N-flation
7/21/2005
707 citations (699 excluding self-citations). One of the most-cited papers connecting string theory to observational cosmology.
The Problem
Inflation, the brief period of exponential expansion in the early universe, requires a scalar field rolling slowly down a very flat potential. Observations of the cosmic microwave background constrain the shape of that potential. The Lyth bound establishes that if inflation produces detectable gravitational waves (a tensor-to-scalar ratio r measurable by CMB experiments), the inflaton field must have traversed a distance greater than the Planck mass during inflation. This is “large-field” inflation.
Large-field inflation is difficult to embed in string theory. The Planck mass is the scale at which quantum gravity effects become important, and controlling a scalar field potential over super-Planckian distances requires understanding physics at that scale. In string compactifications, the natural scalar fields are axions: pseudoscalar particles with periodic potentials whose range is set by the decay constant, typically well below the Planck mass. A single axion cannot drive large-field inflation because it runs out of room. By 2005, embedding inflation in string theory in a controlled, calculable way remained one of the central open problems in theoretical cosmology.
The Key Idea
N-flation solves the field-range problem by distributing inflation across many axion fields. String compactifications naturally produce large numbers of axions, one for each cycle in the compactification geometry, often hundreds. Each individual axion traverses only a sub-Planckian distance, of order Mp/√N, well within the regime where the potential is calculable and radiatively stable. But the collective displacement of N fields in field space is √N times larger than any single field’s excursion. With enough axions, the collective trajectory covers a super-Planckian distance while every individual field stays in the controlled regime.
The mechanism relies on a central-limit-theorem-like effect: the random phases and decay constants of many independent axions average out, producing an effectively smooth inflationary trajectory from microscopically complicated ingredients. The resulting predictions for the spectral index and tensor-to-scalar ratio resemble those of simple single-field chaotic inflation, but the underlying physics is a collective effect of string theory’s natural particle content rather than a single ad hoc scalar field.
Impact
N-flation became one of the standard references in string inflation. The WMAP five-year analysis (5,699 citations) cites it as a benchmark multi-field inflationary scenario. Baumann and McAllister’s Inflation and String Theory (650 citations), the standard textbook on the subject, treats N-flation as one of the primary mechanisms alongside axion monodromy and brane inflation. Baumann’s TASI lectures on inflation (1,369 citations) use it to illustrate how string theory’s many moduli fields can be turned from a problem (the moduli stabilization challenge) into a feature (collective inflation).
The paper motivated two important lines of follow-up work. The first was axion monodromy (Silverstein and Westphal, 1,017 citations; McAllister, Silverstein, and Westphal, 956 citations), which found a different way to achieve large-field axion inflation by unwinding the axion’s periodic potential through brane dynamics. Axion monodromy and N-flation are complementary approaches to the same problem: one extends a single field’s range, the other uses many fields with short ranges.
The second line was the interaction between N-flation and the swampland program. The swampland conjectures (Vafa, 1,266 citations; Ooguri, Palti, Shiu, Vafa, 1,185 citations) propose constraints on which effective field theories can be consistently embedded in quantum gravity, and several conjectures directly constrain the field range and potential shape available to inflatons. N-flation became a test case: can many sub-Planckian fields collectively evade swampland bounds? Rudelius’s “Bounds on Slow Roll and the de Sitter Swampland” (729 citations) explicitly analyzes N-flation in this context. The question remains open and is one of the active frontiers in string cosmology.
Observationally, CMBPol/CMB-S4 mission studies (429 citations) cite N-flation as a target scenario for B-mode polarization measurements that would detect or rule out gravitational waves from inflation. The axion cosmology review by Marsh (1,979 citations) discusses N-flation as part of the broader landscape of axion physics connecting string theory to observation.
Recollections
I was at the end of my postdoc at the Stanford Institute for Theoretical Physics, about to join the SLAC/Stanford faculty. Shamit Kachru and John McGreevy had what they themselves described as a stupid idea to evade the super-Planckian field value problem in large-field inflation. They were discussing it with Savas Dimopoulos when I walked in and quipped that Pythagoras saves chaotic inflation: no individual field takes a super-Planckian excursion, but the full displacement in field space is larger due to the Pythagorean combination of field values. These multi-field axions appear generically in string compactifications, so the ingredients were free.
We realized immediately that this was very cheap and likely not a true solution. We spent most of our time trying to break it. We found a generic string theory correction that would prevent arbitrarily large collective excursions, but the correction kicked in just slowly enough to produce roughly 60 e-folds of inflation, the minimum necessary. We thought this was sufficient to make the paper publishable, though none of us truly felt great about it.
We posted it as a preprint and didn’t bother to submit it to a journal for nearly two years. By the time I realized we should probably publish it officially, it already had more than 100 citations. We submitted to JCAP, a relatively low-stakes journal. John and I laughed about it. The paper that none of us felt great about became one of the most-cited papers any of us have written.