REVIEW
Inflation, Supergravity and Superstrings
High Energy Physics - Phenomenology · Submitted 26 May 1994 · 21 references
Pith's verdict: The central derivation is correct and the K = -ln(1-|φ|²) result V = V1(1-e^{-√2σ}) is the Starobinsky shape derived years early; the soft spots are ψ stability asserted rather than computed and a typo in Eq. 35.
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Referee report
The paper
The paper claims
A Kähler potential that keeps the inflaton light in supergravity
The reading
What carries the argument
The load-bearing device is the factorisation of the Kähler metric between the inflaton sector and the $\psi$ sector. Writing $K=A(\phi,\bar\phi)+\bar\psi B(\phi,\bar\phi)\psi+\cdots$ and demanding $B^{-1}=f(\phi,\bar\phi)C^{-1}(\chi,\bar\chi)$ with $A=-\ln f+g$ is exactly what makes the exponential prefactor cancel against the inverse metric, leaving the flat potential of Eq. (30). The paper notes that all the required conditions follow automatically from an R-parity $\psi\to-\psi$, $\phi\to\phi$, $W\to-W$, $K\to K$, which also stabilises $\psi$ at the origin because $V_\psi=0$ becomes automatic. The worked examples are the coset Kähler potentials $-\ln(1-|\phi|^2)$ for $SU(m,1)/(SU(m)\times U(1))$ and the $SO(m,2)/(SO(m)\times SO(2))$ form, both of which appear in the untwisted sectors of orbifold compactifications, where target-space duality symmetries supply the needed R-parity.
Extended reading notes
Core claim
The central claim is that the supergravity mass problem for the inflaton is not intrinsic but a consequence of generic choices, and that it disappears for a specific class of Kähler potentials. Expanding about a point where the fields are canonical, the overall factor $e^{K}$ in the $F$-term potential contributes $V$ to every scalar mass squared, so that $V''/V = 1 + \text{other terms}$ and slow roll fails unless the terms inside the bracket cancel it. Stewart takes $W=W_\phi=0$ and $W_\psi\neq0$ during inflation, assumes no mixed kinetic terms $K_{\phi\bar\psi}|_{\psi=0}=0$, and requires the Kähler potential to take the form $K=-\ln[f(\phi,\bar\phi)-\bar\psi C(\chi,\bar\chi)\psi]+g(\chi,\bar\chi)$. The $e^{K}$ factor then cancels against $f$, and the $F$-term potential collapses to $V=e^{g}W_\psi C^{-1}\bar{W}_{\bar\psi}$, which is independent of the inflaton except through the kinetic term. Section 6.3 pushes this further: with $K=-\ln(1-|\phi|^2)$ the canonically normalised inflaton is $|\phi|=\tanh(\sigma/\sqrt{2})$, and an $f(\phi)$ that gives no inflation in global supersymmetry yields $V=V_1(1-e^{-\sqrt{2}\tilde\sigma})$ in supergravity, with spectral index $n\simeq1-2/N\simeq0.96$ and a tensor-to-scalar ratio of order $10^{-3}$.
Load-bearing premise
The cancellation requires the inflaton sector and the $\psi$ sector to have no mixed kinetic terms, $K_{\phi\bar\psi}|_{\psi=0}=0$, together with exactly vanishing $W$ and $W_\phi$ during inflation; the paper introduces the decoupling 'for simplicity' and does not show that it survives in a realistic compactification where the inflaton mixes with moduli.
Editorial extensions
If this is right
- Constructing an inflationary model in supergravity reduces to constructing one in global supersymmetry subject to $W=W_\phi=\psi=0$ and $W_\psi\neq0$ during inflation, with no fine tuning of the troublesome mass term.
- Superstring compactifications supply the required Kähler potentials in their untwisted sectors, and target-space duality symmetries supply the R-parities that enforce the superpotential conditions automatically.
- A globally supersymmetric model that does not inflate can still inflate after the supergravity kinetic terms stretch the potential, giving $n\simeq0.96$ and a tensor-to-scalar ratio around $10^{-3}$ for the simplest single-field case.
- Small departures from the idealised limit are calculable rather than fatal: a shift $K=K_0+\delta(\phi,\bar\phi)$ contributes $\delta'$ to $V'/V$ and $\delta''$ to $V''/V-(V'/V)^2$, so the slope of the inflaton potential can be dialled by the deviation.
- The D-term can provide an alternative route, since with $n=1$ and $g\Lambda/\lambda_2<1$ the energy density is dominated by $\frac{1}{2}g^2\Lambda^4$.
Reading between the lines
- The construction suggests a general recipe for any modulus-driven inflation: identify the field that plays the role of $\psi$ and check whether the duality group of the compactification contains an R-parity acting on it, rather than searching the whole Kähler moduli space.
- Because the inflaton potential in the stretched-kinetic-term scenario depends on $f(\phi)$ only through its behaviour as $|\phi|\to1$, the observable predictions are largely insensitive to the detailed form of the superpotential, which makes the near-$0.96$ spectral index a sharper target than it first appears.
- The same cancellation mechanism could be applied to other light scalars that supergravity tends to give Hubble-scale masses, such as fields controlling the end of inflation or reheating, provided they sit in a sector carrying the same R-parity.
- The paper's own Section 6.2 concedes that exact $W=W_\phi=0$ holds only in an idealised limit; a natural follow-up is to quantify how small $|W|^2/|W_\psi|^2$ must be for the spectral index to stay within observational bounds.
Original abstract
The positive potential energy required for inflation spontaneously breaks supersymmetry and in general gives any would-be inflaton an effective mass of order the inflationary Hubble parameter thus ruling it out as an inflaton. In this paper I give simple conditions on the superpotential that eliminate some potential sources for this mass, and derive a form for the Kahler potential that eliminates the rest. This reduces the problem of constructing a model of inflation in supergravity to that of constructing one in global supersymmetry with the extra conditions $W=W_\varphi=\psi=0$ during inflation (where $W$ is the superpotential, the inflaton $\in\varphi$, and $W_\psi\neq0$). I then point out that Kahler potentials of the required form often occur in superstrings and that the target space duality symmetries of superstrings often contain R-parities which would make $W=W_\varphi=0$ automatic for $\psi=0$.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
The positive potential energy required for inflation spontaneously breaks supersymmetry and in general gives any would-be inflaton an effective mass of order the inflationary Hubble parameter thus ruling it out as an inflaton. In this paper I give simple conditions on the superpotential that eliminate some potential sources for this mass, and derive a form for the Kahler potential that eliminates the rest. This reduces the problem of constructing a model of inflation in supergravity to that of constructing one in global supersymmetry with the extra conditions $W=W_\varphi=\psi=0$ during inflation (where $W$ is the superpotential, the inflaton $\in\varphi$, and $W_\psi\neq0$). I then point out that Kahler potentials of the required form often occur in superstrings and that the target space duality symmetries of superstrings often contain R-parities which would make $W=W_\varphi=0$ automatic for $\psi=0$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The positive potential energy required for inflation spontaneously breaks supersymmetry and in general gives any would-be inflaton an effective mass of order the inflationary Hubble parameter thus ruling it out as an inflaton. | background 1.00 | 0.08 | 0.05 | asserted 1.00 |
| s1 | In this paper I give simple conditions on the superpotential that eliminate some potential sources for this mass, and derive a form for the Kahler potential that eliminates the rest. | contribution 0.92 | 0.95 | 0.17 | asserted 1.00 |
| s2 | This reduces the problem of constructing a model of inflation in supergravity to that of constructing one in global supersymmetry with the extra conditions $W=W_\varphi=\psi=0$ during inflation (where $W$ is the superpotential, the inflaton $\in\varphi$, and $W_\psi\neq0$). | result 0.62 | 0.89 | 0.58 | asserted 1.00 |
| s3 | I then point out that Kahler potentials of the required form often occur in superstrings and that the target space duality symmetries of superstrings often contain R-parities which would make $W=W_\varphi=0$ automatic for $\psi=0$. | interpretation 0.44 | 0.58 | 0.12 | asserted 0.51 |
What was examined, and what was not
- arXiv listing abstract v2
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References
How the literature uses it
33 arXiv papers cite this in 52 sentences
Papers citing it · 36 sentences without a confident label
Cites it as background 14 papers
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Let us begin by reviewing the $D$-term inflation model [ref] in light of recent cosmological observations.
Low-Scale D-term Inflation and the Relaxion · $D$-term inflation model · 1.00
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The models of the $D$-term inflation [ref] use the Fayet-Illiopoulos (FI) term [ref] which can be written when the gauge symmetry is $U(1)$.
Inflation and long-range force from clockwork $D$-term · Introduction · 0.85
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At the level of effective field theories, for instance in supergravity, one may postulate approximately conserved symmetries like a `shift symmetry' [ref] or a `Heisenberg symmetry' [ref] to solve the $η$-problem.
Towards Matter Inflation in Heterotic String Theory · Introduction · 0.85
Discussion
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