Section 9: Inflation in String Theory
Because inflation is sensitive to MPlanck-suppressed corrections, physicists must either make strong assumptions about Planck-scale physics or propose and compute with models of inﬂation in theories where they can calculate such gravity eﬀects. String theory provides one class of theories of quantum gravity well developed enough to oﬀer concrete and testable models of inflation—and sometimes additional correlated observational consequences. A string compactification from 10D to 4D often introduces interesting scalar fields. Some of those fields provide intriguing inﬂationary candidates.
Figure 31: A brane and an anti-brane moving toward one another and colliding could have caused inflation and the Big Bang.
Source: © S.-H. Henry Tye Laboratory at Cornell University. More info
Perhaps the best-studied classes of models involve p-branes of the sort we described earlier in this unit. So just for concreteness, we will brieﬂy describe this class of models. Imagine the cosmology of a universe that involves string theory compactification, curling up six of the extra dimensions to yield a 4D world. Just as we believe that the hot gas of the Big Bang in the earliest times contained particles and anti-particles, we also believe that both branes and anti-branes may have existed in a string setting; both are p-dimensional hyperplanes on which strings can end, but they carry opposite charges under some higher-dimensional analog of electromagnetism.
The most easily visualized case involves a 3-brane and an anti-3-brane filling our 4D spacetime but located at different points in the other six dimensions. Just as an electron and a positron attract one another, the brane and anti-brane attract one another via gravitational forces as well as the other force under which they are charged. However, the force law is not exactly a familiar one. In the simplest case, the force falls oﬀ as 1/r4, where r is the distance separating the brane and anti-brane in the 6D compactified space.
Models with sufficiently interesting geometries for the compactified dimensions can produce slow-roll inflation when the brane and anti-brane slowly fall together, under the influence of the attractive force. The inflaton field is the mode that controls the separation between the brane and the anti-brane. Each of the branes, as a material object filling our 4D space, has a tension that provides an energy density filling all of space. So, a more accurate expression for the inter-brane potential would be V(r) ~ 2T3 − 1/r4, where T3 is the brane tension. For sufficiently large r and slowly rolling branes, the term 2T3 dominates the energy density of the universe and serves as the effective cosmological constant that drives inflation.
As the branes approach each other and r ~ , this picture breaks down. This is because certain open strings, now with one end on each of the branes as opposed to both ends on a single brane, can become light. In contrast, when r >> , such open strings must stretch a long distance and are quite heavy. Remember that the energy or mass of a string scales with its length. In the regime where r is very small, and the open strings become light, the picture in terms of moving branes breaks down. Instead, some of the light open strings mediate an instability of the brane configuration. In the crudest approximation, the brane and anti-brane simply annihilate (just as an electron and anti-electron would), releasing all of the energy density stored in the brane tensions in the form of closed-string radiation. In this type of model, the Big Bang is related to the annihilation of a brane with an anti-brane in the early universe.
Other consequences of inflation
Any well-specified model of cosmic inflation has a full list of consequences that can include observables beyond just the density fluctuations in the microwave background that result from inflation. Here, we mention some of the most spectacular possible consequences.
Quantum jiggles: We do not know the energy scale of inflation directly from data. In many of the simplest theories, however, this energy scale is very high, close to the Grand Unified Theory scale of 1016 GeV. It is therefore quite possible that inflation is probing energies 13 orders of magnitude higher than we'll see at the LHC.
Figure 32: A cosmic string could produce a double image of a galaxy behind it.
Source: © NASA/ESA Hubble Space Telescope. More info
If the scale of inflation is high enough, we may see further corroborating evidence beyond the solution of the horizon problem and the explanation of density fluctuations. The density fluctuations we discussed in the previous section came from the quantum jiggles of the inflaton field itself. But during inflation, quantum jiggles also originate in the other fields present, including the gravitational field. Future cosmological experiments exploring those phenomena could pin down the scale of inflation to just a few orders of magnitude shy of the Planck scale.
Cosmic strings: Very particular models often come with their own smoking-gun signatures. Take, for example, the class of speculative models we discussed earlier based on the slow attraction and eventual annihilation of a 3-brane and an anti-3-brane. The annihilation process involves the dynamics of open strings that stretch between the 3-brane and its partner, and that eventually "condense." This condensation process creates cosmic strings as the branes annihilate, which can be thought of as 1-branes or even fundamental strings that thread our 4D spacetime, and have grown to macroscopic size. If these tension-bearing cosmic strings really were created at the end of inflation, they should be present in the universe today, with tell-tale signatures in experiments that study the distribution of matter through gravitational lensing. Future experiments should rule out the presence of such strings or detect them for a wide range of values of the possible scale of inflation.
Figure 33: The Planck satellite will make precise measurements of fluctuations in the CMB.
Source: © ESA. More info
Density fluctuations: The slow-roll approach to inflation, with an inflaton field moving on a flat potential, is the class of models we focused on in the last section, but is not the only model of inflation. In some more modern theories, again partially inspired by branes in superstring theory, the inﬂaton undergoes rapid motion. Instead of the flatness of the potential, a delicate interplay between the potential and the complicated structure of the inflation kinetic terms produces the inflation. If any such model captures a grain of truth, then the pattern of density fluctuations would bear tell-tale structural signatures. Measurements by the Planck satellite that the European Space Agency launched in May 2009 should put constraints on the validity of those models.