1001.0008.txt

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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