arXiv:1001.0008v2  [hep-th]  6 Jan 2010Multi-Stream Inﬂation: Bifurcations and Recombinations i n the Multiverse
Yi Wang∗
Physics Department, McGill University, Montreal, H3A2T8, Canada
In this Letter, we brieﬂy review the multi-stream inﬂation s cenario, and discuss its implications in
the string theory landscape and the inﬂationary multiverse . In multi-stream inﬂation, the inﬂation
trajectory encounters bifurcations. If these bifurcation s are in the observable stage of inﬂation, then
interesting observational eﬀects can take place, such as do main fences, non-Gaussianities, features
and asymmetries in the CMB. On the other hand, if the bifurcat ion takes place in the eternal stage
of inﬂation, it provides an alternative creation mechanism of bubbles universes in eternal inﬂation,
as well as a mechanism to locally terminate eternal inﬂation , which reduces the measure of eternal
inﬂation.
I. INTRODUCTION
Inﬂation [1] has become the leading paradigm for the
very early universe. However, the detailed mechanism
for inﬂation still remains unknown. Inspired by the pic-
ture of string theory landscape [2], one could expect that
the inﬂationary potential has very complicated structure
[3]. Inﬂation in the string theory landscape has impor-
tantimplicationsinbothobservablestageofinﬂationand
eternal inﬂation.
The complicated inﬂationary potentials in the string
theory landscape open up a great number of interest-
ing observational eﬀects during observable inﬂation. Re-
searchesinvestigatingthecomplicatedstructureofthein-
ﬂationary potential include multi-stream inﬂation [4, 5],
quasi-single ﬁeld inﬂation [6], meandering inﬂation [7],
old curvaton [8], etc.
Thestringtheorylandscapealsoprovidesaplayground
for eternal inﬂation. Eternal inﬂation is an very early
stage of inﬂation, during which the universe reproduces
itself, so that inﬂation becomes eternal to the future.
Eternal inﬂation, if indeed happened (for counter ar-
guments see, for example [9]), can populate the string
theory landscape, providing an explanation for the cos-
mological constant problem in our bubble universe by
anthropic arguments.
In this Letter, we shall focus on the multi-stream inﬂa-
tion scenario. Multi-stream inﬂation is proposed in [4].
And in [5], it is pointed out that the bifurcations can
lead to multiverse. Multi-stream inﬂation assumes that
during inﬂation there exist bifurcation(s) in the inﬂation
trajectory. For example, the bifurcations take place nat-
urally in a random potential, as illustrated in Fig. 1. We
brieﬂy review multi-stream inﬂation in Section II. The
details of some contents in Section II can be found in
[4]. We discuss some new implications of multi-stream
inﬂation for the inﬂationary multiverse in Section III.
∗wangyi@hep.physics.mcgill.ca
FIG. 1. In this ﬁgure, we use a tilted random potential to
mimic a inﬂationary potential in the string theory landscap e.
One can expect that in such a random potential, bifurcation
eﬀects happens generically, as illustrated in the trajecto ries
in the ﬁgure.
FIG. 2. One sample bifurcation in multi-stream inﬂation.
The inﬂation trajectory bifurcates into AandBwhen the
comoving scale k1exits the horizon, and recombines when
the comoving scale k2exits the horizon.
II. OBSERVABLE BIFURCATIONS
In this section, we discuss the possibility that the bi-
furcation of multi-stream inﬂation happens during the
observable stage of inﬂation. We review the production
of non-Gaussianities, features and asymmetries [4] in the2
FIG. 3. In multi-stream inﬂation, the universe breaks up
into patches with comoving scale k1. Each patch experienced
inﬂation either along trajectories AorB. These diﬀerent
patches can be responsible for the asymmetries in the CMB.
CMB, and investigate some other possible observational
eﬀects.
To be explicit, we focus on one single bifurcation, as
illustrated in Fig. 2. We denote the initial (before bifur-
cation) inﬂationary direction by ϕ, and the initial isocur-
vature direction by χ. For simplicity, we let χ= 0 before
bifurcation. When comoving wave number k1exits the
horizon, the inﬂation trajectory bifurcates into Aand
B. When comoving wave number k2exits the horizon,
the trajectories recombines into a single trajectory. The
universe breaks into of order k1/k0patches (where k0de-
notes the comoving scale of the current observable uni-
verse), each patch experienced inﬂation either along tra-
jectories AorB. The choice of the trajectories is made
by the isocurvature perturbation δχat scale k1. This
picture is illustrated in Fig. 3.
We shall classify the bifurcation into three cases:
Symmetric bifurcation . If the bifurcation is symmetric,
in other words, V(ϕ,χ) =V(ϕ,−χ), then there are two
potentially observable eﬀects, namely, quasi-single ﬁeld
inﬂation, and a eﬀect from a domain-wall-like objects,
which we call domain fences.
As discussed in [4], the discussion of the bifurcation
eﬀect becomes simpler when the isocurvature direction
has mass of order the Hubble parameter. In this case,
except for the bifurcation and recombination points, tra-
jectoryAand trajectory Bexperience quasi-single ﬁeld
inﬂation respectively. As there are turnings of these tra-
jectories, the analysis in [6] can be applied here. The
perturbations, especially non-Gaussianities in the isocur-
vature directions are projected onto the curvature direc-
tion, resultingin a correctionto the powerspectrum, and
potentially large non-Gaussianities. As shown in [6], the
amount of non-Gaussianity is of order
fNL∼P−1/2
ζ/parenleftbigg1
H∂3V
∂χ3/parenrightbigg/parenleftBigg˙θ
H/parenrightBigg3
, (1)
whereθdenotes the angle between the true inﬂation di-
rection and the ϕdirection.
As shown in Fig. 3, the universe is broken into patches
during multi-stream inﬂation. There arewall-likebound-
aries between these patches. During inﬂation, theseboundaries are initially domain walls. However, after
the recombination of the trajectories, the tensions of
these domain walls vanish. We call these objects domain
fences. As is well known, domain wall causes disasters
in cosmology because of its tension. However, without
tension, domain fence does not necessarily cause such
disasters. It is interesting to investigate whether there
are observational sequences of these domain fences.
Nearly symmetric bifurcation If the bifurcation is
nearly symmetric, in other words, V(ϕ,χ)≃V(ϕ,−χ),
but not equal exactly, which can be achieved by a spon-
taneous breaking and restoring of an approximate sym-
metry, then besides the quasi-single ﬁeld eﬀect and the
domain fence eﬀect, there will be four more potentially
observable eﬀects in multi-stream inﬂation, namely, the
features and asymmetries in CMB, non-Gaussianity at
scalek1and squeezed non-Gaussianity correlating scale
k1and scale kwithk1< k < k 2.
The CMB power asymmetries are produced because,
as in Fig. 3, patches coming from trajectory AorBcan
have diﬀerent power spectra PA
ζandPB
ζ, which are de-
termined by their local potentials. If the scale k1is near
to the scale of the observational universe k0, then multi-
stream inﬂation provides an explanation of the hemi-
spherical asymmetry problem [10].
The features in the CMB (here feature denotes extra
large perturbation at a single scale k1) are produced as
a result of the e-folding number diﬀerence δNbetween
two trajectories. From the δNformalism, the curvature
perturbation in the uniform density slice at scale k1has
an additional contribution
δζk1∼δN≡ |NA−NB|. (2)
These features in the CMB are potentially observable
in the future precise CMB measurements. As the addi-
tional ﬂuctuation δζk1does not obey Gaussian distribu-
tion, there will be non-Gaussianity at scale k1.
Finally, there are also correlations between scale k1
and scale kwithk1< k < k 2. This is because the ad-
ditional ﬂuctuation δζk1and the asymmetry at scale k
are both controlled by the isocurvature perturbation at
scalek1. Thus the ﬂuctuations at these two scales are
correlated. As estimated in [4], this correlation results in
a non-Gaussianity of order
fNL∼δζk1
ζk1PA
ζ−PB
ζ
PA
ζP−1/2
ζ. (3)
Non-symmetric bifurcation If the bifurcation is not
symmetric at all, especially with large e-folding number
diﬀerences (of order O(1) or greater) along diﬀerent tra-
jectories, the anisotropy in the CMB and the large scale
structure becomes too large at scale k1. However, in
this case, regions with smaller e-folding number will have
exponentially small volume compared with regions with
larger e-folding number. Thus the anisotropy can behave
in the form of great voids. We shall address this issue in
more detail in [11]. Trajectories with e-folding number3
diﬀerence from O(10−5) toO(1) in the observable stage
of inﬂation are ruled out by the large scale isotropy of
the observable universe.
At the remainderof this section, we would like to make
several additional comments for multi-stream inﬂation:
The possibility that the bifurcated trajectories never re-
combine. In this case, one needs to worry about the do-
main walls, which do not become domain fence during
inﬂation. These domain walls may eventually become
domain fence after reheating anyway. Another prob-
lem is that the e-folding numbers along diﬀerent tra-
jectories may diﬀer too much, which produce too much
anisotropies in the CMB and the large scale structure.
However, similar to the discussion in the case of non-
symmetric bifurcation, in this case, the observable eﬀect
could become great voids due to a large e-folding number
diﬀerence. The case without recombination of trajectory
also has applications in eternal inﬂation, as we shall dis-
cuss in the next section.
Probabilities for diﬀerent trajectories . In [4], we con-
sidered the simple example that during the bifurcation,
the inﬂaton will run into trajectories AandBwith equal
probabilities. Actually, this assumption does not need to
be satisﬁed for more general cases. The probability to
run into diﬀerent trajectories can be of the same order
of magnitude, or diﬀerent exponentially. In the latter
case, there is a potential barrier in front of one trajec-
tory, which can be leaped over by a large ﬂuctuation of
theisocurvatureﬁeld. Alargeﬂuctuationoftheisocurva-
ture ﬁeld is exponentially rare, resulting in exponentially
diﬀerent probabilities for diﬀerent trajectories. The bi-
furcation of this kind is typically non-symmetric.
Bifurcation point itself does not result in eternal inﬂa-
tion. As is well known, in single ﬁeld inﬂation, if the
inﬂaton releases at a local maxima on a “top of the hill”,
a stage of eternal inﬂation is usually obtained. However,
at the bifurcation point, it is not the case. Because al-
though the χdirection releases at a local maxima, the ϕ
direction keeps on rolling at the same time. The inﬂa-
tiondirectionisacombinationofthesetwodirections. So
multi-stream inﬂation can coexist with eternal inﬂation,
but itself is not necessarily eternal.
III. ETERNAL BIFURCATIONS
In multi-stream inﬂation, the bifurcation eﬀect may ei-
ther take place at an eternal stage of inﬂation. In this
case, it provides interesting ingredients to eternal inﬂa-
tion. These ingredients include alternative mechanism to
producediﬀerentbubble universesandlocalterminations
for eternal inﬂation, as we shall discuss separately.
Multi-stream bubble universes . The most discussed
mechanisms to produce bubble universes are tunneling
processes, such as Coleman de Luccia instantons [12] and
Hawking Moss instantons [13]. In these processes, the
tunneling events, which are usually exponentially sup-
pressed, create new bubble universes, while most parts
FIG. 4. Cascade creation of bubble universes. In this ﬁgure,
we assume trajectory Ais the eternal inﬂation trajectory, and
trajectory Bis the non-eternal inﬂation trajectory.
of the spatial volume remain in the old bubble universe
at the instant of tunneling.
If bifurcations of multi-stream inﬂation happen dur-
ing eternal inﬂation, two kinds of new bubble universes
can be created with similar probabilities. In this case,
at the instant of bifurcation, both kinds of bubble uni-
verseshavenearlyequalspatialvolume. Withachangeof
probabilities, the measures for eternal inﬂation should be
reconsideredformulti-streamtype bubble creationmech-
anism.
If the inﬂation trajectories recombine after a period of
inﬂation, the diﬀerent bubble universes will eventually
have the same physical laws and constants of nature. On
the other hand, if the diﬀerent inﬂation trajectories do
not recombine, then the diﬀerent bubble universes cre-
ated by the bifurcation will have diﬀerent vacuum ex-
pectation values of the scalar ﬁelds, resulting to diﬀerent
physical laws or constants of nature. It is interesting
to investigate whether the bifurcation eﬀect is more ef-
fective than the tunneling eﬀect to populate the string
theory landscape.
Note that in multi-stream inﬂation, it is still possi-
ble that diﬀerent trajectorieshaveexponentiallydiﬀerent
probabilities, as discussed in the previous section. In this
case, multi-stream inﬂation behaves similar to Hawking
Moss instantons during eternal inﬂation.
Local terminations for eternal inﬂation . It is possible
that during multi-stream inﬂation, a inﬂation trajectory
bifurcates in to one eternal inﬂation trajectory and one
non-eternal inﬂation trajectory with similar probability.
Inthiscase,theinﬂatonintheeternalinﬂationtrajectory
frequently jumps back to the bifurcation point, resulting
in a cascade creation of bubble universes, as illustrated
in Fig. 4. This cascade creation of bubble universes, if4
realized, is more eﬃcient in producing reheating bubbles
than tunneling eﬀects. Thus it reduces the measure for
eternal inﬂation.
There are some other interesting issues for bifurcation
in the multiverse. For example, the bubble walls may
be observable in the present observable universe, and the
bifurcations can lead to multiverse without eternal inﬂa-
tion. These possibilities are discussed in [5].
IV. CONCLUSION AND DISCUSSION
To conclude, webrieﬂy reviewedmulti-stream inﬂation
during observable inﬂation. Some new issues such as do-main fences and connection with quasi-single ﬁeld inﬂa-
tion are discussed. We also discussed multi-stream inﬂa-
tion in the context of eternal inﬂation. The bifurcation
eﬀect in multi-stream inﬂation provides an alternative
mechanism for creating bubble universes and populating
the string theory landscape. The bifurcation eﬀect also
provides a very eﬃcient mechanism to locally terminate
eternal inﬂation.
ACKNOWLEDGMENT
We thank Yifu Cai for discussion. This work was sup-
ported by NSERC and an IPP postdoctoral fellowship.
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