Emergently Thermalized Islands in the Landscape

0.12 MB·English

by  Yun-Song Piao

Emergently Thermalized Islands in the Landscape
Yun-Song Piao
College of Physical Sciences, Graduate School of Chinese Academy of Sciences, Beijing 100049, China
In this note, we point out that in the eternal inflation driven by the metastable vacua of the
landscape,itmightbepossiblethatsomelargeandlocalquantumfluctuationswiththenullenergy
condition violation can stride over the barriers between different vacua and straightly create some
islands with radiation and matter in new vacua. Then these thermalized islands will evolve with
thestandardcosmology. Weshowthatsuchislandsmaybeconsistentwithourobservableuniverse,
while has some distinctly observable signals, which may betested in coming observations.
PACSnumbers:
8 Recently, the string landscape with large number will be nucleated, while in other regions the jump to the
0 of metastable vacua has received increased attentions topofthe potentialbarrierwilloccur,whichis mediated
0 [1, 2, 3], which generally exhibit the cosmological dy- by the HM instanton, and then the field will rapidly roll
2 namicsaseternalinflation[4,5]. Thetunnellingbetween down along the another side of barrier to new vacuum

n variousvacuaoflandscapemaybemediatedbytheCDL andthe same reheating as that after inflation will occur,
a instanton [6], which behaves as a bubble nucleating in seeFig.1. Thusitmaybeexpectedthatthisthermalized
J
new vacuum. However, the bubble universe nucleated is region will be followed by a standard evolution of radia-

8 generallyemptyandnegativelycurved,whichcanhardly tiondomination. Inthisnotewewillcheckwhethersome

becomeourrealworld. Thustoproduceanuniversecon- of them are able to become our observable universe.
]
c tainingthestructure,aperiodofslowrollinflationinside These thermalized regions are generally different from
q the bubble is needed [7], which reduces the curvature, the bubbles nucleated by the CDL instanton. The bub-
- provides the primordial density perturbation and large blesareeitheremptyordominatedbythevacuumenergy
r
g number of entropy required by the observable universe. of new vacua, while the thermalized regions are those
[ Theoccurrenceofinflationwithenoughefoldingnumber filled with radiation and matter, which thus more like

requires that the potential above new minimum should someislandsemergingfromthedSbackgroundsea. Note
2
v have a nearly flat and long plain, which actually means that in this note what is referred as the “island” is such
9 a fine tuning, since the regions with nearly flat potential anemergentlythermalizedregionwhichisgeneratedbya
2 aregenerallyexpectedtobequiterareinthestringland- largefluctuationinoriginalvacuumbutemergesinadif-
6 scape. Thus it will be interesting to check whether there ferent vacuum, see the green dashed line in Fig.1, which
1
are other possibilities to lead to an universe like ours. is actually slightly not in its originalmeaning [10]. Thus
.
6 Thelandscapewillbepopulatedduringtheeternalin- herethe emergenceofislandwillbe inevitablyrelatedto
0 flation. Thus in principle there may be many quantum the tunneling in the landscape, especially the HM tun-
7
fluctuations with various spatial and temporal scales in neling, as has been mentioned. In this sense, the emerg-
0
eachvacuaofthe landscape,whichmayviolatethe NEC ing probability of islands in new vacua may be given by
:
v [8], see also Ref. [9] for discussions. However, most of that of the HM instanton between different vacua, while
i these fluctuations are cosmological irrelevant, since gen- in Refs. [10, 11] with the cosmological constant back-
X
erally they will be generated and then return rapidly ground, it is not clear in what case we can calculate the
r
a to their original vacuum. The significant fluctuations emerging probability of island.
are those enough large, as studied in the island universe The“thermalized”heremeansthattheresultingstate
model [10, 11], in which the vacuum background is that is a thermal state with radiation and matter, in which
with the observedvalue of the current cosmologicalcon- all components are assumed to be in thermal equilib-
stant. Thus with diverse environments of string land- rium. Thusthispartofregionis similartothatafterthe
scape, we may expect that it is possible that some large reheatingfollowing a slow rollinflation. This can be dis-
and local quantum fluctuations with the NEC violation tinguished from the discussion in Ref. [15], in which the
canstrideoverthe barrierbetweendifferentvacuainthe stateafterthefluctuationisassumedtobeanobservable
landscapeandstraightlycreatesomethermalizedregions universewithstructures,andalsofromtherecyclinguni-
innewvacua. Thismightbephenomenallyillustratedby verse proposed in Ref. [16], see also Ref. [17], in which
applying the HM instanton [12], since the HM instanton the state after the fluctuation is another dS spacetime.
maybe regardedasathermalfluctuationwhoseratecan Though the spawning of island in the landscape is ac-
be given by the difference in the entropies between the tually a quantum process, it may be regarded phenome-
fluctuation and the equilibrium state [2]. In a normal nally or semiclassically as a NEC violating evolution to
landscape in which the barriers between various minima study,aswasdoneinRef. [11],seealsoRef. [18]. Thusin
are neither sharpnor broad[13], the CDL instanton and thissensetheislandactuallysharessomeremarkablesuc-
HM instantonwill be expected to coexist [14]. This sug- cessesofinflationmodel. Thereasonisthattheinflation
gests that in some regions the bubbles with new vacua can be generally regarded as an accelerated stage, and
2
be vanishingly small, where the subscript ‘i’ and ‘e’ de-
note the initial and end value of the NEC violating fluc-
a
tuation respectively, and thus h is determined by the
i
energy scale of original vacuum. To make dt → 0, we
b need hi or |ǫ| is quite large. The larger hiRis the more
violent the fluctuation is anticipative. However, as will
d be showed here, h is required to be so small as to solve
i
c thehorizonproblemofstandardcosmology. ThusEq.(1)
suggests |ǫ| ≫ 1, which means that though during the
fluctuation the change of h is drastic, the expansion of
the scale factor is extremely slow, since a∼h1/|ǫ|.
The scale of the NEC violating region is generally re-
quiredto be largerthanthe Hubble scaleoforiginalvac-
FIG. 1: The landscape of simple potential used to illustrate
the islands proposed here. In the eternal inflation, the bub- uum [10], see also Refs. [19, 20, 21]. This is also consis-
bles with vacuum ‘c’ can be produced by the CDL instanton tentwithapplicationoftheHMinstantonactioninwhich
from either ‘a’ or ‘d’ vacua. The endpoint of tunnellingfrom theregiontunnellingtothetopofthebarriercorresponds
‘a’ would lie on a plain region ‘b’ of potential, and thus the totheHubblescaleoftheoriginalvacuum,whichmaybe
bubbleof‘a’→‘c’willhaveaperiodofinflationandthenmay understood by using the stochastic approachto inflation
evolvetoourrealworld,whereasthebubbleof‘d’→‘c’willbe [22, 23, 24, 25]. This result sets the initial value of local
empty, as is given by the black dashed lines. However, here
evolution of a, and since it is nearly unchanged during
we pointed out that ‘d’→‘c’ may be also induced by some
the fluctuation,
large and local quantum fluctuation with theNEC violation,
which produces some thermalized islands, as is given by the
a ≃a ≃1/h , (2)
green dashed line. These islands will evolve with the stan- e i i
dardFRWcosmology andsomeofthemmaylikeourobserv-
maybe deduced, whichmeans thatthe smallerh is, the
i
ableuniverses. Notethatitisalsopossible thatsomeislands
largerthe scaleoflocalthermalizedregionafter the fluc-
are generated in ‘b’ dueto the NEC violating fluctuationsin
tuation is. To obtain enough efolding number for solv-
‘a’. However, in this case the radiation and matter will be
ing the horizon problem of standard cosmology,we need
diluted quickly and the island universewill then enter an in-
a ≫1/h , thus h ≪h is required, which may be also
flation phase dominated by the vacuum energy in ‘b’, which e e i e
will have the same results as usual slow roll inflation. Here seen as follows.
what we concern is the case of the green dashed line, which
a h
mightprovidesadifferentavenuetoanobservableuniversein N ≡ln e e (3)
thelandscape. (cid:18) ah (cid:19)
is the efolding number of mode with some scale ∼ 1/k,
where k = ah, which leaves the horizon before the end
so may defined as an epoch when the comoving Hubble of the NEC violating fluctuation, and thus k is the last
e
length decreases, which actually occurs equally during mode to be generated. When taking ah = a h , where
0 0
a NEC violating expansion. When the island emerges, the subscript ‘0’ denotes the present time, we generally
the change of local background may be depicted by the haveN ∼50,whichisrequiredbyobservablecosmology,
drastic evolution of local Hubble parameter ‘h’, where seeRef. [26]foradiscussiononthevalueofN. Byusing
the “local” means that the quantities, such as the scale Eq.(2) and (3), we may obtain approximately
factor ‘a’ and ‘h’, only character the values of the NEC
violating region. We begin with introducing ‘ǫ’ defined h T 2
e e
−h˙/h2, which can be regarded as ǫ ≃ 1 ∆h, and thus N ≃ln(cid:18)h (cid:19)≃ln(cid:18)Λ (cid:19) , (4)
h∆t h i i
actuallydescribesthechangeofhinunitofHubbletime.
During the NEC violating fluctuation, h˙ >0, thus ǫ<0 where Λ ≃ h1/2 is the energy scale of original vacuum
i i
can be deduced. We assume here that ǫ is constant for 1/2
and T ≃ h is the thermalized temperature after the
simplicity. Thus after making the integral for the defini- e e
NEC violating fluctuation, which may be the same as
tion of ǫ, we have a∼h1/|ǫ|.
the reheating temperature after inflation, and m2 = 1
The more rapid the fluctuation is, in principle the p
has been taken. When taking N ≃ 50 and T ∼ 1015
stronger it can be, which in some sense is also a reflec- e
Gev, we have Λ ∼ Tev. For a lower T , Λ is required
tion of the uncertainty relation between the energy and i e i
to be smaller. Thus unlike the case in Refs. [10, 11],
time in quantum dynamics. Therefore to make the NEC
it seems that here the efolding number required to solve
violating fluctuate be so strong as to be able to create
the horizon problem of standard cosmology can not be
the islands of our observable universe, we should take
always obtained. To have an enough efolding number,
the time scale of the NEC violation
an enough low original vacuum should be selected. The
e dh 1 reason is that the smaller Λ is, the larger the Hubble
dt= ≃ (1) i
Z Zi |ǫ|h2 |ǫ|hi scale of corresponding dS vacuum is, and thus the size
3
of local universe after the fluctuation and the efolding value, however, in this case to have a proper amplitude,
number. This result also suggests that if our universe the thermalization temperature T should be smaller.
e
is actually such an island originated from last vacuum, Here the fine tuning for ǫ is actually similar to that ap-
then the observations made in our universe might have pearing in the inflation model. They are related to each
recordedsomeinformationonlastvacuum. Forexample, other by a dual transformation |ǫ| ↔ 1/|ǫ| [27], which
for the efolding number N > 50, we may know that its also corresponds to a duality between their background
energy scale should be low, and further if the thermal- evolutions, i.e. between the nearly exponent expansion
ization temperature T is about 1015 Gev, our universe with |ǫ|≃0 and the slow expansion |ǫ|→∞.
e
must not be originate from a fluctuation of those vacua Thecalculationsofprimordialtensorperturbationdur-
with the energy density larger than Tev scale. In prin- ingtheevolutionwiththeNECviolationhavebeendone
ciple, the T should be required to be lower than that in Ref. [30]. The case corresponding to the island uni-
e
the monopoles production needs, while higher than Tev. verse is given in Eq.(13) of Ref. [11]. The spectrum is
This can not only avoidthe monopole problem afflicting
the standard cosmology, but helps to provide a solution v(e) 2 k2
toTthheemcaalctutelartgioennsesoisf.primordialscalarperturbationdur- PT ∼=k3(cid:12)(cid:12)(cid:12) ake (cid:12)(cid:12)(cid:12) ≃ a2e, (6)
ingtheevolutionwiththeNECviolationhavebeendone (cid:12) (cid:12)
where the gauge invariant(cid:12) vari(cid:12)able v is related to the
in Refs. [11, 18, 27, 28]. The emergence of island in the k
tensor perturbation h by v = ah . It can be seen
landscape corresponds to the limit case with ǫ≪−1. In k k k
that the tensor spectrum is quite blue, which is actually
Ref. [11], which firstly calculates the curvature pertur-
a reflection of the rapid increase of background energy
bation of island universe, in which the background vac-
density during the NEC violating fluctuation, see Ref.
uum is taken as the observed value of cosmological con-
[30] for details. This resultindicates that the tensor am-
stant, it has been shown that the spectrum of Bardeen
plitudeintheislandwillbeintenselysuppressedonlarge
potential Φ before the thermalization is dominated by
scale. Tocalculatethe tensoramplitude, wemayreplace
an increasing mode and is nearly scale invariant, which
a by using k =a h in Eq.(6) and have
under some conditions may induce scale invariant cur- e e e e
vature perturbation. Whether the resulting spectrum is
k 2
scaleinvariantis determinedby the physics atthe epoch P ≃h2 , (7)
of thermalization. Thus in this case there is generally T e(cid:18)ke(cid:19)
anuncertainty,seethediscussionsinRef. [11]. However,
which gives the value of tensor amplitude in various
lateritwasnotedthatthecurvatureperturbationmaybe
scales. Thus the value on large scale may be obtained
also induced by the entropy perturbation [27], or by the
by combining Eqs.(3) and (4), and then substituting the
perturbation of test scalar field [28] before the thermal-
result into Eq.(7), which is P ∼= h2. This is quite low,
ization,whichunder certainconditionsmay hasanearly T i
for example, taking h2 ≃Λ ∼Tev, we have P ∼10−60.
scale invariant spectrum and proper amplitude required i i T
NotethatthisresultonlargescaleissameasthatinRef.
bytheobservations. Thecurvatureperturbationinduced
[10], in which the dS sea phase is suddenly matched to
bytheentropyperturbationhassameformasthatbythe
a radiationdominated phase and the NEC violating me-
Bardeenpotential[27],onlyuptoanumericalfactorwith
diated phase is neglected. However, in their paper they
unite order, which, more importantly, is not dependent
argued that the tensor spectrum is scale invariant and
of the physical detail of thermalized surface. Thus un-
thushas sameamplitude inallscales,while inourcalcu-
like the case in Refs. [10, 11], when with more freedom
lationthe spectrumisactuallystrongblue, whichonlyis
degrees provided by the landscape, since in low energy
sameasthatinRef. [10]onlargescale. Thediscrepancy
the landscape may be approximated as the space of a
between Refs. [10] and [11] lies in the NEC violating
set of fields with a complicated and rugged potential, it
phase neglected by Ref. [10], which it is that tilts the
may be more natural to consider the curvature pertur-
tensor spectrum. In small scale, which corresponds to
bation induced by the entropy perturbation, which will
take k = k , we have P ≃ h2, which is actually quite
definitely give the scale invariant spectrum with proper e T e
large for a high thermalization scale.
amplitude. In Refs. [11, 27], the amplitude of curvature
In summary, we point out that in the eternal inflation
perturbation is given by
driven by the metastable vacua of the landscape, it may
P ∼=|ǫ|h2, (5) be possible that some large and local quantum fluctua-
s e
tionswith the NECviolationcanstrideoverthe barriers
which only depends on |ǫ| and h . When |ǫ| → ∞, the between different vacua in the landscape and straightly
e
perturbation amplitude will be divergent. Thus for our create some thermalized islands in new vacua. We show
purpose, the value of |ǫ| seems to require a litter fine that these islands may be consistent with our observable
tuning. For example, having taken T ∼ 1015 Gev as universe. This result suggests that with the landscape
e
the thermalization temperature and |ǫ| ∼ 102, we have the observable universe may be some of many thermal-
P ∼10−10 for (5), which is just the observedamplitude ized regions, which appear either by a slow roll inflation
s
of CMB [29]. In principle, ǫ may be taken as a larger afterthenucleationofbubbles,followedbythereheating,
4
or by a straightly thermalization in new vacua without generally negatively curved, and thus the corresponding
the slow roll inflation, spawned within the eternally in- universe is an open universe, while the island leaded to
flating background. by the NEC violating fluctuation may be closed. Thus
The observations in principle can determined whether in principle if the cosmologicaldynamics is actually con-
we live in an emergently thermalized island or in a re- trolled by a landscape with many metastable vacua, the
heating region after inflation inside bubble. It has been curvaturemeasurementofouruniversewillbesignificant
shown that in the island the tensor amplitude is negligi- to make clear where we live in.
bleonlargescale,whilethereexistsalargeclassinflation It should be fairly said that the discussions here is in-
model, such as large field inflation model, with moder- evitably slightly speculative, since a full description for
ate amplitude of tensor perturbation, see e.g. Ref. [31] the phenomena with the NEC violationis still lackedfor
for the various inflation models. Thus it seems that the the moment. However, the results showed here might
detectionofa stochastictensorperturbationwillbe con- havecaptured some essentials of emergently thermalized
sistent with the inflation model, while rule out the pos- island in the landscape, which might be interesting and
sibility that an straightly thermalized region is regarded significant to phenomenological study of landscape cos-
as our real world. The low tensor amplitude on large mology.
scale is also not conflicted with the inflation model, e.g. Acknowledgments This work is supported in part
somesmallfieldinflationmodels. Thusinthiscaseother by NNSFC under Grant No: 10405029, in part by the
distinguishabilities need to be considered. The bubble Scientific Research Fund of GUCAS(NO.055101BM03),
after the nucleation described by the CDL instanton is in part by CAS under Grant No: KJCX3-SYW-N2.
[1] R.Bousso and J. Polchinski, JHEP 0006, 006 (2000). [17] K.M. Lee, E.J. Weinberg, Phys.Rev. D36, 1088 (1987).
[2] S. Kachru, R. Kallosh, A. Linde and S.P. Trivedi, Phys. [18] Y.S. Piao and E Zhou, Phys.Rev. D68, 083515 (2003).
Rev.D68, 046005 (2003). [19] E. Farhiand A.H.Guth, Phys.Lett. B183 149 (1987).
[3] L. Susskind,hep-th/0302219. [20] T.VachaspatiandM.Trodden,Phys.Rev.D61,023502
[4] A.Vilenkin, Phys.Rev. D27, 2848 (1983). (2000).
[5] A.D.Linde, Phys.Lett. B175, 395 (1986). [21] A. Borde, M. Trodden, and T. Vachaspati, Phys. Rev.
[6] S.R. Coleman, F. De Luccia, Phys. Rev. D21, 3305 D59, 043513 (1999).
(1980). [22] A.A. Starobinsky, in “Current Topics in Field The-
[7] B. Freivogel, M. Kleban, M.R. Martinez, L. Susskind, ory, Quantum Gravity and Strings,” edited by H.J. de
JHEP 0603, 039 (2006). Vega and N. Sanchez, Lecture Notes in Physics, Vol.26
[8] S. Winitzki, gr-qc/0111109; T. Vachaspati, (Springer, Heidelberg, 1986), 107.
astro-ph/0305439. [23] A.S. Goncharov, A.D. Linde, V.F. Mukhanov, Int. J.
[9] S.Winitzki, gr-qc/0612164. Mod. Phys.A2, 561 (1987).
[10] S. Dutta and T. Vachaspati, Phys. Rev. D71, 083507 [24] A.D. Linde, “Particle Physics and Inflationary Cosmol-
(2005); S. Dutta,Phys.Rev. D73(2006) 063524.. ogy”, (Harwood, Chur, Switzerland, 1990); Contemp.
[11] Y.S.Piao, Phys. Rev. D72, 103513 (2005). Concepts Phys.5, 1 (2005), arXiv:hep-th/0503203.
[12] S.W.Hawking,I.G.Moss,Nucl.Phys.B224,180(1983). [25] A.D. Linde, Nucl.Phys. B372, 421 (1992).
[13] When the barrier is broad enough, the regions of space [26] A.R. Liddle and S.M. Leach, Phys. Rev. D68 103503
which tunnel to the top of the barrier will undergo the (2003).
slow roll eternal inflation. In this case below the slow [27] Y.S. Piao, arXiv:0706.0981, Phys.Rev. D,in press.
roll eternally inflating regime a period of usual slow roll [28] Y.S. Piao, Phys. Rev. D74, 043509 (2006).
inflationisstillneeded.Therelevantcosmologyhasbeen [29] D.N. Spergel et al. (WMAP Collaboration),
discussed in P. Batra and M. Kleban, hep-th/0612083. astro-ph/0603449.
[14] A.Linde, hep-th/0611043. [30] Y.S. Piao, Phys. Rev. D73, 047302 (2006).
[15] L. Dyson, M. Kleban and L. Susskind, JHEP 0210, 011 [31] D.H. Lyth,A.Riotto, Phys.Rept. 314, 1 (1999).
(2002).
[16] J.GarrigeandA.Vilenkin,Phys.Rev.D57,2230(1998).