩX , a minimalcondition for stasis is dΩX/dt = 0. From the third lineof Eq. (5.3) we then see that this will only happen if

wX =Ωγ

3(ΩM + Ωγ). (5.4)

Equivalently, inverting this relation, we see that for anyvalue of ΩX with equation-of-state parameter wX , thecorresponding stasis solutions for this system must alltake the general form

ΩM = (1− 3wX)(1− ΩX)

Ωγ = 3wX(1− ΩX) .(5.5)

It is easy to interpret the condition in Eq. (5.4). In gen-eral, the quantity in Eq. (5.4) is nothing but the equation-of-state parameter for the combined matter + radiationsubsystem during stasis. Thus, Eq. (5.4) tells us that wecan append any additional component ΩX onto a com-bined matter + radiation subsystem without destroyingits stasis property so long as the equation-of-state param-eter wX of this additional component matches the stasisequation of state of the original matter + radiation sub-system. This ensures that the total system — with theX-component included — continues to have the same sta-sis equation of state wX as the original matter + radiationsubsystem and thus remains wX -dominated. The X-abundance ΩX then remains constant under cosmolog-ical redshifting — even though it is unaffected by thedecays of the matter components — simply as a resultof the general property that the abundance of any quan-tity with a given equation-of-state parameter w alwaysremains constant in a fully w-dominated universe.

Eq. (5.4) came from the final equation in Eq. (5.3) andthus represents only one condition for stasis. The otherremaining condition comes from the first two equationsin Eq. (5.3). Indeed, demanding dΩM/dt = dΩγ/dt = 0we obtain the additional condition

∑

`

Γ`Ω` = HΩMΩγ

ΩM + Ωγ= 3wXHΩM . (5.6)

This result is the analogue of Eq. (2.9).Of course, we wish to ensure that Eq. (5.6) holds not

only at one instant but over an extended stasis time in-terval. Within such an interval we see from Eqs. (5.1)

16

and (5.4) that the Hubble parameter H now takes thesimple form

H(t) =2

3(1 + wX)

1

t, (5.7)

whereupon we find that

Ω`(t) = Ω∗`

(t

t∗

)2wX/(1+wX)

e−Γ`(t−t∗) (5.8)

for any fiducial time t∗ during stasis. From Eq. (5.6)we thus have two additional conditions beyond that inEq. (5.4) which must also be satisfied simultaneously inorder to have an extended period of stasis:

∑

`

Ω`(t) = ΩM

∑

`

Γ`Ω`(t) =2wXΩM1 + wX

1

t(5.9)

where Ω`(t) is given in Eq. (5.8). Eq. (5.9) is of coursethe analogue of Eq. (2.15), and leads to the condition

∑` Γ`Ω`∑` Ω`

=2wX

1 + wX

1

t, (5.10)

which is the analogue of Eq. (2.16).It turns out that our model from Sect. III — in con-

junction with an additional energy component X — fur-nishes us with a realization of this three-component sta-sis as well. Indeed, the only required modification to ourmodel is that we no longer assert ΩM = 1 as an initialcondition at the production time t(0). Because of theassumed presence of the additional X-component withinour system, we shall instead leave the initial value of ΩMarbitrary. However, proceeding exactly as in Sect. III,we once again obtain the result given in Eq. (3.14) — aresult which did not depend on the initial value of ΩM .Comparing with Eq. (5.10) we thus identify

2wX1 + wX

=α+ 1/δ

γ, (5.11)

or equivalently

wX =1 + αδ

2γδ − (1 + αδ). (5.12)

We thus see that the parameters (α, γ, δ) of our modelmust be chosen appropriately for the equation-of-stateparameter wX of the desired X-component that we wishto add. Our model then yields constant stasis valuesΩM ,Ωγ ,ΩX satisfying Eq. (5.5), with the same provi-sos as discussed in Sect. III for the two-component stasis.

Note that our model does not yield specific values forΩM , Ωγ , or ΩX until specific initial values of ΩM and

Ωγ are chosen at the production time t(0). In princi-ple this is no different from the simpler two-component

case we have already considered, given that even in thetwo-component case we also chose a specific initial valueΩM = 1 (with an implied corresponding initial choiceΩγ = 0). Indeed, this choice precluded any room for aninitial additional energy component ΩX . In this sense,allowing more general initial values ΩM < 1 with Ωγ = 0is tantamount to allowing an initial value ΩX > 0. Af-ter the initial transient edge effects have died away, thisthen ultimately leads to a particular non-zero stasis valueΩX , whereupon we then obtain specific predicted valuesfor ΩM and Ωγ satisfying Eq. (5.5).

Following the steps in Sect. IV, we can also demon-strate that our three-component stasis continues to bean attractor like its two-component cousin. Of courseour phase space is now described by four independent dy-namical variables, namely ΩM , 〈ΩM 〉, Ωγ , and 〈Ωγ〉. Onepotentially surprising feature is that the corresponding4× 4 Jacobian matrix [analogous to the Jacobian matrixin Eq. (4.9)] actually has only three negative eigenvaluesand one zero eigenvalue. However, this is completely inkeeping with our expectation that our stasis solution isno longer an attractor point in the corresponding phasespace, but rather an attractor line. This attractor line isanalogous to a flat direction in the sense that all pointsalong the line correspond to equally valid solutions. In-deed, moving along this “flat direction” corresponds toshifting the value of ΩX , with the corresponding values ofΩM and Ωγ tracking the line of stasis solutions given inEq. (5.5). It is therefore not a surprise that it ultimatelycomes down to a particular choice of initial conditions forΩX (or equivalently for ΩM and Ωγ) which determineswhere along this line our resulting stasis is eventuallyrealized.

We see, then, that the two-component stasis whichhas been the focus of this paper is not an isolated phe-nomenon, existing for universes containing only mat-ter and radiation. Rather, we now see that our two-component stasis is actually the endpoint of an entire lineof possible stasis solutions in which a variety of additionalenergy components X with varying abundances ΩX andequations of state wX are also possible. This observa-tion once again reinforces our conclusion that stasis is ageneric feature in these sorts of theories.

Of course, not all types of energy components X maybe introduced. From Eq. (5.4) it follows that

0 < wX < 1/3 . (5.13)

As discussed in Ref. [6], this includes, for example, theenergies associated with scalar theories in which thescalars φ oscillate coherently within monomial potentialsV (φ) ∼ |φ|n where 2 < n < 4. However, this precludesenergy components with wX < 0, such as vacuum en-ergy Λ with wΛ = −1. This result implies that stasis isnot possible when ΩΛ 6= 0. However, even with ΩΛ 6= 0,it is possible that the ratio of ΩM and Ωγ might never-theless remain constant, thereby yielding what might beconsidered a weaker form of stasis. Of course, even whenΩΛ 6= 0, an extended period of approximate stasis can

17

exist during cosmological epochs for which ΩΛ — thoughgrowing — is still small. Indeed, as we shall discuss inSect. VI, this is the most likely context in which any phe-nomenologically realistic model of stasis might appear.Moreover, as long as ΩΛ 6= 0 at some initial time, theinevitable growth of ΩΛ provides an additional naturalmechanism for exiting the period of stasis and resuminga more traditional period of cosmological evolution.

At first glance, given the result in Eq. (5.4), it mightseem that our extra X-component could itself be com-posite, consisting of two subcomponents A and B, solong as this composite X-sector has the required totalabundance ΩX = 1 − ΩM − Ωγ as well as the requiredtotal equation-of-state parameter wX given in Eq. (5.4).If so, one could imagine that as ΩA grows, ΩB wouldshrink to compensate and thereby keep ΩA+ΩB and wXfixed. One could even further speculate that the grow-ing component could be vacuum energy Λ. However, itis easy to verify that even though we might carefully setΩX and wX to the required values at a given time, theensuing A/B dynamics will generally keep neither thetotal abundance of the X-sector fixed at ΩX nor the to-tal equation-of-state parameter fixed at wX , as requiredfor stasis within the matter + radiation sector. Indeed,the only way to have our X-sector retain both its abun-dance ΩX and its equation-of-state parameter wX is tohave the X-sector consist of only a single component Xwhose abundance is then naturally fixed as a result ofits equation-of-state parameter wX matching that of thematter + radiation sector.

VI. DISCUSSION AND COSMOLOGICALIMPLICATIONS

In this paper we have demonstrated the existence ofa new theoretical possibility for early-universe cosmol-ogy: epochs of cosmological stasis during which the rel-ative abundances Ωi of the different energy componentsremain constant despite cosmological expansion. Suchstasis epochs therefore need not be radiation-dominatedor matter-dominated, and need not be dominated byany particular component at all. We demonstrated thatsuch epochs emerge naturally in many extensions to theStandard Model and that the stasis state even servesas a global attractor within the associated cosmologi-cal frameworks. As a result, within these frameworks,the universe will naturally evolve towards such periodsof stasis for a wide variety of initial conditions, even ifthe system does not begin in stasis. Moreover, as we haveseen, each period of stasis comes equipped with not only anatural beginning but also a natural ending. Dependingon the parameters of the underlying theory, such stasisepochs can nevertheless persist for arbitrary lengths oftime.

Needless to say, our results give rise to a host of newtheoretical possibilities for physics across the entire cos-mological timeline. Indeed, an epoch of cosmological sta-

sis can be expected to provide non-trivial modificationsto the evolution of primordial density perturbations aswell as the dynamics of cosmic reheating. The existenceof a stasis epoch can also affect dark-matter production,structure formation, and even estimates of the age of theuniverse. However, in order to study such possibilities,we must first understand where and how our stasis epochmight arise within what might otherwise be consideredthe standard cosmological timeline.

Within the standard ΛCDM cosmology, the universefirst undergoes a phase of accelerated expansion knownas cosmic inflation. Immediately after this inflationaryepoch, the energy density of the universe is typicallydominated by the coherent oscillations of the inflatonfield. The subsequent decays of this field then reheatthe universe, thereby giving rise to a radiation-dominated(RD) era. Since the energy density of radiation is dilutedby cosmic expansion more rapidly than that of matter,the relative abundance of matter rises over time, reachesparity with the abundance of radiation at the point ofmatter-radiation equality (MRE), and then exceeds theabundance of radiation, eventually coming to dominatethe universe. The ensuing matter-dominated (MD) erathen persists until very late times, at which point vacuumenergy becomes dominant.

Although this timeline is relatively simple and com-pelling, there is considerable room for modification with-out running afoul of experimental or observational data.In particular, it is possible to imagine “splicing” an epochof stasis into this timeline, either as an additional seg-ment inserted into the timeline or as the replacement fora segment which is removed.

To see how this might occur, let us first recall thatour stasis scenario is one in which the universe passesfrom a matter-dominated epoch into a period of stasisand then finally into a radiation-dominated epoch. Ingeneral, this initial MD epoch begins at the time t(0)

at which our φ` states are produced — provided the φ`are produced non-relativistically and with a sufficientlylarge abundance. Of course, if these fields are relativisticat the production time t(0), or if there exists a significantabundance of radiation at that time, then the universemight be radiation-dominated for a few e-folds after t(0).However, even in such cases, our φ` states will eventuallycome to dominate the universe and the universe will en-ter a matter-dominated epoch. Thus, in either case, ourstasis scenario generically begins with a pre-stasis MDepoch. However, once our φ` particles begin to decay, wethen enter the stasis period. Indeed, as we have demon-strated in Sect. IV, the stasis state is a global attractorregardless of the particular initial conditions. Thus, aslong as the φ` states are produced with appropriatelyscaled abundances and lifetimes as in Sect. III, we willnecessarily enter into a period of stasis. Finally, as wereach the time at which the last φ` states decay, we thenexit the stasis era and enter a radiation-dominated epochin which no φ` particles remain.

Given these observations, there are many locations

18

along the standard ΛCDM timeline during which a stasisepoch might occur. However, for concreteness, we shallhighlight two scenarios that naturally stand out whenconstructing a cosmological model of stasis.

• Stasis spliced into RH: A minimal approach tosplicing a period of stasis into the standard cosmo-logical timeline involves choosing a location wherethere is already a transition from a MD universe toa RD universe, accompanied by an episode of parti-cle production during which our φ` states might beinitially populated. Within the standard cosmol-ogy, there is only one such location: during reheat-ing. Indeed, assuming that the inflaton oscillatescoherently within a nearly quadratic potential, theuniverse is effectively matter-dominated during thisperiod, evolving with an equation-of-state param-eter w ≈ 0. The decay of the inflaton, which isusually assumed to reheat the universe, would in-stead produce the φ` states. Such states wouldthen quickly become non-relativistic (if they werenot already produced non-relativistically), therebyextending the reheating period into a longer MDepoch. The subsequent stasis epoch and the φ`decays therein would then ultimately provide anew environment for reheating [7]. Finally, oncethe φ` decays have concluded, the universe willbe radiation-dominated, with a traditional ΛCDMevolution beyond that point. Thus, schematically,this scenario amounts to an insertion into the RHepoch of the form

RH −→ RH + stasis . (6.1)

• Stasis spliced into RD: Given that our sta-sis scenario leads to a radiation-dominated epoch,another possibility is to splice the stasis scenariodirectly into the usual RD era. This would re-quire that t(0), the production time for our φ`states, occur at a time when the universe is al-ready radiation-dominated. If the φ` states behaveas massive matter, then their energy density willeventually dominate the total energy density of theuniverse even though the universe was radiation-dominated at t(0). This scenario thus induces anew early matter-dominated epoch (EMDE) imme-diately prior to the onset of stasis, with the subse-quent transition to stasis only occurring once theφ` states begin to decay. During the ensuing sta-sis epoch, of course, the universe consists of anadmixture of matter and radiation with unchang-ing relative abundances. Finally, after the last φ`particles have decayed, the universe is once againradiation-dominated, with subsequent time evolu-tion proceeding in the traditional ΛCDM manner.Thus, this scenario schematically amounts to an in-sertion into the RD epoch of the form

RD −→ RD + EMDE + stasis + RD . (6.2)

In either of these scenarios, the equation-of-state pa-rameter w for the stasis epoch can take essentially anyvalue within the range 0 < w < 1/3. Indeed, within thestasis scenario this is achieved through the emergence ofa stable mixed state rather than through the introduc-tion of a pure state involving a new type of cosmologicalfluid. As a result of this unorthodox value of w, the evo-lution of the universe during the stasis era is unlike itsevolution during any other cosmological epoch, expand-ing more rapidly than it does during a RD epoch butmore slowly than it does during a MD epoch.

In Fig. 6 we illustrate the traditional ΛCDM cosmologyas well as the two alternative scenarios itemized above.In the top panel, we show the traditional ΛCDM cosmol-ogy, sketching the relative cosmological abundances Ωiassociated with vacuum energy (i = Λ, in red), matter(i = M , in blue), and radiation (i = γ, in orange) as theyevolve through an initial inflationary epoch followed by areheating epoch, a radiation-dominated epoch, and ulti-mately a matter-dominated epoch that is only now givingway to an epoch dominated again by vacuum energy. Wehave also shaded each region according to the dominantenergy component during that epoch. By contrast, inthe middle and bottom panels, we sketch the new sce-narios in which a stasis interval with ΩM = Ωγ = 1/2occurs within the reheating epoch (middle panel) or theradiation-dominated epoch (lower panel). As shown, thelatter possibility requires the introduction of a new earlymatter-dominated epoch (EMDE) immediately prior tothe emergence of the stasis state. Of course, it is onlyfor simplicity that we have chosen to illustrate these sta-sis states as having ΩM = Ωγ = 1/2 in Fig. 6; any stasis

configuration with ΩM = 1−Ωγ would have been equallyvalid for either of the two cases shown.

In either case, we see from Fig. 6 that the interveningstasis period has the effect of delaying the entire cosmo-logical timeline. Thus, while the universe eventually re-turns to the traditional ΛCDM script in each case, it doesso at an age which is advanced relative to that normallyascribed to it within the ΛCDM framework. We also ob-serve that during these stasis epochs, the universe is notdominated by any particular energy component. It is forthis reason that the stasis epochs are not shown in Fig. 6with any shaded background. Normally it is not possibleto have such an extended unshaded epoch; the differentabundances Ωi are constantly in flux and it is only dur-ing the relatively brief transition periods between epochsthat the universe might fail to have a dominant compo-nent. However, during stasis, this situation can persistacross an arbitrary number of e-folds.

In Fig. 7, we illustrate these same two alternate sce-narios, this time plotting not the corresponding cosmo-logical abundances Ωi but rather the corresponding co-moving Hubble radius (aH)−1. Within each panel ofFig. 7, we show not only the standard ΛCDM cosmology(black lines) but also the corresponding modified cosmol-ogy (red lines) in which a stasis epoch arises. Note thatthe sketches in Figs. 6 and 7 correspond to the regime in

19

1

0

Λ M γ M Λ

Inflation RH RD MDN

Ω

ΛCDM

1

0

Λ M γ M Λ

stasis

Inflation RH RD MDN

Ω

Stasisspliced

into RH

1

0

Λ M M γγ M Λ

stasis

Inflation RH RD EMDE stasis RD MDN

Ω

Stasisspliced

into RD

FIG. 6. Sketch of the traditional ΛCDM cosmology (top panel) as well as the two scenarios itemized in the text and describedschematically in Eqs. (6.1) and (6.2) in which a stasis epoch with ΩM = Ωγ = 1/2 is inserted into the cosmological timeline(middle and bottom panels). Abundances associated with vacuum energy (red), matter (blue), and radiation (orange) areplotted as functions of the number N of e-folds, with corresponding background shadings indicating the dominant componentin each epoch. In the “stasis spliced into RH” scenario (middle panel), reheating occurs during the stasis epoch and resultsfrom the decays of the φ` states [7]. By contrast, in the “stasis spliced into RD” scenario (bottom panel), reheating has alreadyconcluded but the insertion of the stasis scenario induces the existence of an early matter-dominated era (labeled ‘EMDE’).

t∼τ N−

1 t∼τ 0

Reh

eati

ng

Infl.

end

t=t(

0)

present horizon scaleIn

flation

Osc.Inflaton

RD

MD

Stas

is

RD

MD

MD

DE

DE

N

ln(aH

)−1

Stasis spliced into RH

t∼τ N−

1 t∼τ 0

Reh

eati

ng

Infl.

end

t=t(

0)

present horizon scale

Inflation

Osc.Inflaton

RD

MD

Stas

is

RD

MD

MD

DE

DE

N

ln(aH

)−1

Stasis spliced into RD

FIG. 7. The same scenarios as illustrated in Fig. 6, only now sketched in terms of the evolution of the comoving Hubbleradius (aH)−1. The present-day horizon scale is also indicated. In each panel the black lines indicate the standard ΛCDMcosmology, while the red lines indicate the new cosmologies that result in the “stasis spliced into RH” and “stasis spliced intoRD” scenarios (left and right panels, respectively). In each case we have explicitly indicated the time t(0) at which the φ` statesare initially produced as well as the times τN−1 and τ0 which respectively approximate the onset and cessation of the stasisstate. In both cases we see that the insertion of a stasis period delays the subsequent timeline relative to traditional ΛCDMexpectations.

20

which H(0) ΓN−1, thereby ensuring that all of the φ`states behave like matter before they start decaying.

One important feature illustrated in these figures isthat the introduction of a stasis period either prior toor during the traditional RD era requires a shorteningof the duration of the RD era as a whole. For example,we observe from the sketches in Fig. 6 that the periodduring which Ωγ dominates (shaded orange) is far longerin the ΛCDM case than it is in either of two cases thatinvolve a prior period of stasis. We stress that this short-ening of the radiation-dominated era is not imposed inorder to preserve the age of the universe; indeed, as al-ready noted above, both of the cosmologies that includethe stasis epoch reach the present day only after a longertime interval has elapsed. Rather, this shortening of thesubsequent Ωγ-dominated epoch is required in order toguarantee not only a fixed horizon scale today, but also afixed number of e-folds since MRE. Indeed, both of thesequantities are measured through observational data, im-plying that the final ΛCDM-like portions of the cosmo-logical timelines after MRE may be shifted horizontallyin Fig. 7 but never vertically.

Introducing an epoch of stasis into the standard cos-mological timeline has a number of effects. First, withinmodified cosmologies involving a stasis epoch, the co-moving Hubble radius grows more slowly than it doesin the standard cosmology. This in turn can have nu-merous consequences for observational cosmology. Forexample, perturbations in the matter density reenter thehorizon at a later time in cosmologies involving a stasisepoch than they do in the standard cosmology. Consis-tency with observational data therefore typically requiresthat the number of e-folds between horizon crossing andthe end of inflation must be smaller in such cosmologies.This in turn affects the predictions for inflationary ob-servables [8–11].

Another potential observational consequence of sta-sis stems from its effect on structure formation. Dur-ing a RD epoch, primordial density perturbations whichhave already entered the horizon grow only logarithmi-cally with the scale factor. By contrast, during a stasisepoch, these perturbations could potentially grow muchmore rapidly, as is the case within an EMDE [12–14].Such rapid growth could therefore likewise result in theformation of compact objects such as primordial blackholes [14, 15] or ultra-compact minihalos [12, 16].

Unlike the scenarios illustrated in Figs. 6 and 7, it isalso possible to imagine scenarios in which a period of sta-sis replaces an unorthodox modification to the standardcosmology but otherwise places us back on the same cos-mological timeline. For example, as illustrated in Fig. 8,one particularly well-known modified cosmology consistsof introducing an EMDE entirely within the usual RDera. Such an EMDE may be inserted either earlier withinthe RD era (as shown in purple in Fig. 8) or later (asshown in dark cyan). Nevertheless, as sketched in Fig. 8,it is possible to imagine splicing a stasis epoch into thisregion instead (red). Even though the stasis scenario and

t∼τ N−

1 t∼τ 0

Reh

eati

ng

Infl.

end

t=t(

0)

present horizon scale

Inflation

Osc.Inflaton

RD

Stas

is

MD

EMDE

RD

MD

DE

N

ln(aH

)−1

EMDE

Stasis vs. EMDE

FIG. 8. Sketch of a scenario in which the insertion of a periodof stasis (red) into the standard ΛCDM cosmology (black) re-places either of two possible EMDE insertions (green or pur-ple). This replacement nevertheless leaves the universe onthe same subsequent cosmological timeline. Such a scenariotherefore provides a testing ground for comparing the phe-nomenological effects of an EMDE insertion versus those of astasis insertion.

the EMDE scenario place the universe on the same cos-mological timeline for all subsequent times all the wayto the present day, it would be interesting to explore theextent to which one can use present-day observational in-formation in order to determine which path the universeultimately followed. The scenario in Fig. 8 thus providesa framework within which one can directly compare theeffects of an EMDE insertion against those of a stasis in-sertion. Density perturbations with different wavenum-bers enter the horizon at different times during the differ-ent cosmologies depicted in Fig. 8. Moreover, once theyenter the horizon, they do not scale the same way withthe scale factor a during a period of stasis as they do ina MD epoch. As a result, one would in general expectthese different cosmologies to yield different perturbationspectra, with possible implications for small-scale struc-ture.

A stasis epoch could also impact particle-physics pro-cesses in the early universe in a number of ways. Indeed,any model involving the out-of-equilibrium production ofparticles would be affected by the corresponding modifi-cation of the expansion history. The abundance of darkmatter in our present universe typically depends on theequation of state of the universe both during and afterthe epoch in which that abundance is initially generated(see, e.g., Refs. [17–26]). Likewise, the manner in which alepton or baryon asymmetry evolves with time within theearly universe is often sensitive to the expansion historyof the universe and the extent to which energy can betransferred between different cosmological components.For example, within the context of electroweak baryoge-nesis, the washout of the baryon asymmetry by sphaleron

21

processes is mitigated in scenarios in which the universeexpands faster during the electroweak phase transitionthan it does during radiation domination [27–31]. More-over, entropy production from the late decays of heavyfields, such as those needed to sustain a stasis epoch,can also serve to dilute a pre-existing baryon or leptonasymmetry [32]. This can be advantageous [33] in mod-els such as Affleck-Dine baryogenesis [34–36] in which theinitial baryon asymmetry is typically too large. Modify-ing the expansion history of the universe can also modifythe dynamics which governs the evolution of light scalarfields such as the QCD axion and other, axion-like par-ticles. For example, it has been shown that an EMDEcan render phenomenologically viable certain regions ofparameter space for low-mass axion dark matter whichwould otherwise have been excluded [37]. It is reasonableto expect that a period of cosmological stasis would havesimilar consequences for the dynamics of such fields. Sim-ilarly, if a stochastic gravitational background can be pro-duced prior to or during the stasis epoch, its power spec-trum is also expected to be modified due to the change inthe expansion history and the injection of entropy fromsuccessive φ` decays [38–40].

The splicing of a stasis epoch into the cosmologicaltimeline could also have implications for dark-matterphysics. In this paper, we have not assumed any par-ticular relation between the dark matter and the dynam-ics involved in cosmic stasis. However, most canonicalmechanisms for establishing a cosmological abundanceof dark-matter particles turn out to be compatible withmodified cosmologies which include a stasis epoch.

The extent to which the presence of a stasis epochimpacts the properties of the dark matter ultimately de-pends on the time at which the dark-matter abundanceis established. One possibility is that the dark-matterabundance is generated only after stasis has concluded.In this case, the stasis epoch has essentially no impacton the dark matter. Another possibility is that the dark-matter abundance is established prior to the stasis epochbut remains negligible throughout this epoch and there-fore does not disrupt the stasis itself. In this case, therate at which the dark-matter abundance redshifts as afunction of time is modified because of the different back-ground cosmology — in particular the different equation-of-state parameter w — involved in stasis.

Yet another possibility is that the dark-matter abun-dance is established during the stasis epoch itself. Thispossibility is particularly intriguing, as the dynamics ofdark-matter production can be modified in several waysif this production occurs during a stasis epoch. For ex-ample, when thermal freeze-out occurs during a stasisepoch, the annihilation cross-section required in order toachieve an appropriate late-time abundance for a givendark-matter particle is smaller than it is when freeze-outoccurs in the standard ΛCDM cosmology. This is becausethe production of entropy from φ` decays dilutes thedark-matter relic abundance. Indeed, this is similar tothe case when freeze-out occurs during an EMDE [11, 41–

46]. It is also possible that a non-thermal population ofdark-matter particles could be produced directly fromthe decays of the φ`. The late-time velocities of suchdark-matter particles can be non-negligible, leading to asuppression of power on small scales. Indeed, this is alsoknown to occur in situations in which a significant num-ber of dark-matter particles are produced by the decaysof the oscillating scalar or massive particle species whichdominates the universe during an EMDE [47]. How-ever, the dark-matter velocity distributions which arisefrom φ` decays during a stasis epoch can be expected tobe even more complicated than they would be within anEMDE, as these velocity distributions receive contribu-tions from a large number of states decaying at differenttimes. In such scenarios, methods such as those devel-oped in Ref. [48] could potentially be employed in orderto extract information about the φ`.

While the dark matter need not be related to the φ`, itis also interesting to consider the possibility that the par-ticle species which constitutes the dark matter is in factsimply the most long-lived of these φ` tower states. Con-sistency with the standard cosmology at late times wouldof course require that the lifetime of this lightest state beparametrically longer and its initial abundance paramet-rically smaller than those of the other φ`. However, suchseparations of scales can occur naturally in a number ofnew-physics scenarios, including scenarios wherein the φ`are the Kaluza-Klein modes of a field which propagatesin the bulk in a theory involving extra spacetime dimen-sions.

Given the results of this paper, many avenues for futureresearch suggest themselves. For example, in addition tothe issues discussed above, our work in this paper implic-itly rested on certain assumptions which can be relaxed.One assumption which has been implicit throughout ouranalysis is that the states φ` are all non-relativistic whenthey are produced at t(0). While this is possible if thesestates are produced through a freeze-out mechanism, thiswould typically not be the case if the different states areproduced through the decays of a heavier particle. Asdiscussed in Ref. [48], the phase-space distribution of theφ` particles could therefore be rather non-trivial at thetime t(0). Indeed, this distortion of the phase-space dis-tribution could be viewed as introducing an additionalinitial “edge” effect that would also have to be takeninto account in our analysis.

Another assumption implicitly made throughout ourwork is that the different φ` states remain out of equi-librium until they completely decay. Within a specificparticle-physics model, this assumption would need to beverified. In particular, if stasis takes place while the tem-perature of the thermal bath is sufficiently large, thermalprocesses could induce the thermalization of the lighterstates within the tower and thereby modify their relativeabundances, while the heavier states remain decoupledand act as massive matter.

In our work we also assumed that the φ` states onlydecay into radiation. However, it is possible that the

22

heavy states might decay into lighter φ` states withinthe tower or other light particles beyond the tower. Ofcourse, if the daughter states are sufficiently light, ormore generally if the mass differences between parentsand daughters are sufficiently large, such decays will behighly exothermic. The daughters will then be producedrelativistically and continue to act as radiation. As such,these decays will continue to effectively transfer energydensity from matter back to radiation, and can thereforecontinue to serve as the counterbalance to cosmologicalexpansion which tends to push the relative abundancesthe other way and induce stasis. However, as the universecontinues to cool, these daughter states will eventuallybecome non-relativistic — an effect which then flows inthe opposite direction, effectively transferring radiationback to matter. Indeed, this reversal might serve as an-other means of exiting a stasis epoch. In either case, suchnon-trivial dynamics will produce a non-trivial time evo-lution for the background cosmology. This in turn couldaffect all sectors of the corresponding theory, including itsdark sector, and thereby leave interesting signatures inthe matter power spectrum [48] and corresponding halo-mass distributions [49].

In this context, we emphasize that we have not as-sumed in this work that our φ` states are scalars. In-deed, these fields may well be fermionic, which wouldthen permit decays into final states which are collec-tively fermionic and which might include light matterfields such as neutrinos which could potentially act asradiation, as discussed above. Of course, the spins of theparent and daughter states can potentially affect the setof relevant exponents γ which govern the scalings of thedecay widths Γ` in Eq. (3.1) and which ultimately feedinto the matter abundance ΩM during stasis.

In a similar vein, in this work we have implicitly as-sumed that the energy density ρ` associated with eachmassive field φ` results from particle-like excitations ofthat field. However, if the φ` states are scalars, thesefields may also have homogeneous zero-modes whose co-herent oscillations give rise to energy densities whichscale in exactly the same way. (An example of this phe-nomenon is the well-known coherent oscillation of the ax-ion field.) It therefore follows that a collection of scalarfields φ` with coherently oscillating zero-modes can like-wise support stasis. This is an intriguing possibility, sincelarge numbers of scalar fields are a generic prediction ofstring theory and a variety of other extensions of theStandard Model. Moreover, vacuum-misalignment pro-duction provides a natural mechanism through which aspectrum of energy densities ρ` with power-law scalingscan be generated for such scalars at early times [1].

The non-trivial dynamics of such φ` zero-modes cannevertheless play an interesting role in determining theassociated scaling exponents. Recall that if the misalign-ment production time at which ρ` is produced is suffi-ciently early that H(t) >∼ 2m`/3 for a given `, the cor-responding φ` zero mode will be overdamped and thecorresponding energy density ρ` will behave like vacuum

energy. Indeed, it is only after we reach a time at whichH(t) ∼ 2m`/3 that the φ` zero-mode will “turn on” andbecome underdamped; the corresponding energy densityρ` is then associated with the resulting zero-mode os-cillations and begins to scale like massive matter. Thisnon-trivial process by which such states turn on can havenon-trivial implications for stasis. If the production timeoccurs at a late time when H(t) <∼ 3m`/2 for all ` in thetower, all of the φ` will immediately behave as matterand contribute to ΩM (t) from the moment when theyare produced. We would therefore have realized the ini-tial conditions for our stasis model, and a stasis epochwould merge as long as the relevant parameters satisfythe appropriate constraints, such as those in Eq. (3.19).However, if the production time occurs earlier, it is pos-sible that only the heaviest φ` states will be “turned on”and immediately contribute to ΩM (t) at the productiontime. By contrast, the lighter states will only experiencea subsequent “staggered” turn-on as time progresses [1].This would then have three effects: the effective num-ber of states in the tower contributing to ΩM (t) wouldincrease with time as increasing numbers of states turnon; our system would initially contain a vacuum-energycomponent which eventually drops as a function of time,ultimately becoming relatively small; and the contribu-tions Ω`(t) from the lighter states in the tower will be en-hanced relative our usual expectations at the time whenthey turn on and begin to contribute to the total matterabundance ΩM (t). This enhancement is the result of thefact that these Ω`(t) would have grown as vacuum energyrather than as matter during the intervening time afterproduction but prior to turning on.

At times after all φ` states have turned on, our entiretower acts as matter and there is no remaining vacuum-energy component. At such times, we would also expectto have stasis, but with a modified scaling exponent αwhich reflects the enhancement of the light abundancesΩ` that occurred during the period wherein the lighterφ` were still experiencing their staggered turn-ons. How-ever, if the heavier φ` begin decaying during this stag-gered turn-on phase, the situation is much more compli-cated. At late times during this phase, we expect thevacuum-energy component, though non-zero, to be rel-atively small. It is possible that this will therefore notdisrupt the emergence of stasis during this period, eventhough the effective number of matter states in the toweris still evolving. This case requires further investigation.

Another assumption made in our work is that the val-ues of the scaling exponents (α, γ, δ) and mass param-eters (m0,∆m) remain constant throughout our towerof φ` states. While this is true for the towers of stateswhich emerge in many extensions to the Standard Model,there do exist scenarios in which these exponents them-selves deform from one value to another as one passes be-tween the high-mass and low-mass portions of the sameφ` tower [1, 50]. In such cases, the properties of the corre-sponding stasis epoch can also shift slightly as the stasisproceeds through successive φ` decays. For example, one

23

stasis value of ΩM may persist for many e-folds beforegradually shifting to another value which then persiststhrough an additional extended epoch. It is also pos-sible that some portions of the φ` will have parameters(α, γ, δ) that lie outside the limits in Eq. (3.19). In suchcases, stasis would persist only during the decays of thosestates for which Eq. (3.19) is valid, and thus the effectivesize of the tower may appear truncated relative to N .

In this paper, we have paid significant attention to the“edge” effects which occur at early and late times andwhich are ultimately responsible for the entrance intoas well as the exit from our stasis epoch. However, itmay also be possible to exploit these edge effects in or-der to construct variations of the cosmologies that wehave discussed here. For example, in our “stasis splicedinto RD” scenario in Eq. (6.2), we indicated that anEMDE must immediately precede our stasis epoch. In-deed, given that our stasis model in Sect. III begins witha fully matter-dominated universe as an initial condition,such an EMDE is required if we wish to adopt the modelin Sect. III wholesale when splicing it into the RD epochof the ΛCDM timeline. However, we have seen that theedge effects associated with this model can quickly de-form such an early matter-dominated initial conditioninto one in which most of the initial matter abundanceis quickly dissipated. Indeed, we have already seen fromthe orange/yellow curves in Fig. 5 that such a rapid de-pletion can occur as an initial edge effect for sufficientlylarge values of ΓN−1/H

(0). One could therefore contem-plate splicing a previous RD-dominated epoch directlyonto the time at which much of the initial matter abun-dance ΩM is already depleted. Alternatively (but withsimilar phenomenological consequences), we can imaginethat the creation of the φ` states occurs at the begin-ning of the inserted epoch, but that the chosen value ofΓN−1/H

(0) is sufficiently large that any interval of earlymatter domination is extremely short.

As discussed in Sect. III, theories involving large ex-tra spacetime dimensions naturally give rise to infinitetowers of Kaluza-Klein states which can serve as the φ`.The spectrum of such states depends on the dimension-ality and geometry of the compactification manifold, andthe simplest case in which a single extra dimension iscompactified on a circle (or a Z2 orbifold thereof) re-sults in KK spectra with (m0,∆m, δ) = (m, 1/R, 1) or(m, 1/(2m2R), 2), where R is the compactification radiusand m is the four-dimensional mass of the compactifiedfield. Indeed, the first possibility occurs if mR 1, whilethe second arises if mR 1. Both of these values of δare within the ranges that produce viable stasis values ofΩM .

Clearly, the space of models involving such large ex-tra dimensions is huge, and thus the set of correspond-ing values of α and γ is also huge, depending on themodel-specific details of how these states are producedin the early universe and how they decay to the visi-ble sector. However, one critical model-independent is-sue concerns the extent to which a viable period of stasis

can even arise within such a framework. As we haveseen in Eq. (3.28), a bona-fide period of stasis requiresa relatively large tower of φ` states — i.e., a relativelylarge value of N . Given this, and given that KK theoriesare essentially effective field theories (EFTs), one mighttherefore worry that there might be an intrinsic upperlimit to the size of N . However, this worry ultimatelyturns out to be unfounded because the ultraviolet (UV)cutoff for such theories is set not by the compactificationradius but by other factors which relate to the onset ofnew physics (such as the possible emergence of a grand-unified theory or quantum gravity). Indeed, there existnumerous explicit examples in the literature in which thecompactification scale can be separated by many ordersof magnitude from the scale of new physics. For exam-ple, there is a vast literature, initiated in Refs. [51–55],in which the size of extra spacetime dimensions is setfar below such scales in a self-consistent way. Of course,one might worry that the emergence of KK states woulddrive gauge couplings towards Landau poles (through thesame couplings to the visible sector that induce the φ`decays), but there exist numerous examples where thisdoes not happen, even within theories whose low-energylimits include the Standard Model [56, 57].

Within such frameworks, then, the number of KKstates is therefore essentially unbounded and there is nomodel-independent obstruction to having a large tower ofstates and thus a stasis epoch of long duration. Of course,it will be interesting to actually construct phenomeno-logically viable models of KK-induced stasis. Doing sowill ultimately depend on a number of further model-specific factors, such as the desired placement of thisepoch within the cosmological timeline and the assump-tion of an appropriate theory of particle physics duringthat time (be it the Standard Model or a supersymmetricor grand-unified extension thereof).

Another framework for physics beyond the StandardModel which naturally gives rise to large towers of statesis string theory. String theories generally have criticalspacetime dimensions exceeding four, and thus typicallyinvolve geometric compactifications down to four dimen-sions. For this reason, many of the different string states(particularly those in the so-called “bulk”) are endowedwith towers of additional KK excitations. Moreover,these KK states will generally be unstable and decay,although their particular decay phenomenologies dependon the properties of the specific string model under dis-cussion. All of this is therefore consistent with the possi-bility of achieving KK-induced stasis, as discussed above.

However, string theories include not only towers of KKstates but also towers of Regge excitations. These exci-tations are independent of the spacetime compactifica-tion and are a consequence of the extended nature ofthe string itself. Unlike the KK states (and their closed-string cousins, the winding states), these Regge stateshave mass spectra which correspond to the scaling expo-nent δ = 1/2. In principle, this is not a problem for stasis;we have kept δ arbitrary in our analysis, and nothing pre-

24

cludes δ = 1/2. However, these towers of Regge excita-tions also have a degeneracy of states at each mass levelwhich grows exponentially with the mass of the state.This is the well-known Hagedorn phenomenon [58]. As aresult, such theories tend to have effective energy densi-ties ρ` and abundances Ω` which grow exponentially withthe masses m` before other effects (such as those due toBoltzmann suppression [4]) are included. It would there-fore be interesting to explore the extent to which stasiscan arise in such scenarios, or more generally in theorieswith non-power-law scaling relations.

The above comments regarding Regge excitations areindependent of the relevant scales in these theories. Forthis reason we have not worried about states which mightbe super-Planckian and which might therefore form blackholes. However, one important special class of string the-ories consists of those strings whose radii of compact-ification are very large compared to the inverse stringscale. These are precisely the strings that are capableof yielding reduced GUT [52, 54], Planck [51, 53, 55],and string [52, 59, 60] scales. Depending on the detailsof their constructions, such strings may even be effec-tively stable without spacetime supersymmetry [57, 61].In general, such string theories have densely populatedtowers of light KK states whose masses lie well belowthese reduced GUT, Planck, or string scales. Such stringsare thus prime candidates for producing not only a KK-induced stasis, as described above, but one which is re-alized within a full string-theoretic (and therefore UV-complete) framework. Moreover, the Regge excitationswithin such frameworks play no role at scales below thestring scale and can therefore be disregarded as far asstasis is concerned.

A third framework for physics beyond the StandardModel which gives rise to an infinite tower of φ` statesconsists of QCD-like theories at strong coupling. Insuch cases, the fundamental QCD-like degrees of free-dom (e.g., “quarks”) are bound into an infinite spec-trum of composite objects (“hadrons”). It turns outthat such theories can be analyzed via the so-calledAdS/CFT correspondence [62]. Through this correspon-dence, such strongly-coupled theories map onto five-dimensional gravitational theories whose low-energy lim-its are KK theories on anti-de Sitter (AdS) spacetimes.For example, the scale ΛIR below which the strongly-coupled theory becomes confining and the cutoff scaleΛUV above which the effective theory involving the“quarks” breaks down are related by ΛIR = ΛUVe

−πkR,where R is the radius of the extra dimension and k isthe AdS curvature scale. Thus, within the πkR 1regime, there is a significant separation ΛUV ΛIR be-tween these scales. Moreover, within this same regime,the mass m0 ∼ πmKK of the lightest composite state inthe theory and the difference m`+1 − m` ∼ πmKK be-tween the masses of adjacent states in the tower are bothset by the scale mKK ≡ ke−πkR [63]. Thus, one finds thatfor πkR 1, there can exist a large number of stateswith masses below the cutoff scale ΛUV, as required for

stasis.

Vacuum-misalignment production provides a naturalmechanism for generating a spectrum of abundances forthe composite states in strongly-coupled theories of thissort. In particular, if there also exists a fundamentalscalar in the theory which dynamically acquires a mass asthe result of a phase transition at a critical temperatureTc ΛIR, the composite states can acquire abundancesby mixing with this fundamental scalar [64]. The scal-

ing behavior of Ω(0)` with m` depends on the details of

the model and the background cosmology. For example,one finds that α = −1 in situations in which all of themass-eigenstate fields of the theory at temperatures be-low Tc are sufficiently heavy that their zero-modes beginoscillating immediately at the time of the phase transi-tion. By contrast, in situations in which the zero-modesfor these fields begin oscillating only at later times, onefinds that α < −1.

In general, in order to assess whether a cosmologicalmodel involving a large tower of states might potentiallygive rise to a stasis epoch, one must examine whetherthe effective scaling exponents α, γ, and δ obtained forthat cosmology satisfy the criterion in Eq. (3.19). Thisremains true even when these states are composite. Inthe regime in which the dynamically-generated mass ofthe fundamental scalar is small compared to all otherrelevant scales in the theory, the decay widths typicallyscale with m` such that γ ≈ 4 [64]. On the other hand,mixing with the fundamental scalar does not dramati-cally alter the mass spectrum of the theory within theπkR 1 regime, whereupon m`+1−m` remains roughlyconstant [65]. This implies that δ ≈ 1. However, a moredetailed analysis reveals that this mass splitting decreasesslightly with `, implying that the effective value of δ forsuch a tower of states is actually slightly less than 1. Asa result, for the case in which α = −1, one finds that atower of partially composite “hadrons” is at least approx-imately consistent with the criterion in Eq. (3.19) andthus appears promising as a potential model for cosmicstasis. It will be interesting to further explore possiblerealizations of stasis along these lines.

Finally, let us discuss the possibility that the φ` areactually primordial black holes (PBHs). The abun-dances of PBHs scale as matter, and thus an earlyPBH-dominated epoch can serve as the initial matter-dominated epoch required for stasis. Of course, suchPBHs do not produce radiation via particle-like decay —they do so via evaporation. This is nevertheless a processthat effectively converts matter into radiation, and thusit is possible that this too could yield a stasis-like solutionthat counterbalances the redshifting effects of cosmic ex-pansion. One novel feature is that lighter PBHs tend toevaporate more rapidly than heavier PBHs. However, aslong as the abundances of these lighter PBHs are greaterthan those of the heavier PBHs, one could still poten-tially achieve a stasis; this stasis would simply proceedup the tower, from lighter to heavier PBHs, rather thandown. Another novel feature is that black-hole evapora-

25

tion does not follow an exponential decay law; rather, themass of an evaporating black hole (and therefore the en-ergy density of the corresponding PBH population) dropswith time according to a non-trivial function which canin various regimes be approximated as a power-law. Itis nevertheless possible that stasis can be achieved evenwith such a change in this functional form. We leave thisquestion for further study [7].

Many aspects of our work are tangentially related toideas which have already been discussed in the literature.For example, the idea that the Friedmann equations canadmit attractor solutions which cause certain cosmologi-cal components to evolve in predictable ways is of coursenot a new one. Within the context of quintessence, for ex-ample, so-called “tracker solutions” have been identifiedwherein the equation-of-state parameter for the scalarsector evolves toward the equation-of-state parameter forthe dominant background component, be it either matteror radiation [66]. Certain types of interactions betweenthe dark-energy sector and the matter sector at the levelof Friedmann equations have also been exploited in or-der to engineer attractor solutions to these equations,with potential implications for the cosmic coincidenceproblem [67–70]. It can also be shown that there ex-ist certain types of interactions which can be added tothe equations of motion for the cosmological componentswithout disturbing the Lotka-Volterra structure of theseequations [71, 72]. Such “jungle universe” models oftenexhibit non-trivial attractor solutions.

Despite these similarities, the stasis scenario presentedin this paper differs from these other cosmological sce-narios in several crucial ways. First, we do not assumeany unorthodox equation of state for any of the particlespecies involved in our scenario. We also do not introduceany ad hoc suppositions concerning the form of the scalarpotential in our theory, nor do we posit the existence ofany additional interaction terms in the Friedmann equa-tions. Instead, we demonstrated that stasis emerges asthe result of a subtle and complex interplay between theeffects of cosmic expansion and the conversion of mat-ter to radiation through particle decays. This is typi-cally not the case for the “tracking” solutions that arisein quintessence models, wherein the scalar sector simplymimics the background rather than modifying the equa-tion of state of the universe as a whole. Jungle-universemodels, by contrast, are more akin to our stasis scenarioin this respect. Nevertheless, the structure of the dynam-ics which gives rise to a stasis epoch is fundamentallydifferent from the dynamics which governs such mod-els. Likewise, the dynamics underlying stasis does notrequire the introduction of any interaction terms withinthe Friedmann equation that do not, a priori , originatefrom fundamental particle interactions.

The above ideas concerning the physical implicationsof stasis are likely only the tip of the iceberg, and newphenomenological possibilities involving stasis are likelyto continue to present themselves. Of course, this is to beexpected. In general, the expansion of the universe has

far-reaching implications for almost everything containedwithin it. As a result, there are many possible “clocks”that can be used to measure the passage of cosmologicaltime. One of these clocks, for example, is based directlyon the expansion itself, tracking the number of e-foldsof growth in the cosmological scale factor. However, an-other clock is based on the abundances of the different en-ergy components and the passage between cosmologicalepochs. Viewed from this perspective, stasis representsa way of suspending the passage of time for the secondclock while allowing the first clock to keep ticking. Thisin turn causes the different clocks to fall out of alignment,implying that standard abundance-based time-markers(such as the moment of matter-radiation equality) maynow occur within a universe whose overall size is quitedifferent than normally assumed. Even more impor-tantly, this method of misaligning the clocks introduceslong-lived mixed-component epochs whose equations-of-state parameters w lie between 0 and 1/3 and remainfixed throughout the entire interval. Indeed, as we haveseen, this implies that we can have long-lived epochswhich are not radiation-dominated or matter-dominated,and in fact are not dominated by any particular compo-nent at all. This decoupling of the different clocks, to-gether with the existence of such stable mixed-componentepochs, thus introduces new degrees of flexibility intoearly-universe model-building, and it is likely that thesefeatures can be exploited to address a number of cosmo-logical puzzles. The implications of stasis are thus ripefor future exploration.

ACKNOWLEDGMENTS

The research activities of KRD are supported in partby the U.S. Department of Energy under Grant DE-FG02-13ER41976 / DE-SC0009913, and also by the U.S.National Science Foundation through its employee IR/Dprogram. The work of LH is supported in part by theU.K. Science and Technology Facilities Council (STFC)under Grant ST/P001246/1. The work of FH is sup-ported in part by the International Postdoctoral Ex-change Fellowship Program; by the National NaturalScience Foundation of China under Grants 12025507,11690022, 11947302, 12022514, and 11875003; by theStrategic Priority Research Program and Key ResearchProgram of Frontier Science of the Chinese Academyof Sciences under Grants XDB21010200, XDB23010000,and ZDBS-LY-7003; and by the CAS project for YoungScientists in Basic Research YSBR-006. The work of DKis supported in part by the U.S. Department of Energyunder Grant DE-SC0010813. The work of TMPT is sup-ported in part by the U.S. National Science Foundationunder Grant PHY-1915005. The research activities ofBT are supported in part by the U.S. National ScienceFoundation under Grant PHY-2014104. The opinionsand conclusions expressed herein are those of the authors,and do not represent any funding agencies.

26

QUANTUM MECHANICS COSMOLOGY

wavepacket broadening increasing ΩM from cosmological expansion (due to Ωγ → ΩM )confining harmonic-oscillator potential counterbalancing effect ΩM → Ωγ due to particle decays

width/shape of wavepacket abundances ΩM , Ωγcoherent state (fixed width/shape) stasis state (fixed ΩM ,Ωγ)

tower of oscillator states |`〉 tower of decaying states φ`specific linear combination in Eq. (A1) specific scaling relations in Eq. (3.1)

lowering operator a time evolution of sequential decays down the towercoherent state as eigenstate of a stasis state invariant under time evolution of sequential decays

true (stable) ground state (η = 0) state with vanishing ΩM or Ωγ (trivially stable epoch)excited coherent state (η 6= 0) stasis state with both ΩM ,Ωγ 6= 0

infinite tower of |`〉 states no edge effectswith |` = 0〉 properly annihilated by a at top or bottom of tower

=⇒ exact coherent state =⇒ stasis state eternaltruncate |`〉 tower at top initial edge effects

=⇒ small-scale fluctuations of packet shape =⇒ stasis state not valid to arbitrarily early timesstable orbit in phase space local attractor in (ΩM , 〈ΩM 〉) space

minimum uncertainty relation global attractor in (ΩM , 〈ΩM 〉) space

TABLE I. A proposed analogy between quantum-mechanical coherent states and cosmological stasis states.

Appendix A: Stasis as a cosmological coherentstate?

In this Appendix we present a “bonus track” — aninteresting analogy between stasis states and quantum-mechanical coherent states. Our purpose here is merelyto highlight some similarities which, although purelyspeculative, might eventually be developed into a morerigorous correspondence.

One hallmark of quantum-mechanical systems is thatwavepackets broaden. This expectation is true in freespace, but it is possible to counterbalance this broadeningby considering our system under the influence of an ap-propriate confining potential. With this potential takento be that of a harmonic oscillator, one can constructa wavepacket in which this tendency towards broaden-ing is exactly cancelled. This wavepacket is that of aso-called “coherent” state. Indeed, if a† and a are theharmonic-oscillator raising and lowering operators, thecoherent state |η〉 parametrized by any η ∈ C is given as

|η〉 ∼∞∑

`=0

η`√`!|`〉 (A1)

where |`〉 ≡ (a†)`|0〉/√`!. One defining property of a

coherent state is that it is an eigenstate of the loweringoperator: a|η〉 = η|η〉. Another is that this state is aspatially translated version of the true ground state (η =0). Just like the true ground state of the system, coherentstates have stable orbits in phase space and satisfy theminimum uncertainty relations.

Ultimately the coherent state maintains its coherenceas a result of the careful balancing of the different con-tributing states |`〉 within Eq. (A1). The lowering opera-tor a takes us from each state |`〉 to the state immediatelybelow it within the tower, namely |` − 1〉. The particu-

lar combination of |`〉 states within Eq. (A1) is then aneigenstate of a because our tower of |`〉 states is infinite,with the lowest state annihilated by a.

Given these observations, it is possible to proposean analogy, as shown in Table I, between quantum-mechanical coherent states and our cosmological stasisstates. Of course, such an analogy is at best highly spec-ulative and awaits more rigorous development.

27

[1] K. R. Dienes and B. Thomas, Phys. Rev. D 85, 083523(2012), arXiv:1106.4546 [hep-ph].

[2] K. R. Dienes and B. Thomas, Phys. Rev. D 85, 083524(2012), arXiv:1107.0721 [hep-ph].

[3] K. R. Dienes and B. Thomas, Phys. Rev. D 86, 055013(2012), arXiv:1203.1923 [hep-ph].

[4] K. R. Dienes, F. Huang, S. Su, and B. Thomas, Phys.Rev. D 95, 043526 (2017), arXiv:1610.04112 [hep-ph].

[5] K. R. Dienes, J. Fennick, J. Kumar, and B. Thomas,Phys. Rev. D 97, 063522 (2018), arXiv:1712.09919 [hep-ph].

[6] M. S. Turner, Phys. Rev. D 28, 1243 (1983).[7] K. R. Dienes, L. Heurtier, F. Huang, D. Kim, J.-C. Park,

S. Shin, T. M. P. Tait, and B. Thomas, to appear.[8] R. Easther, R. Galvez, O. Ozsoy, and S. Watson, Phys.

Rev. D 89, 023522 (2014), arXiv:1307.2453 [hep-ph].[9] K. Das, K. Dutta, and A. Maharana, Phys. Lett. B 751,

195 (2015), arXiv:1506.05745 [hep-ph].[10] R. Allahverdi, K. Dutta, and A. Maharana, JCAP 10,

038 (2018), arXiv:1808.02659 [astro-ph.CO].[11] L. Heurtier and F. Huang, Phys. Rev. D 100, 043507

(2019), arXiv:1905.05191 [hep-ph].[12] A. L. Erickcek and K. Sigurdson, Phys. Rev. D 84,

083503 (2011), arXiv:1106.0536 [astro-ph.CO].

[13] J. Fan, O. Ozsoy, and S. Watson, Phys. Rev. D 90,043536 (2014), arXiv:1405.7373 [hep-ph].

[14] J. Georg, G. Sengor, and S. Watson, Phys. Rev. D 93,123523 (2016), arXiv:1603.00023 [hep-ph].

[15] A. M. Green, A. R. Liddle, and A. Riotto, Phys. Rev. D56, 7559 (1997), arXiv:astro-ph/9705166.

[16] G. Barenboim and J. Rasero, JHEP 04, 138 (2014),arXiv:1311.4034 [hep-ph].

[17] F. D’Eramo, N. Fernandez, and S. Profumo, JCAP 05,012 (2017), arXiv:1703.04793 [hep-ph].

[18] S. Hamdan and J. Unwin, Mod. Phys. Lett. A 33,1850181 (2018), arXiv:1710.03758 [hep-ph].

[19] M. Drees and F. Hajkarim, JCAP 02, 057 (2018),arXiv:1711.05007 [hep-ph].

[20] F. D’Eramo, N. Fernandez, and S. Profumo, JCAP 02,046 (2018), arXiv:1712.07453 [hep-ph].

[21] R. Allahverdi and J. K. Osinski, Phys. Rev. D 99, 083517(2019), arXiv:1812.10522 [hep-ph].

[22] P. Arias, N. Bernal, A. Herrera, and C. Maldonado,JCAP 10, 047 (2019), arXiv:1906.04183 [hep-ph].

[23] R. Allahverdi and J. K. Osinski, Phys. Rev. D 101,063503 (2020), arXiv:1909.01457 [hep-ph].

[24] M. Sten Delos, T. Linden, and A. L. Erickcek, Phys. Rev.D 100, 123546 (2019), arXiv:1910.08553 [astro-ph.CO].

[25] M. A. G. Garcia, K. Kaneta, Y. Mambrini, andK. A. Olive, Phys. Rev. D 101, 123507 (2020),arXiv:2004.08404 [hep-ph].

[26] A. Cheek, L. Heurtier, Y. F. Perez-Gonzalez, andJ. Turner, (2021), arXiv:2107.00016 [hep-ph].

[27] M. Joyce, Phys. Rev. D 55, 1875 (1997), arXiv:hep-ph/9606223.

[28] M. Joyce and T. Prokopec, Phys. Rev. D 57, 6022 (1998),arXiv:hep-ph/9709320.

[29] S. Davidson, M. Losada, and A. Riotto, Phys. Rev. Lett.84, 4284 (2000), arXiv:hep-ph/0001301.

[30] G. Servant, JHEP 01, 044 (2002), arXiv:hep-ph/0112209.[31] G. Barenboim and J. Rasero, JHEP 07, 028 (2012),

arXiv:1202.6070 [hep-ph].[32] G. Nardini and N. Sahu, (2011), arXiv:1109.2829 [hep-

ph].[33] J. Bramante and J. Unwin, JHEP 02, 119 (2017),

arXiv:1701.05859 [hep-ph].[34] I. Affleck and M. Dine, Nucl. Phys. B 249, 361 (1985).[35] M. Dine, L. Randall, and S. D. Thomas, Nucl. Phys. B

458, 291 (1996), arXiv:hep-ph/9507453.[36] M. Dine and A. Kusenko, Rev. Mod. Phys. 76, 1 (2003),

arXiv:hep-ph/0303065.[37] A. E. Nelson and H. Xiao, Phys. Rev. D 98, 063516

(2018), arXiv:1807.07176 [astro-ph.CO].[38] N. Bernal and F. Hajkarim, Phys. Rev. D 100, 063502

(2019), arXiv:1905.10410 [astro-ph.CO].[39] R. Allahverdi et al., (2020), 10.21105/astro.2006.16182,

arXiv:2006.16182 [astro-ph.CO].[40] H.-K. Guo, K. Sinha, D. Vagie, and G. White, JCAP

01, 001 (2021), arXiv:2007.08537 [hep-ph].[41] G. F. Giudice, E. W. Kolb, and A. Riotto, Phys. Rev.

D 64, 023508 (2001), arXiv:hep-ph/0005123.[42] G. B. Gelmini and P. Gondolo, Phys. Rev. D 74, 023510

(2006), arXiv:hep-ph/0602230.[43] G. Gelmini, P. Gondolo, A. Soldatenko, and C. E.

Yaguna, Phys. Rev. D 74, 083514 (2006), arXiv:hep-ph/0605016.

[44] A. L. Erickcek, Phys. Rev. D 92, 103505 (2015),arXiv:1504.03335 [astro-ph.CO].

[45] A. Berlin, D. Hooper, and G. Krnjaic, Phys. Lett. B760, 106 (2016), arXiv:1602.08490 [hep-ph].

[46] A. Berlin, D. Hooper, and G. Krnjaic, Phys. Rev. D 94,095019 (2016), arXiv:1609.02555 [hep-ph].

[47] C. Miller, A. L. Erickcek, and R. Murgia, Phys. Rev. D100, 123520 (2019), arXiv:1908.10369 [astro-ph.CO].

[48] K. R. Dienes, F. Huang, J. Kost, S. Su, and B. Thomas,Phys. Rev. D 101, 123511 (2020), arXiv:2001.02193[astro-ph.CO].

[49] K. R. Dienes, F. Huang, J. Kost, K. Manogue, andB. Thomas, (2021), arXiv:2101.10337 [astro-ph.CO].

[50] K. R. Dienes, J. Kost, and B. Thomas, Phys. Rev. D95, 123539 (2017), arXiv:1612.08950 [hep-ph].

[51] N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali,Phys. Lett. B 429, 263 (1998), arXiv:hep-ph/9803315.

[52] K. R. Dienes, E. Dudas, and T. Gherghetta, Phys. Lett.B 436, 55 (1998), arXiv:hep-ph/9803466.

[53] I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, andG. R. Dvali, Phys. Lett. B 436, 257 (1998), arXiv:hep-ph/9804398.

[54] K. R. Dienes, E. Dudas, and T. Gherghetta, Nucl. Phys.B 537, 47 (1999), arXiv:hep-ph/9806292.

[55] N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali,Phys. Rev. D 59, 086004 (1999), arXiv:hep-ph/9807344.

[56] K. R. Dienes, E. Dudas, and T. Gherghetta, Phys. Rev.Lett. 91, 061601 (2003), arXiv:hep-th/0210294.

[57] S. Abel, K. R. Dienes, and E. Mavroudi, Phys. Rev. D97, 126017 (2018), arXiv:1712.06894 [hep-ph].

[58] R. Hagedorn, Nuovo Cim. Suppl. 3, 147 (1965).[59] J. D. Lykken, Phys. Rev. D 54, R3693 (1996), arXiv:hep-

th/9603133.[60] Z. Kakushadze and S. H. H. Tye, Nucl. Phys. B 548, 180

(1999), arXiv:hep-th/9809147.[61] S. Abel, K. R. Dienes, and E. Mavroudi, Phys. Rev. D

28

91, 126014 (2015), arXiv:1502.03087 [hep-th].[62] J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998),

arXiv:hep-th/9711200.[63] T. Gherghetta and A. Pomarol, Nucl. Phys. B 586, 141

(2000), arXiv:hep-ph/0003129.[64] Y. Buyukdag, K. R. Dienes, T. Gherghetta, and

B. Thomas, Phys. Rev. D 101, 075054 (2020),arXiv:1912.10588 [hep-ph].

[65] B. Batell and T. Gherghetta, Phys. Rev. D 76, 045017(2007), arXiv:0706.0890 [hep-th].

[66] I. Zlatev, L.-M. Wang, and P. J. Steinhardt, Phys. Rev.Lett. 82, 896 (1999), arXiv:astro-ph/9807002.

[67] L. P. Chimento, A. S. Jakubi, and D. Pavon, Phys. Rev.D 62, 063508 (2000), arXiv:astro-ph/0005070.

[68] E. J. Copeland, M. Sami, and S. Tsujikawa, Int. J. Mod.Phys. D 15, 1753 (2006), arXiv:hep-th/0603057.

[69] S. Tsujikawa, (2010), 10.1007/978-90-481-8685-3-8,arXiv:1004.1493 [astro-ph.CO].

[70] G. M. Kremer and O. A. S. Sobreiro, Braz. J. Phys. 42,77 (2012), arXiv:1109.5068 [gr-qc].

[71] J. Perez, A. Fuzfa, T. Carletti, L. Melot, andL. Guedezounme, Gen. Rel. Grav. 46, 1753 (2014),arXiv:1306.1037 [gr-qc].

[72] A. Simon-Petit, H.-H. Yap, and J. Perez (2016)arXiv:1603.02267 [gr-qc].