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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

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