1. Introduction

Inflation (Guth 1980, Linde 1981, Albrecht et al. 1982) has become the leading paradigm for the very early universe. Specifically, it is believed that at the end of the Grand Unification Epoch, 10^-36 seconds after the Big Bang, the universe began to exponentially expand, fueled by a negative-pressure vacuum energy density. The assumption is that the currently observable universe originated as a causally connected singularity. This was followed by the Big Bang and the beginning of the Grand Unification Epoch in which three of the four fundamental interactions, i.e., electromagnetism, and the weak and the strong interaction were unified as the electronuclear force. According to inflation theory, 10^-36 seconds after the Big Bang, the strong force separated from the other fundamental forces with the inflaton field (a hypothetical particle) providing the mechanism to drive a period of rapid expansion. That is, it is believed that prior to expansion the inflaton field was at a high energy state and due to random quantum fluctuations following the big bang, a phase transition was triggered and the inflaton field released its potential energy thereby generating a repulsive force that drove the following expansion.

Implicit in expansion theory is the belief that portions of the universe expanded faster than the speed of light. Although this concept appears to violate Special Relativity it is concordant with General Relativity in which objects may move faster than the speed of light with respect to each other. Therefore, the universe can be divided into an observable and a non-observable universe and which are unable to communicate with each other due to the constraints of Special Relativity and the speed of light. That is, since parts of the universe are moving away faster than the speed of light and will inflate eternally, they cannot exchange information with those regions moving much slower, and those slower regions include Earth, our solar system, and the Milky Way Galaxy.

Those parts of the universe which can't be viewed are therefore beyond the cosmological horizon which marks the boundary between the observable and non-observable universe which is still moving faster than the speed of light and thus too fast to be viewed from Earth. However, as the cosmological horizon expands, new regions of space become visible as they fall inside this side of the horizon, and here lies the conundrum: the newly visible portions of the universe are no different from those which had already been positioned on this side of the event horizon. They have the same temperature, space-time curvature, and so on. If these regions were moving away too fast to communicate or to influence one another, then how did they come to be similar? Inflation provides several answers, and explains why the universe appears homogeneous and flat, instead of highly curved and heterogeneous as predicted by the physics of a big bang. That is, all the regions come from an earlier era with a big vacuum energy, i.e. the cosmological constant.

However, the detailed mechanism for inflation still remains unknown. Inspired by the picture of string theory landscape (Bousso et al. 2000, Giddings et al. 2001, Kachru et al. 2003, Douglas 2003), one could expect that the inflationary potential has very complicated structure (Huang et al. 2008). Inflation in the string theory landscape has important implications in both observable stage of inflation and eternal inflation.

String theory is based on the premise that the universe is composed of what could be likened to vibrating filaments and branes (membranes) which consist of energy. These energetic branes (membranes) could also be likened to bubbles, with our observable universe attached to the theoretical walls of one of these bubbles. Therefore, the portion of the universe which cannot be observed would be part of a different brane. Different branes would represent different parallel universes. The string theory landscape in fact allows for multiple parallel universes, the multiverse.

The string theory landscape (also known as the anthropic landscape) refers to the possibility of a large number of vacua (or bubbles) which may take on many different possible configurations, only some of which may support life--hence, the "anthropic landscape." In quantum field theory, these vacua are nucleated and potentially unstable, due to quantum fluctuations, and can rapidly fall to a low energy state. However, these fluctuations are compensated by the tension of the bubble walls (the surrounding brane) which will expand at a rate approaching the speed of light; and thus the universe also expands and inflates. Unfortunately, if two adjacent bubbles rapidly expand, they may collide.

The complicated inflationary potentials in the string theory landscape opens up a great number of interesting observational effects during observable inflation. Researchers investigating the complicated structure of the inflationary potential include scenarios such as multi-stream inflation (Li et al. 2009a,b), quasi-single field inflation (Chen et al. 2009a, Chen et al. 2009b), meandering inflation (Tye et al. 2009), old curvaton (Gong et al. 2008, Gong et al. 2009), etc.

The string theory landscape also provides a playground for eternal inflation. Eternal inflation is a very early stage of inflation, during which the universe reproduces itself, so that inflation becomes eternal to the future. Eternal inflation, if indeed it is happening (for counter arguments see, for example Mukhanov et al. 1996, Cai et al. 2007, Huang et al. 2007, Wang 2008), can populate the string theory landscape, providing an explanation for the cosmological constant problem in our bubble universe by anthropic arguments.

The anthropic argument can be summarized thusly: life exists in this universe (or brane), and we can observe this universe, because this universe and the laws which govern it are compatible with life. The same would not be true of those universes which are not amenable to sustaining life as we know it. The anthropic principle also implies that the fundamental physics of this universe are not applicable to other universes, but instead have these values not for physical reasons but because they are necessary for life. Therefore, it could be said that this specific universe exists only for the purpose of creating observers which can observe this universe. Yet, if other universes exist which do not support life, why then is this universe unique? The point is: this universe is not unique but is just one of many possible universes; multiple universes which may have been created by multi-stream inflation scenarios.

In this article, we shall focus on the multi-stream inflation scenario as proposed by Li et al. (2009a). In Li et al. (2009b), it is pointed out that the bifurcations can lead to multiverse. The idea of Multi-stream inflation (Li et al. 2009a) assumes that during inflation there exist bifurcation(s) in the inflation trajectory, such that inflationary streams split apart (forming patches of universes) and flow along parallel or quite divergent courses in potentially random ways as illustrated in Fig. 1. and detailed in Section 2. The implication of multi-stream inflation for the inflationary multiverse is explained in Section 3.

Fig. 1: In this figure, we use a tilted random potential to mimic a inflationary potential in the string theory landscape. One can expect that in such a random potential, bifurcation effects happens generically, as illustrated in the trajectories in the figure.

In this section, we discuss the possibility that the bifurcation of multi-stream inflation happens during the observable stage of inflation. We review the production of non-Gaussianities, features and asymmetries (Li et al. 2009a) in the cosmic microwave background (CMB), and investigate some other possible observational effects. In physical cosmology, the fluctuations of the CMB are known to be approximately Gaussian and symmetric or curved in shape. However, non-Gaussianity fluctuations are believed to have prevailed in the primordial density field thereby contributing to inflation.

Fig. 2: One sample bifurcation in multi-stream inflation. The inflation trajectory bifurcates into A and B when the comoving scale k₁ exits the horizon, and recombines when the comoving scale k₂ exits the horizon.

To be explicit, we focus on one single bifurcation, as illustrated in Fig. 2. We denote the initial (before bifurcation) inflationary direction by φ, and the initial isocurvature direction by χ. For simplicity, we let χ=0 before bifurcation. When comoving wave number k₁ exits the horizon, the inflation trajectory bifurcates into A and B. When comoving wave number k₂ exits the horizon, the trajectories recombines into a single trajectory. The universe breaks into of order k₁/k_o patches where k_o denotes the comoving scale of the current observable universe), each patch experienced inflation either along trajectories A or B. The choice of the trajectories is made by the isocurvature perturbation δχ at scale k₁. This picture is illustrated in Fig. 3.

Fig. 3: In multi-stream inflation, the universe breaks up into patches with comoving scale k₁. Each patch experienced inflation either along trajectories A or B. These different patches can be responsible for the asymmetries in the CMB.

We shall classify the bifurcation into three cases:

Symmetric bifurcation. If the bifurcation is symmetric, in other words, V(φ x)=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 (Li et al. 2009a,b), 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, trajectory A and trajectory B experience quasi-single field inflation respectively. As there are turnings of these trajectories, the analysis in (Chen et al. 2009a, Chen et al. 2009b) can be applied here. The perturbations, especially non-Gaussianities in the isocurvature directions are projected onto the curvature direction, resulting in a correction to the power spectrum, and potentially large non-Gaussianities. As shown in (Chen et al. 2009a, Chen et al. 2009b), the amount of non-Gaussianity is of order

where θ denotes the angle between the true inflation direction and the φ direction.

As shown in Fig. 3, the universe is broken into patches during multi-stream inflation. There are wall-like boundaries between these patches. During inflation, these boundaries 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 -X) ∼ V(φ,−χ), but not equal exactly, which can be achieved by a spontaneous breaking and restoring of an approximate symmetry, 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 scale k_{1k1 and scale k with k1k2P^A and P^B, which are determined by their local potentials. If the scale k1k0 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 δN between two trajectories. From the δN formalism, the curvature perturbation in the uniform density slice at scale k1 has an additional contribution}

Finally, there are also correlations between scale k₁ and scale k with k₁k_{2 and the asymmetry at scale k are both controlled by the isocurvature perturbation at scale k1. Thus the fluctuations at these two scales are correlated. As estimated in (Li et al. 2009a), this correlation results in a non-Gaussianity of order}

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 trajectories, the anisotropy in the CMB and the large scale structure becomes too large at scale k_{1-5) to O(1) in the observable stage of inflation are ruled out by the large scale isotropy of the observable universe.}

At the remainder of this section, we would like to make several additional comments for multi-stream inflation:

The possibility that the bifurcated trajectories never recombine. In this case, one needs to worry about the domain walls, which do not become domain fence during inflation. These domain walls may eventually become domain fence after reheating anyway. Another problem is that the e-folding numbers along different trajectories 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 discuss in the next section.

Probabilities for different trajectories. In Li et al. (2009), we considered the simple example that during the bifurcation, the inflaton will run into trajectories A and B with 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 trajectory, which can be leaped over by a large fluctuation of the isocurvature field. A large fluctuation of the isocurvature field is exponentially rare, resulting in exponentially different probabilities for different trajectories. The bifurcation of this kind is typically non-symmetric.

Bifurcation point itself does not result in eternal inflation. 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 although the χ direction releases at a local maxima, the φ direction keeps on rolling at the same time. The inflation direction is a combination of these two directions. So multi-stream inflation can coexist with eternal inflation, but itself is not necessarily eternal.

3. Eternal Bifurcations

In multi-stream inflation, the bifurcation effect may either take place at an eternal stage of inflation. In this case, it provides interesting ingredients to eternal inflation. These ingredients include alternative mechanism to produce different bubble universes and local terminations 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 (Coleman et al. 1980) and Hawking Moss instantons (Hawking et al. 1982).

The string theory landscape refers to the possibility of a large number of vacua (or bubbles) which are essentially false vacuums. That is, immediately after the big bang our universe was in a state that can be described as a vacuum which is classically stable but quantum mechanically unstable. In quantum theory, matter in a false vacuum may "tunnel" to its true vacuum state; it shrinks (or decays) in size. False vacuum decay proceeds via the nucleation of bubbles in the false vacuum. However, each bubble is an infinite open universe in which inflation may occur. Coleman-De Luccia instanton describes the creation of an open universe via this bubble nucleation.

Hawking Moss instantons also give rise to open universes but do not require the existence of a false vacuum or other very specific properties of the excited matter state. However, Hawking Moss instantons begin as singularities where the resulting curvature becomes infinite.

Both Hawking Moss and Coleman-De Luccia instanton involve tunneling events, which are usually exponentially suppressed, thereby creating new bubble universes, while most parts of the spatial volume remain in the old bubble universe at the instant of tunneling.

If bifurcations of multi-stream inflation happen during 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 universes have nearly equal spatial volume. With a change of probabilities, the measures for eternal inflation should be reconsidered for multi-stream type bubble creation mechanism.

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 created by the bifurcation will have different vacuum expectation 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 effective than the tunneling effect to populate the string theory landscape.

Note that in multi-stream inflation, it is still possible that different trajectories have exponentially different 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. In this case, the inflaton in the eternal inflation trajectory 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, if realized, is more efficient in producing reheating bubbles than tunneling effects. Thus it reduces the measure for eternal inflation.

Fig. 4a and 4b: Cascade creation of bubble universes. In this figure, we assume trajectory A is the eternal inflation trajectory, and trajectory B is the non-eternal inflation trajectory.

There are some other interesting issues for bifurcation in the eternal stage of inflation. For example, the bubble walls may be observable in the present observable universe, and the bifurcations can lead to multiverse without eternal inflation. This possibility is discussed in (Li et al. 2009b).

4. Conclusion and Discussion

To conclude, we briefly reviewed multi-stream inflation during observable inflation. Some new issues such as domain fences and connection with quasi-single field inflation are discussed. We also discussed multi-stream inflation 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.