Late-time vacuum phase transitions: Connecting sub-eV scale physics with cosmological structure formation

0.76 MB·

by  Amol V. Patwardhan

Late-time vacuum phase transitions: Connecting sub-eV scale physics with
cosmological structure formation
Amol V. Patwardhan∗
Department of Physics, University of California,
San Diego, La Jolla, California 92093-0319, USA
George M. Fuller†
Department of Physics, University of California,
San Diego, La Jolla, California 92093-0319, USA and
Kavli Institute for Cosmological Physics, The University of Chicago, Chicago, Illinois 60637, USA
4
(Dated: September29, 2014)
1
0 Weshowthataparticularclassofpostrecombinationphasetransitionsinthevacuumcanleadto
2 localizedoverdenseregionsonrelativelysmallscales,roughly106 to1010M⊙,potentiallyinteresting
p fortheorigin oflargeblackholeseedsandfordwarfgalaxyevolution. Ourstudysuggests thatthis
e mechanism could operateoverarangeofconditions which areconsistent with currentcosmological
S andlaboratorybounds. Onebyproductofphasetransitionbubble-walldecaymaybeextraradiation
energy density. This could provide an avenue for constraint, but it could also help reconcile the
5
2 discordant values of the present Hubble parameter (H0) and σ8 obtained by cosmic microwave
background (CMB) fits and direct observational estimates. We also suggest ways in which future

probes, including CMB considerations (e.g., early dark energy limits), 21-cm observations, and
]
O gravitational radiation limits, could provide more stringent constraints on this mechanism and the
sub-eV scale beyond-standard-model physics, perhaps in the neutrino sector, on which it could be
C
based. Late phase transitions associated with sterile neutrino mass and mixing may provide a way
h. toreconcile cosmological limits and laboratory data, should a future disagreement arise.
p
- PACSnumbers: 95.36.+x,14.60.Pq,05.30.Rt,98.80.Es
o
r
t
s I. INTRODUCTION a standard ΛCDM cosmologicalmodel (i.e., a cosmolog-
a ical constant + cold dark matter) with Gaussian initial
[
Inthispaperweinvestigatethepotentialconsequences fluctiations (presumably from inflation), remains a chal-

3 ofnewsub-eVscalephysics,specificallythecosmological lenge, and there have been several attempts to address
v implicationsofavacuumphasetransitionoccurringafter this question [18–28].
3
2 photon decoupling. The experimental revelation of neu- Wasserman,in Ref. [9], suggesteda novelway of deal-
9 trino mass and flavor mixing physics, and the puzzle of ing with this problem, although the primary motivation
1 the originofneutrino masses,providespeculative license behind that study was an attempt to explain the orga-
1. for this investigation [1–7], and lower energy-scale phase nization of large-scale structure. That work outlined a
0 transitions in the early Universe have been considered mechanism through which a first-order vacuum phase
4 before [2–4, 8–13]. While our considerations are generic transition could gravitationally bind comoving regions
1 and need not pertain exclusively to the neutrino sector, with scales small compared to the horizon size. In this
v: the work presented here attempts to connect this specu- paper we revisit this Wasserman mechanism. We mod-
i lative vacuum physics both with emerging observational ify this mechanism, highlight the role played by the cur-
X probes and with unresolved problems in cosmology, in rent observed vacuum (dark) energy Λ, and show how
r particular the origin of the seeds for supermassive black it renders the binding process significantly more difficult
a
holes. to accomplish. Nevertheless, we demonstrate that this
Advances in observational astronomy in the last few mechanismcanproducenonlinearregimefluctuationson
decades have allowed us to probe objects and structure scalesroughly106–1010M ,athighredshifts(z ∼ 3–10,
⊙
at high redshifts, opening up opportunities to examine for a phase transition occurring at z ∼ 50–500), and is
thestateofthe Universeatremoteepochs. Forexample, subjecttoconstraintbycurrentandfuture observations.
observations have provided evidence for the existence of We limit our analysis to phase transitions in the postre-
ultra-massive black holes (∼109–1010M⊙) at high red- combinationerasoastobypassthecomplicationsassoci-
shifts (z ≈ 5–7) [14–17], corresponding to epochs where atedwithdampingofdensityperturbationsviaradiation
the Universe was only of order a billion years old. Ex- diffusion.
plaining the existence of these objects, in the context of
In Sec. II we provide an overview of the local dy-
namics and conserved quantities in the expansion of the
Universe, and in Secs. III and IV we build on this and
∗ [email protected] describe the Wasserman mechanism and the physics of
† [email protected] cosmic vacuum phase transition nucleation in this con-
2
text. Issues surrounding fluctuation binding and growth Dividing by mr2/2, we obtain the familiar Friedmann
are discussed in Sec. V. Observational and experimental equation
constraints and probes are outlined in Sec. VI, and con-
clusions are given in Sec. VII, along with speculations a˙2(t)− 8πGρa2(t)=−k, (2)
about possible connections to neutrino physics. 3
where we take k =−2E/mr2. The constant k is related
tothespatialRiccicurvaturescalarandcantaketheval-
II. BACKGROUND
ues±1or0. Theenergydensityρ includescontributions
fromnonrelativistic(ρ ),relativistic(ρ ), andvacuum
NR R
We take the Universe prior to the phase transition
(ρ ) energy densities. Observational data suggest that
to be homogeneous and isotropic and described by the vac
our Universe is “critically dense” [29–33], corresponding
Friedmann-Lemaˆıtre-Robertson-Walker(FLRW) metric.
to k =0 inEq. (2),i.e., zerototalenergyonanycomov-
At any time t, the proper distance d(t) of a point on
ing spherical shell. We can thus write
an imaginary spherical shell of comoving radius r (e.g.,
Fig. 1), from its center, is given by d(t) = ra(t), where 8πG
a(t) is the scalefactor. The Hubble parameteris H(t)≡ a˙2(t)− (ρNR+ρR+ρvac)a2(t)=0. (3)
3
a˙(t)/a(t), where a˙ ≡ da/dt is the derivative of the scale
factor with respect to FLRW coordinate time t. The lo- Thevariouscomponentsofenergydensitydifferinthe
cationofthe center of the shell canbe chosenarbitrarily mannerinwhichthey dependonthe scalefactor: ρ ∝
NR
because of the spacetime symmetry, and is conveniently a−3 (follows frommass conservation),whereasρ ∝a−4
R
taken to be at the origin of our coordinate system. (a consequence of Stefan-Boltzmann law), and ρ does
vac
The evolutionof the scale factor with time is given by not depend on the scale factor at all. Consequently, if
the Friedmann equation, which can be derived from the we define the scale factor to be a(t ) ≡ 1 at some initial
i
homogeneityandisotropysymmetryofthisspacetimevia time t , then at any subsequent time t we have
i
Birkhoff’s theorem. In the context of an FLRW space-
time, for regions small compared to the causal horizon a˙2(t)− 8πG ρNR(ti) + ρR(ti) +ρ a2(t)=0. (4)
length, i.e., d(t) ≪ H−1(t), Birkhoff’s theorem implies 3 (cid:20) a3(t) a4(t) vac(cid:21)
that the “totalmechanical energy” of a comoving spher-
icalshellis conserved. Forasphericalshellofcoordinate The relative mix of the various energy densities con-
radius r, this condition can be written as tributing to the gravitational potential will then dictate
how a comoving volume evolves with time.
1 G(4/3)πr3a3(t)mρ
mr2a˙2(t)+ − =E, (1)
2 (cid:18) ra(t) (cid:19)
III. TRANSITION DYNAMICS
where G≡1/m2 is the gravitational coupling constant,
P
andmP ≈1.22×1022 MeV is the Planckmass. The two Following Wasserman[9], we assume that a first-order
terms on the left-hand side of the equation can be in- phasetransitioninthevacuumtakesplaceatsomeepoch
terpretedasthekineticandthe“gravitationalpotential” after photon decoupling. The transition causes separa-
energies of a test particle of negligible mass m on the tion of phases via a bubble nucleation process [34–36],
spherical shell. Here ρ is the total mass-energy density. leadingtorelativelysmall,initiallysphericaldensityfluc-
tuations, distributed more or less evenly in space. The
vacuum energy density ρ is assumed to drop across
vac
the bubble wall,from aninitial value ρ inthe unbroken
v
phase to its currently observed value ρ ≈ 3.5 keV/cm3
Λ
in the broken phase.
Consider the evolution of a bubble that nucleates at
time t=t , atthecenterofacomovingsphericalshell,
nuc
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:2)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:7) i.e., at r = 0. As shown in Refs. [34–37], the spherical
bubble containing the broken phase expands outwards,
quickly ramping up to relativistic speed, and “sweeping
up” the vacuum energy (ρ −ρ ) from the surrounding
v Λ
unbrokenphase onto its boundary (e.g., see Fig. 2). An-
other, perhaps more correct, interpretation of this phe-
nomenonwouldbe to identify the difference betweenthe
vacuum energies of the unbroken and the broken phase
asakineticenergy,whichresidesatthe expandingphase
FIG. 1. A comoving coordinate sphere expanding with the boundary/bubble wall (not to be confused with the ki-
Hubbleflow prior to theonset of the phasetransition netic energy of the comoving shells).
3
side in Eq. (5). Upon simplification, we obtain the evo-
lution equation for the modified scale factor a(t;r),
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:2)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:7)
8πG ρ ρ
a˙2(t;r)= NR,n + R,n +ρ a2(t;r)
3 (cid:20)a3(t;r) a4(t) Λ(cid:21)
(6)
8πG
+ (ρ −ρ )a2(r),
3 v Λ c
(cid:13)(cid:4)(cid:5)(cid:6)(cid:7)(cid:2)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:7)
where a (r)≡a(t (r))=a(t (r);r).
c c c
B. Energy redistribution
Suppose the expansion of the bubble wall stops at
t = t > t (r) when it collides with an adjacent ex-
f c
FIG. 2. A bubble of the broken phase (shaded) nucleates
panding bubble (e.g., Fig. 3). Let us assume (again, in
inside a comoving spherical shell (dashed line) and expands
tunewithRef.[9])thatthebubblewallsdisintegrate,and
relativistically. As it expands, it sweeps up most of the vac-
theswept-upenergyisredistributedviaemissionofsome
uumenergyfromtheunbrokenphaseontoitswall(thickgray
kind of radiation that quickly fills up the fluctuation re-
line).
gion uniformly with a relativistic energy density ρ −ρ
v Λ
(which then begins to redshift away with the scale fac-
tor). We shall(for the mostpart)remainagnosticabout
A. Shell crossing
the identity of this radiation, only assuming that all its
couplingsareweakenoughtoallowustoneglectanynon-
Suppose the expanding bubble wall crosses a comov-
gravitational effects. Admittedly, this is just one among
ing spherical shell of coordinate radius r at some time
several possible outcomes of such a vacuum phase tran-
t (r) > t . For t < t (r), the equation of motion for
c nuc c sition,thedynamicsofwhicharegovernedbythe under-
the comoving spherical shell has exactly the same form
lying physics. References [38–40] have explored some of
as Eq. (4) with t = t [let us define a(t ) ≡ 1], and
i nuc nuc the other possibilities through numerical simulations of
ρ = ρ . However, for t > t (r) the expanding bubble
vac v c bubble collisions. Once the fluctuation region has been
wall carrying the swept-up vacuum energy has escaped
filled up, we can use Birkhoff’s theorem again and write
the interior of the comoving sphere. The equation of
motion of the comoving sphere therefore changes, as its
kineticenergyisthesame,butitsgravitationalpotential
energy is now smaller in magnitude. We now define the
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:2)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:7)
proper distance from the origin to the comoving shell as
d(t) = ra(t;r), where a(t;r) is the modified scale factor
forthatcomovingshell,inside the bubble volume. Using
Birkhoff’s theorem,the equationfor energyconservation
can now be written as (suppressing the mr2 factors)
1a˙2(t;r)− 4πGa3(t;r) ρNR,n + ρR,n +ρ (cid:13)(cid:4)(cid:5)(cid:6)(cid:7)(cid:2)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:7)
2 3a(t;r) (cid:20)a3(t;r) a4(t) Λ(cid:21)
(5)
4πGa3(t (r);r)(ρ −ρ )
c v Λ
= ,
3a(t (r);r)
c
where the symbols labeled with the subscript “,n” are
beingevaluatedatt=t ,i.e.,attheonsetofthephase
nuc
transition. As the energy swept up on the bubble wall
escapes the interior of the comoving sphere, the vacuum FIG. 3. The expanding bubble crosses the comoving sphere
energy term in the gravitationalpotential drops from ρ and collides with an adjacent bubble. We assume that the
v
to ρ , making the potential less negative. The comov- bubble walls disintegrate as they collide, radiating away the
Λ
ing shell thus becomes temporarily unbound, acquiring swept-upvacuumenergy(i.e.,thekineticenergyofthebubble
walls).
a positive total energy which appears on the right-hand
4
the condition for energy conservation as be shown to be of O(1) or bigger (we have used B = 1
1
in all our calculations). Equation (9), in its stated form,
1 4πG ρ ρ
a˙2(t;r)− NR,n + R,n isstrictly validonlyforradiation-dominatedepochs, but
2 3 (cid:20)a3(t;r) a4(t) the differences between that and the more correct ex-
a4(t ) pressionoccur inthe argumentofthe logarithm,andare
+(ρv−ρΛ) a4(tf) +ρΛ(cid:21)a2(t;r) (7) therefore rendered insignificant because mP ≫ Tc. The
ratio of the typical bubble size to the Hubble length can
4πGa2(r)(ρ −ρ ) 4πGa3(t )(ρ −ρ )
= c v Λ − f v Λ . thus be seen to scale only logarithmically with the criti-
3 3a(tf) cal temperature, and for Tc ∼ 0.01–0.1 eV (which is the
rangeweareinterestedin), thesuppressionfactoristyp-
The energy density (ρ − ρ )a4(t )/a4(t) represent-
v Λ f ically δ ∼ 1/300B . The typical spatial extent of the
1
ing this leftover radiation now starts contributing to the
vacuum bubbles at the end of the phase transition can
gravitationalpotential, making it more negative. There-
then be identified as
fore, to balance the books, the total energy, i.e., the
right-hand side of Eq. (7), also picks up an additional m −1 8πG −1/2
negative contribution −4πG(ρv − ρΛ)a2(tf)/3. Since Rf ≡δHc−1 ∼(cid:20)4B1ln TP(cid:21) (cid:20) 3 ρc(cid:21) , (10)
a(t ) > a (r), it is immediately obvious that the to- c
f c
tal energy is now negative, which opens up the possibil-
where the total energy density ρ at the time of phase
c
ityofthe sphericalfluctuations becominggravitationally
transitionsets the Hubble scaleatthatepoch. The total
bound and evolving away from the background FLRW
massenclosedwithin the fluctuationregionis thengiven
metric, growing significantly overdense with time. Sim- by M ≈ (4/3)πR3ρ , setting a conservative upper
plifying the above equation yields f f NR,n
limit on the mass that one might expect to collapse via
such a mechanism. It turns out that the typical mass
8πG ρ ρ a4(t )
a˙2(t;r)= NR,n + R,n +(ρ −ρ ) f +ρ withinabubblevolumerangesfromM ∼5×108M to
3 (cid:20)a3(t;r) a4(t) v Λ a4(t) Λ(cid:21) f ⊙
M ∼3×1011M acrossour parameterspace. The rea-
f ⊙
×a2(t;r)+ 8πG(ρ −ρ )(a2(r)−a2(t )). sonforthisbeingaconservativelimitisthat,inpractice,
3 v Λ c f only a fraction of this mass may become gravitationally
(8) bound, as demonstrated in the following section.
It may be noted that, in each of our equations of mo-
tion, all relativistic energy densities are seen to redshift
V. FLUCTUATION BINDING AND GROWTH
with the universal scale factor a(t), rather than the lo-
cal scale factor a(t;r). This is because as long as the
radiationissufficiently weaklycoupled,anybubble-sized In Sec. III, we described how the process of energy re-
inhomogeneitiesinradiationdensityquicklygetsmeared distribution via bubble nucleation can modify the equa-
out on time scales much shorter than the Hubble time. tions of motion of a comoving spherical shell within the
The relationshipbetweenthe typicalbubble sizeandthe fluctuation region. At the end of the phase transition,
Hubble scale is elucidated in the following section. the Friedmann equation was seen to pick up the addi-
tionalterm(8πG/3)(ρ −ρ )(a2(r)−a2(t )),whichwith
v Λ c f
Birkhoff’s theorem could be interpreted as a binding en-
IV. NUCLEATION SCALE ergy on account of it being negative.
Itcanbeshownthatunderspecificcircumstances,this
Previousstudieshaveidentifiedacharacteristicsizefor binding energy can cause the expansion of the comoving
thebubblesofthebrokenphaseattheendofafirst-order regions to slow down and eventually stop. The effective-
relativisticcosmologicalphasetransition[34–36,41],pro- ness of this mechanism can be characterized by identi-
vided certain conditions are met (such as the nucleating fying a time scale over which locally overdense regions
action depending on cosmologicalbackgroundquantities are formed. We can choose this to be the value of the
andhavingacertainfunctionalform). Ithasbeenshown FLRW time coordinate at which the expansion of a par-
thatthe ratioofthe typicalbubble size atthe endofthe ticular comoving shell stops, i.e., when the time deriva-
phasetransitiontotheHubblelengthH−1atthatepoch, tiveofthelocalscalefactor[Eq. (8)]associatedwiththat
c
is given by shell drops to zero. We call this the “halting time” for
that comoving shell—this is analogous to the definition
m −1 of “turnaround time” in conventional models of struc-
P
δ ≈ 4B ln , (9)
(cid:20) 1 T (cid:21) ture growth. Alternatively, we could look for time scales
c
over which fluctuations of different sizes become nonlin-
where T is the critical temperature (i.e., the tempera- ear,by finding the value of the time coordinate at which
c
ture of the photon background at the onset of the phase δρ/ρ = 1. Here, δρ is the density perturbation, de-
NR
transition). B is the logarithmic derivative of the nu- finedasthedifferencebetweenthenonrelativisticmatter
1
cleating action in units of cosmological time t, and can density inside a bound comoving sphere and the average
5
nonrelativistic matter density throughout the Universe,
(cid:2) (cid:6)(cid:1)(cid:7)
i.e., (cid:3)(cid:4)(cid:1)(cid:5)
(cid:1)
ρ ρ (cid:1)
δρ(t;r)≡ρ′ −ρ = NR,n − NR,n. (11) (cid:13)
NR NR a3(t;r) a3(t)
(cid:6)(cid:16)(cid:7)
(cid:2) (cid:11)(cid:12)(cid:11)(cid:2) (cid:6)(cid:1) (cid:7)
It is evident that the binding energy depends on the (cid:8)(cid:9)(cid:8)(cid:1)(cid:10) (cid:3)(cid:4)(cid:1)(cid:5) (cid:13)
(cid:6)(cid:16)(cid:16)(cid:7)
comoving radius r [the dependence coming from the (cid:2) (cid:11)(cid:14)(cid:11)(cid:2) (cid:6)(cid:1) (cid:7)
(cid:8)(cid:9)(cid:8)(cid:1)(cid:10) (cid:3)(cid:4)(cid:1)(cid:5) (cid:13)
a2(r)−a2(t ) factor], and therefore the associated col-
c f
lapse time scales (halting time or time-to-nonlinearity) (cid:6)(cid:16)(cid:16)(cid:16)(cid:7)
(cid:2) (cid:11)(cid:15)(cid:11)(cid:2) (cid:6)(cid:1) (cid:7)
are also functions of r. And since r is directly related to (cid:8)(cid:9)(cid:8)(cid:1)(cid:10) (cid:3)(cid:4)(cid:1)(cid:5) (cid:13)
the mass scale of the comoving volume, it allows us to
estimate the time scales over which comoving regions of
different mass tend to become overdense.
A. Toy model analysis
Usingasimpletoymodelwecandemonstratethat,for
therangeoftemperaturesweareinterestedin,bindingby
thismechanismwouldnotbe possibleunlessthevacuum
FIG. 4. Cartoon illustrating the energetics associated with
energydensity priorto the phasetransitionwereatleast fluctuation binding. For binding to be accomplished, the ki-
a few orders of magnitude higher than its present value. netic energy has to go to zero, which can only happen if the
To simplify matters, let us ignore the contributions to total energy E (horizontal dashed line) intersects thepo-
total
the Friedmann equation coming from relativistic energy tential energy curve (Egrav, solid line). Cases (ii) and (iii)
densities (not totally unreasonable, since these redshift represent configurations which allow binding [with (ii) being
awayquicklyandthusbecomeinsignificantatlatetimes). right at the threshold], whereas case (i) depicts a situation
where binding cannot be accomplished in spite of a negative
With that approximation, Eq. (8) reduces to
total energy.
8πG ρ
a˙2(t;r)= NR,n +ρ a2(t;r)
Λ
3 (cid:18)a(t;r) (cid:19)
critical temperature alone. ρ is just a constant energy
Λ
− 8πG(ρ −ρ )[a2(t )−a2(r)] (12) density, whereas ρNR at any time is proportional to the
3 v Λ f c cubeofthetemperature(inversecubeofthescalefactor).
8πG ρ Consequently, we can write
= NR,n +ρ a2(t;r)−ρ ,
Λ BE
3 (cid:18)a(t;r) (cid:19)
3
ρ Ω T
NR,n NR c
= , (15)
where we have defined ρ ≡ (ρ −ρ )[a2(t )−a2(r)]. ρ (cid:20) Ω (cid:21) (cid:18)T (cid:19)
BE v Λ f c Λ Λ 0 0
Now the expansion of the comoving sphere halts when
with the subscript “0” being used to denote the values
the right-hand side of the above equation goes to zero.
at the currentepoch. The minimum requiredρ /ρ in
For that to happen, ρ has to be at least as large as BE Λ
BE
theminimumvalueattainedbyρ /a(t;r)+ρ a2(t;r) terms of the critical temperature is then given by
NR,n Λ
(e.g., see Fig. 4). Taking the derivative of this expres-
ρ 3 Ω 2/3 T 2
sion with respect to a(t;r) and setting it to zero im- BE = NR c . (16)
plies that this minimum value is attained when a(t;r)= (cid:18) ρΛ (cid:19)min 22/3 (cid:20) ΩΛ (cid:21)0 (cid:18)T0(cid:19)
(ρ /2ρ )1/3, whichgives the minimum requiredρ ,
NR,n Λ BE Since T ≈ 0.23 meV, the minimum ρ /ρ required
0 BE Λ
(ρBE)min = 223/3ρ2N/R3,nρ1Λ/3. (13) fRoercbalilnindgintghtautrρnBsEouiststiombpely∼(ρ1v04−fρoΛr)T[ac2∼(tf0).−01a–2c0(.r1)]e,Vit.
immediately follows that the ratio ρ /ρ has to be quite
v Λ
It is more instructive to represent this in terms of the large for this mechanism to bring about binding. It can
ratioρ /ρ . Thecorrespondingminimumvalueofthis be shown that the quantity a2(t )−a2(r) can at most
BE Λ f c
ratio necessary for binding is then given by be of the same order as the ratio of transition width to
Hubbletime,i.e.,δ,asdefinedinSec.IV.Quantitatively,
ρ 3 ρ 2/3 it follows that a2(t )−a2(r) is of order 10−2 or smaller,
BE = NR,n . (14) f c
(cid:18) ρΛ (cid:19)min 22/3 (cid:18) ρΛ (cid:19) implyingthatρv/ρΛhastobeoforder106orbigger. The
numbers change slightly [by some O(1) factor] when rel-
By using the observed closure fractions Ω and Ω ativistic energy densities are included in the calculation,
NR Λ
at the current epoch, the ratio ρ /ρ at the time of but the principle remains the same. This equips us with
NR,n Λ
phase transition can be expressed as a function of the theforesighttoimmediatelyexcludecertainregionsfrom
6
our parameter space where fluctuation binding via this computationsdependonthetwoextraparametersofour
mechanism is theoretically impossible to accomplish. model—thecriticaltemperatureT andtheearlyvacuum
c
energy density ρ . We can then look for regions of the
v
parameter space where the above time scales are of the
orderof1Gyrorlower(a1Gyrcosmologicaltimecorre-
B. Numerical calculations
sponds to a redshift z ≈5.5, whereas 2 Gyr and 0.5 Gyr
correspond to z ≈3 and z ≈10, respectively).
Numerical results can be obtained by first integrating
Figure 5 shows contours of halting time and time-
Eq. (6) from t (r) to t . This allows us to explicitly
c f to-nonlinearity plotted against the critical temperature
compute the quantity a2(r)−a2(t ) which can then be
c f T in eV; and the early vacuum energy density ρ in
c v
used in Eq. (8). Integrating Eq. (8) then allows us GeV/cm3 (for comparison,the current observedvacuum
to compute the time scales defined earlier, i.e., the halt- energydensityisabout3.5×10−6GeV/cm3). Foragiven
ingtime,orthetime-to-nonlinearity. Theresultsofthese
set of parameter values, the aforementioned time scales
may be calculated for different comoving shells, i.e., at
different coordinate radii within the fluctuation volume
[plots(a)and(b)inFig.5]. Theseplotsrevealthatbind-
100
3V/cm] 90 0.2 0.5 1 2 5 ivonabglsueierssvfeoadvfotehraeardtl,yawtviatlcohuwinuemracfleruintceitcruagalyttideomennpvseoitrlyua.tmuAer,elsstohaenitidncnhaeingrhrbeeer-
e
G 80 gions (smaller comoving radii) become bound on shorter
[
nergy 70 0.2 0.5 1 2 timSeinsccealtehsecocomopradrinedatteortahdeiuosutiserdrireegciotlnysr(elalartgeedr rtaodtiih)e.
e
m 5 mass enclosed within the comoving volume [via M ≈
cuu 60 (4/3)πr3ρNR,n], we can associate a collapse time scale
va 50 with a comoving mass at each point in the parameter
y
rl space. Figure 6 shows how halting times vary with co-
a
E 40 moving mass for a few selected combinations of parame-
ter values. It can be seen that comoving regions of mass
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 up to 109M could become significantly overdense on a
⊙
CriticalTemperature[eV] time scale of order 1 Gyr via this mechanism.
(a)r=0.1(δHc−1) It should be emphasized that such time scales calcu-
lated by employing the Friedmann equations are only
100
3m] 30.3 0.5 1 2 5 1010
eV/c 90 0. 0.5 1 2 5 es]
[G 80 ass
y m
g
er 70 ar 109
en sol
vacuum 5600 mass[in
arly ng 108
E 40 vi
o
m
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Co-
CriticalTemperature[eV] 107
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
(b)r=0.5(δHc−1)
Haltingtime[Gyr]
FIG. 5. Contours of halting time (solid) and time-to-
nonlinearity (dashed), labeled in gigayears, across a parame- FIG.6. Haltingtimescorresponding todifferentmass scales,
ter space spanned by critical temperature and early vacuum for certain selected parameter values. The four curves (from
energy density. Plots (a) and (b) correspond to two different left to right) correspond to the following parameter values:
comovingshells,labeledwiththeirrespectivecoordinateradii Tc = 0.03 eV, ρv = 55.3 GeV/cm3; Tc = 0.03 eV, ρv = 38
inrelationtothetypicalnucleationscaleδHc−1. Shadedareas GeV/cm3;Tc =0.05eV,ρv =100.2GeV/cm3;andTc =0.05
indicateregions oftheparameterspacewherecomovingfluc- eV, ρv =76 GeV/cm3, respectively. Evidently, lower critical
tuations of the specified radii cannot become gravitationally temperaturesandhigherearlyvacuumenergydensitiesimply
bound,either within a O(1 Gyr) time scale, or at all. stronger binding.
7
supposedtoserveasaguide to the eye. Once the fluctu- V
0.04 a
ations become significantly overdense,physics at smaller 100 c
u
scales comes into play and can lead to fragmentation of 3m] um
thegravitationallyboundcomovingvolume. Butevenin V/c 90 clo
that case, the viability of this mechanism in helping cre- Ge 80 sur
ateseedblackholesofmass103–106M atearlyredshifts [ e
cannotberuledout. Theobjectivebeh⊙indthisanalysisis ergy 70 0.03 frac
simplytodemonstratethatformationofsmall,overdense en tio
m n
regions is a likely outcome of such a vacuum phase tran- uu 60 at
swititiohnthperofcoersms,atrieosnulotifnsgupinersmmaaslsleivretiombejescctsa,leassacsosmocpiaarteedd yvac 50 0.02 recom
to conventional cosmologicalmodels. Earl 40 bina
t
io
n
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
VI. OBSERVATIONAL CONSTRAINTS
CriticalTemperature[eV]
A. Contributions to closure fraction FIG. 7. Contours labeled with the early vacuum energy clo-
sure fraction at photon decoupling, plotted across a parame-
As demonstrated in the previous section, bringing ter space spanned by critical temperature and early vacuum
about binding via such a mechanism requires that the energy density.
densityofearlyvacuumenergybe severalordersofmag-
nitude bigger than its current value. This immediately
raises the question of how this would affect the observed an imprint on the CMB spectrum in the form of Comp-
closure fractions of the different energy densities at vari- ton y-distortions. However, constraints on other forms
ousepochs. Forinstance,theimpactoftheearlyvacuum of relativistic energy density (“dark radiation”—such as
energycouldbe assessedbasedonitscontributiontothe active/sterile neutrinos, or perhaps something more ex-
closure density at photon decoupling. otic)atthecurrentepocharenotasstrong. Infact,ithas
As can be seen from Fig. 7, in our model, the clo- been argued[49] that extra radiationenergy density can
sure fraction of early vacuum energy at photon decou- reconcile discordant values of H0 and σ8 inferred from
plingisindependentofthecriticaltemperature,andtyp- CMB and other, more direct inferences, respectively.
ically varies between 1–4 percent in the regions of the The Wasserman mechanism that lies at the heart of
parameter space that are of interest to us. There have our analysis requires that the swept-up vacuum energy
been a few attempts to constrain the closure fraction of decay into relativistic particles in order to bring about
early vacuum energy at recombination using CMB data binding. However, this does not preclude the possibility
[29, 30, 33, 42–47]. At present, these limits, although
model dependent, are in the same ballpark as our calcu-
lated numbers (i.e., at the level of a few percent of the
criticaldensity). Future observationswill likely have the 100
potential to impose stronger constraints on our parame- 3m]
tcteioorunlsdppaaacrlesao.mAaetffelearcsrt,gestuhvceahlbuaeesstoN-ffieetffa.vralyluveascoufumotheenrerrgeycodmebnisnitay- gy[GeV/c 8900 0.5 0.20.10.05 0.01 0.005 0.001
As describedinSec.III, weassume thatthe earlyvac- er 70
n
uum energy in our model gets swept up onto the bubble e
m
walls, before disintegrating via conversion to some un- u 60
u
known relativistic particles. It is then possible to calcu- ac
v 50
late the closure fraction of this leftover radiation at the y
rl
present epoch, as illustrated in Fig. 8. a
E 40
There exist very strong constraints on the tempera-
ture and spectral shape (and consequently, the energy 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
density) of the photon background [48], and our calcu- CriticalTemperature[eV]
lated closure fraction throughout the parameter space
appears too high to be consistent with these. This al-
FIG. 8. Contours labeled with the closure fraction, at the
lows us to rule out the possibility of photons constitut- presentepoch,oftheleftoverradiationfrombubblecollisions,
ing a significant fraction of the leftover radiation. The plottedacrossaparameterspacespannedbycriticaltempera-
radiation may also not contain, in appreciable amounts, tureandearlyvacuumenergydensity. Shadedareasrepresent
other standard-modelparticles that interactelectromag- regionsoftheparameterspacethataredisfavored bycurrent
netically (e.g., charged leptons), since that could leave observational data.
8
of these particles becoming nonrelativistic at late times fromitsbest-fitΛCDMvalueof69.32 km s−1Mpc−1,de-
as the Universe cools, in a manner akin to the cosmic termined using data from the above surveys.
background neutrinos. In such a situation, these par- Quiteclearly,thedifferencebetweenthecalculatedH
0
ticles could contribute to the cold dark matter density in our model and the CMB-derived best-fit, is minimum
at late times. Uncertainties associated with the CMB- in the lower right-hand corner of the parameter space,
determined ΩCDM best-fit values hover typically around i.e., at higher critical temperatures and smaller values
1–2 percent of the critical density [29–33, 50] at the 1σ of early vacuum energy density. And as Fig. 8 demon-
level,allowingforsomeroomtomaneuverinthisregard. strates, this also corresponds to a smaller leftover radi-
ation closure fraction. The calculated H values may
0
also be comparedto direct low-redshiftobservationales-
B. Hubble parameter timates using Type Ia supernovae and Cepheid variable
stars [51, 52].
Anotherwaytoexpressthecontributionoftheleftover It is also possible to measure the Hubble parameter
radiationat late times is by looking at the impact it has at relatively high redshifts, e.g., z ≈2.36, using Quasar-
on the Hubble parameter at various epochs. It follows Lyman α forest cross-correlations. Font-Ribera et al., in
fromtheFriedmannequationthattheHubbleparameter Ref.[53],calculateH(z =2.36)=226±8km s−1Mpc−1.
at a redshift of z is related to the closure density as Extrapolating the ΛCDM best-fit value from CMB ob-
servations(WMAP9+eCMB+BAO+H )tothatredshift
0
H2(z)= 8πGρ(z), (17) gives H(z = 2.36) ≈ 236 km s−1Mpc−1, consistent with
3 the above result to within error. The same procedure
canbeappliedtoourcosmologicalmodeltocalculatethe
where ρ(z) is the closure density of the Universe at that
corresponding H(z = 2.36) values across the parameter
epoch, consisting of contributions from nonrelativistic
space, as shown in Fig. 10.
matter, radiation and vacuum energy densities. Conse-
It can be seen that a large chunk of our parameter
quently, the additional energy density coming from the
space is consistent with direct observations at that red-
leftoverradiationcaninfluence the Hubble parameterby
shift to within 2–3σ. However, comparing Figs. 9 and
contributing to the right-hand side of Eq. (17).
10 tells us that the higher we go in redshift, the tighter
Figure 9 shows what the contours of the current Hub-
these constraints get. This is understandable, since the
ble parameter would look like in our model across the
contribution from the leftover radiation is much more
parameter space. Nonrelativistic and vacuum energy
significant at higher redshifts (e.g., see Fig. 13). Other
densities used for this calculation were adopted from
techniques with future very largetelescopes may be able
WMAP9+eCMB+BAO+H results [31, 32]. Neutrinos
0
to obtain the Hubble parameter at even higher redshift,
were approximated as massless. The impact of the left-
overradiationcanbegaugedbyobservingtheshiftinH
0
100
100 3m] 37
3V/cm] 90 05 72 70 69.5 69.4 69.35 GeV/c 8900 300 266 250 242 238 2
Ge 80 87 gy[
nergy[ 70 mener 70
e u 60
m u
u 60 ac
u v 50
rlyvac 50 Early 40
a
E 40
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 CriticalTemperature[eV]
CriticalTemperature[eV]
FIG. 10. Contours labeled with the Hubble parameter, cal-
FIG. 9. Contours labeled with the Hubble parameter, culated at a redshift of z = 2.36 in our model (H2.36, in
calculated at the present epoch in our model (H0, in km s−1Mpc−1),plottedacrossaparameterspacespannedby
km s−1Mpc−1),plottedacrossaparameterspacespannedby critical temperature and early vacuum energy density. The
criticaltemperatureandearlyvacuumenergydensity. Shaded dashed line at 242 km s−1Mpc−1 marks the 95% C.L. limit
areasrepresentregions oftheparameterspacewherethecal- given by Ref. [53], while the shaded areas represent regions
culated H0 values are not in good agreement with current of theparameter space where thecalculated H2.36 values are
observational data. significantly at odds(3σ or more) with that result.
9
z ∼ 5 [54]. Such observations in the future would rep- Redshift(1+z)
resent a promising avenue for further constraining our
1000 100 10 1
model. 1
0.9
0.8
C. Scale factor evolution and age of the Universe ns 0.7
o
ti 0.6
c
a
We haveshownthatthe closurefractionsofearlydark fr 0.5
e
energyatrecombinationandtheleftoverradiationatthe ur 0.4
s
current epoch can be restricted to a few-percent level in Clo 0.3
certainregionsofourparameterspace. However,itturns 0.2
out that, at epochs close to the phase transition, these
0.1
componentsofenergydensitycaninfactbethedominant
0
ones. Figure11showscontoursofclosurefractionofearly 0.1 0.01 0.001
dark energy in our model at T = Tc. The contribution Temperature[eV]
of early dark energyto the closure density at the critical
temperature can be seen to vary across our parameter
FIG. 12. Evolution of closure fractions contributed by the
space from about 20% to more than 99%. various components of energy density with cosmic tempera-
Introducing this epoch of prodigious vacuum energy ture in the standard ΛCDM cosmological model. The solid,
contributionataroundz ∼100affectsthetime-evolution dotted,anddashedlinesrepresentnonrelativistic,relativistic,
of the overall scale factor a(t). Understandably, this ef- and vacuum energy densities, respectively. Neutrinos were
fectissmallerinregionsoftheparameterspacewherethe approximatedasbeingmassless throughoutthecourseofthe
amount of vacuum domination, as well as the duration evolution.
of the vacuum-dominant epoch, are relatively small.
Figure 12 shows how the relative mix of the various
components of energy density changes with time in the plottedagainsttime,ratherthantemperature. Figure14
standard model, whereas Fig. 13 does the same in the comparestheΩNR vstcurveinthe standardmodelwith
context of our cosmological model. It is clear that a the ones in our model, for different parameter values.
higher critical temperature (i.e., an earlier phase transi- Whilematterdominationcanbeseentobequiteheavily
tion) is associatedwith a lowervacuum energycontribu- suppressed early on, this effect tapers off as the leftover
tionattheepochofthetransition,andasmallerduration radiation from the phase transition redshifts away. It is
of vacuum energy domination. notimmediatelyclearhowmuchofanimpactthiswould
While it may seem that matter domination is quite haveonstructure growthatscales largerthan the nucle-
substantiallysuppressedinourmodelascomparedtothe ationscale,e.g.,galaxyformation. Large-scalenumerical
standard model, this effect appears much less dramatic simulations that incorporate these effects may help ad-
when the curves representing the closure fractions are dress this question.
The impact of this new physics can also be analyzed
by comparing the time-evolution of the universal scale
factor a(t) in our model (for various parameter values),
100
with the standard ΛCDM cosmologicalmodel, as shown
3eV/cm] 90 0.99 0.95 0.9 0.8 0.7 0.6 0.5 ihnavFeiga.=15.1Tathethsecaplerefsaecnttorephoacshb.eIetncraenscbaelesdeeinnothrdaetrftoor
[G 80 Tc = 0.05 eV, the deviation of the scale factor evolution
y curvefromitsstandardmodelcounterpartisalotless,as
erg 70 0.4 compared to T = 0.03 eV. Apparently, a higher critical
n c
e temperature serves to mitigate the effect of a large early
m
u 60 vacuum energy density.
u
c
va 50 The plot in Fig. 15 can be used to infer the age of the
rly 0.3 Universe in our model, by reading off the value of the
a
E 40 time coordinate (x axis) when the scale factor becomes
unity. For the case where T = 0.03 eV and ρ = 38
c v
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 GeV/cm3,the agecanbe inferredtobe about12.7×109
CriticalTemperature[eV] years,whereasforT =0.05eVandρ =76GeV/cm3,it
c v
increases to 13.3×109 years. The apparent disharmony
FIG. 11. Contours labeled with the early vacuum energy between these numbers and the best-fit CMB prediction
closure fraction at the onset of the phase transition (i.e., at (i.e., 13.8 Gyr), must not be taken too seriously, for the
T =Tc),plottedacrossaparameterspacespannedbycritical latter measurement is predicated on the assumption of
temperature and early vacuum energy density. the Universe having followed a standard ΛCDM-based
10
Redshift(1+z)
1000 100 10 1
0.9
1 n
o
0.9 ti 0.8
c
a
0.8 fr 0.7
ons 0.7 sure 0.6
acti 0.6 clo 0.5
Closurefr 000...345 relativisic 00..34
0.2 Non 0.2
0.1
0.1
0
0.1 0.01 0.001 0 2 4 6 8 10 12 14
Temperature[eV] Time[Gyr]
(a)Tc=0.03eV,ρv=38GeV/cm3
FIG.14. Evolution oftheclosurefraction contributedbythe
Redshift(1+z) total nonrelativistic energy density (ΩNR) with time in our
modified cosmological model, compared with the standard
1000 100 10 1
model picture. The solid line at the top represents the stan-
1
dard model, whereas the other two curves (from top to bot-
0.9
tom)representevolutionin ourcosmological model,withthe
0.8 following parametervalues: Tc =0.05eV,ρv =76GeV/cm3;
ns 0.7 and Tc =0.03 eV, ρv =38 GeV/cm3, respectively.
o
ti 0.6
c
a
fr 0.5
e 1
ur 0.4
Clos 0.3 0.9
0.2 0.8
0.1 0.7
0 or 0.6
0.1 0.01 0.001 ct
fa 0.5
Temperature[eV] e
(b)Tc=0.05eV,ρv=76GeV/cm3 Scal 0.4
0.3
FIG.13. Evolutionofclosurefractionscontributedbythevar-
0.2
ious components of energy density with cosmic temperature
0.1
in our modified cosmological model, for different parameter
values. Thesolid,dotted,anddashedlinesrepresentnonrela- 0
tivistic, primordial relativistic, and vacuum energy densities, 0 2 4 6 8 10 12 14
respectively, whereas the dot-dashed line represents the left- Time[Gyr]
overradiationfromthephasetransition. Noticehowthevac-
uumenergydensitypeaksatT =Tc,thensuddenlyplummets FIG. 15. Evolution of the scale factor with time in ourmod-
toanearly zero closure fraction (asmost ofit gets converted ified cosmological model, compared with the standard model
to relativistic particles), before eventually becoming signifi- picture. The solid line on the far right represents the stan-
cant again at late times. dardmodel,whereastheothertwocurves(fromlefttoright)
represent evolution in our cosmological model, with the fol-
lowing parameter values: Tc = 0.03 eV, ρv = 38 GeV/cm3;
evolutionary track throughout its history. The consis- and Tc =0.05 eV, ρv =76 GeV/cm3, respectively.
tency ofthese calculatednumberswith independentlim-
its on the age of the Universe, derived using nucleocos-
systematic.
mochronology[55,56],main-sequenceturnoffinglobular
clusters [57], and white dwarf cooling [58, 59], may be
noted. As of today, the tightest of these independent
limits comes from observations of the oxygen-to-iron ra- D. Perturbations on the CMB temperature map
tiointheultra-metal-poorhalosubgiantstarHD140283,
coupled with stellar evolution theory. Bond et al., in The density fluctuations generated in our vacuum
Ref. [60], estimate the age of the star to be 14.5±0.8 phase transition model can leave their imprint on the
Gyr, where the uncertainty is part statistical and part CMB temperature map, in the form of anisotropies aris-