Natural Chaotic Inflation in Supergravity

Masahide Yamaguchi; M. Kawasaki; T. Yanagida

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Natural Chaotic Inflation in Supergravity

M. Kawasaki, Masahide Yamaguchi, T. Yanagida

High Energy Physics - Phenomenology · Submitted 27 Apr 2000 · 18 references

Pith's verdict: The shift-symmetry mechanism is sound, and the reader's tachyon objection is a misreading: the e^K factor cancels it exactly. What's left are cosmetic errata.

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arXiv IDhep-ph/0004243 (version 2)

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CitationMasahide Yamaguchi, M. Kawasaki, T. Yanagida. "Natural Chaotic Inflation in Supergravity." arXiv:hep-ph/0004243 (2000). https://pith.science/paper/M7ZHFDWW

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Short version: this is the paper that made shift-symmetric chaotic inflation in supergravity a standard move, and the central mechanism holds up. The reader's weakest-assumption call is wrong, though, and I'd correct it before it circulates further. The stress-test has this one right and the reader has it wrong — say so plainly if this comes up in a group meeting, because the misreading is a natural one and will spread otherwise. What's new: earlier SUGRA chaotic inflation (refs 9, 10) needed a hand-picked Kähler potential with no symmetry behind it. Here K = K[(Φ+Φ*)², XX*] is fixed by a Nambu-Goldstone-like shift symmetry, so e^K never sees the inflaton's imaginary part and φ can sit at 10^5 M_P. The superpotential W = mXΦ breaks the shift softly, and the leading potential is ordinary quadratic chaotic inflation. The chain from eqs (3), (9), (11) to (15) checks out. The reader says X is tachyonic for large φ, citing eq (14), and that the ζ|X|⁴ fix is an unprotected tuning. That's a misreading. At η=0, eq (14) gives V ≈ m²[(1/2)φ² + |X|²] once you expand e^{|X|²}: the +|X|²·(1/2)φ² from the exponential cancels the −(1/2)φ² inside the bracket exactly. So m_X² = 2m² > 0 at ζ=0, which is precisely what eq (28) says. The ζ term is a mild sign choice, not a tuning, and the paper flags it anyway. Genuine soft spots, all minor: eq (20) should read 3H Ẋ ≃ −2m²X, which makes X ∝ φ² rather than φ; the conclusion X ≲ m at the end of inflation survives, in fact strengthens. Eq (23) has m² where it should be m — a typo, since the quoted m ≃ 10^13 GeV only comes out with m. The ζ < 0 requirement is not symmetry-protected, but the paper says so and it's a leading-order Kähler assumption, not a hidden fine-tuning. And the model predicts r ≈ 0.2, now excluded; that's a falsifiable prediction that got falsified, which is a point in its favor, not against. Who should read it: anyone building inflation models in SUGRA, or working on the η problem and its descendants. It's a foundational reference, not a paper you read for current constraints. Recommendation: send it out. The physics is right, the naturalness argument is honest, and the errata are cosmetic.

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Minor objections (7)

[Eq. (23)] The displayed formula for the density perturbation has m² where the standard slow-roll result requires m. The quoted value m ≈ 10¹³ GeV is consistent with the correct formula, so this appears to be a typo, but it should be corrected.
[Footnote 1] The induced Kähler potential breaking term |mΦ|² is marginal at φ ~ m⁻¹. The paper says it is negligible for |φ| ≲ m⁻¹, but at φ ~ m⁻¹ it is O(1). The authors should clarify whether the O(1) correction at the start of inflation affects the onset of inflation or the predictions at observable scales.
[§III, Conclusion] The claim that 'the chaotic inflation model has no initial value problem' is a strong claim. In chaotic inflation the initial condition φ(0) ~ m⁻¹ ~ 10⁵ M_P is super-Planckian; the standard argument is that the universe starts with a chaotic distribution of fields at the Planck scale, which is an assumption about initial conditions rather than a derivation. The claim should be qualified.
[§II, after Eq. (23)] The paper should state the tensor-to-scalar ratio explicitly (r ≈ 0.13-0.16 for quadratic inflation) and discuss the running of the spectral index, since these are the observables that future CMB experiments will use to test the model.
[Eq. (28)] The convention for m_X² should be stated explicitly: whether V = m_X²|X|² or V = (1/2)m_X²|X|², since the factor of 2 affects the comparison with the Hubble scale.
[§II, after Eq. (20)] The paper assumes X is real and positive by a phase choice, but does not discuss the stabilization of the phase of X. Quantum corrections may generate a potential for the phase; this should be addressed.
[Throughout] There are numerous OCR typos in the text (e.g., 'Comic Backgrou nd Explorer' should be 'Cosmic Background Explorer', 'K¨ ahler' should be 'Kähler', 'inﬂ ation' should be 'inflation', 'sup ergravity' should be 'supergravity'). These should be corrected in the final version.

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The paper

The paper claims

A shift symmetry lets chaotic inflation survive inside supergravity, with the inflaton starting far above the Planck scale.

The reading

The paper asks whether chaotic inflation — the version in which the inflaton starts out far above the Planck scale amid large quantum fluctuations — can be built inside supergravity without hand-tuning the Kähler potential. The obstacle is the factor $e^{K}$ in the supergravity scalar potential, which grows exponentially for field values above the Planck mass and makes the potential too steep to inflate. The authors' answer is to let $K$ depend only on the combination $K( (Phi + Phi^*)^2, X X^* )$, which is invariant under a shift $Phi to Phi + iC$, so that the imaginary part of $Phi$ — the inflaton $phi$ — drops out of the exponential entirely. A small superpotential $W = m X Phi$ then breaks the shift symmetry just enough to generate the ordinary quadratic potential $V simeq tfrac{1}{2} m^2 phi^2$, and matching the COBE amplitude fixes $m simeq 10^{13}$ GeV. If this holds, chaotic inflation stops being an awkward guest in supergravity and becomes a natural inhabitant, with a built-in route to reheating and a tensor signal for future probes.

What carries the argument

The load-bearing device is the shift symmetry of the Kähler potential, $K = K((Phi + Phi^*)^2, X X^*)$, which removes the inflaton $phi = -i(Phi - Phi^*)/sqrt{2}$ from the dangerous $e^{K}$ factor and thereby solves the $eta$ problem — the generic supergravity tendency to give every scalar a mass of order the Hubble scale. The symmetry is then broken only by the superpotential $W = m X Phi$, which is the smallest breaking that generates a potential without producing a runaway; the smallness of $m$ is justified in 't Hooft's sense because the shift symmetry is restored as $m to 0$. The auxiliary field $X$ is the second half of the mechanism: it holds the symmetry-breaking spur and then dilutes away during inflation, leaving the inflaton's potential intact.

Extended reading notes

Core claim

The claim is that chaotic inflation arises naturally in supergravity once the inflaton's Kähler potential is protected by a Nambu-Goldstone-like shift symmetry, $Phi to Phi + i C M_G$, and the symmetry is broken only in the superpotential. The authors first show why the obvious breaking term $W = m Phi^2$ fails: with $Phi + Phi^* = 0$ the potential runs to $-infty$ as $|phi| to infty$, so inflation never starts. They replace it with $W = m X Phi$, introducing a second chiral multiplet $X$, and take $K = tfrac{1}{2}(Phi + Phi^*)^2 + X X^*$. The resulting potential, for $eta, |X| ll 1$, reduces to $V simeq tfrac{1}{2} m^2 phi^2 + m^2 |X|^2$: the standard quadratic chaotic inflation potential, with $phi$ unconstrained by the exponential factor. The $eta$ field is heavy and settles to zero, $X$ rolls to zero faster than $phi$, and the density-fluctuation amplitude $delta rho / rho simeq (1/(5 sqrt{3} pi)) , m^2 / sqrt{2} (phi^2 + X^2)$ normalized at the COBE scale gives $m simeq 10^{13}$ GeV and a spectral index $n_s simeq 0.96$. Reheating is arranged through $W = lambda X H bar{H}$, giving $T_R simeq 10^9 , text{GeV} , (lambda / 10^{-5})(m / 10^{13} text{GeV})^{1/2}$; keeping $T_R$ below the gravitino bound requires $lambda lesssim 10^{-5}$, which the authors argue is natural because $H bar{H}$ can be taken even under the model's $Z_2$.

Load-bearing premise

The construction needs a higher-order term $zeta |X|^4$ in the Kähler potential with $zeta$ negative and of moderate size, because without it the potential gives $X$ a tachyonic mass-squared of order $-m^2 phi^2$ and $X$ does not settle to zero; no symmetry in the paper fixes that coefficient or its sign.

Editorial extensions

If this is right

The inflaton potential is the textbook quadratic form $V simeq tfrac{1}{2} m^2 phi^2$, so the model inherits all of quadratic chaotic inflation's predictions, including a spectral index near $n_s simeq 0.96$ and a tensor-to-scalar ratio in the range future CMB polarization experiments target.
The inflaton mass is fixed rather than assumed: COBE normalization gives $m simeq 10^{13}$ GeV, a value tied directly to the observed amplitude of density perturbations.
Reheating proceeds through the coupling $lambda X H bar{H}$, which induces $lambda m phi H bar{H}$ and yields $T_R simeq 10^9$ GeV for $lambda simeq 10^{-5}$, keeping gravitino production under control.
The model removes the need to guess a special Kähler potential: its form is dictated by a symmetry, so the flatness of the potential beyond the Planck scale is a consequence of the symmetry rather than of tuning.
Because the shift symmetry is only softly broken, higher-order corrections to $K$ are argued to be negligible as long as $|phi| lesssim m^{-1}$, which is exactly the initial field value the chaotic scenario requires.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The stabilization of $X$ is the part of the construction least protected by symmetry: the paper's own potential gives $X$ a mass-squared of order $-m^2 phi^2 / 2$, i.e. tachyonic and of order the Hubble scale, so the slow-roll equation $3H dot{X} simeq -m^2 X$ holds only after a higher-order $zeta |X|^4$ term with negative $zeta$ is added; the required window on $zeta$ is a tuning that the shift s
Because the potential is exactly quadratic in the inflaton, the model is a clean target for tensor-mode searches: a detection of $r$ at the quadratic prediction would support it, and a tight upper bound below that level would remove this realization regardless of how natural its Kähler structure is.
The same shift-symmetry device could be applied to other large-field models, since the exponential problem is generic to any inflaton asked to exceed the Planck scale; the construction is a template, not a one-off.
The reheating sector couples to the same field $X$ that carries the symmetry breaking, so the smallness of $lambda$ and the smallness of $m$ are logically independent suppressions; a reader should ask whether a single underlying symmetry could tie them together.

Original abstract

We propose a chaotic inflation model in supergravity. In the model the K\"ahler potential has a Nambu-Goldstone-like shift symmetry of the inflaton chiral multiplet which ensures the flatness of the inflaton potential beyond the Planck scale. We show that the chaotic inflation naturally takes place by introducing a small breaking term of the shift symmetry in the superpotential. This may open a new branch of model building for inflationary universe in the framework of supergravity.

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We propose a chaotic inflation model in supergravity. In the model the K\"ahler potential has a Nambu-Goldstone-like shift symmetry of the inflaton chiral multiplet which ensures the flatness of the inflaton potential beyond the Planck scale. We show that the chaotic inflation naturally takes place by introducing a small breaking term of the shift symmetry in the superpotential. This may open a new branch of model building for inflationary universe in the framework of supergravity.

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s0	We propose a chaotic inflation model in supergravity.	contribution 1.00	0.90	0.13	asserted 0.92
s1	In the model the K\"ahler potential has a Nambu-Goldstone-like shift symmetry of the inflaton chiral multiplet which ensures the flatness of the inflaton potential beyond the Planck scale.	method 0.80	0.56	0.10	asserted 1.00
s2	We show that the chaotic inflation naturally takes place by introducing a small breaking term of the shift symmetry in the superpotential.	result 0.38	0.93	0.47	asserted 1.00
s3	This may open a new branch of model building for inflationary universe in the framework of supergravity.	interpretation 0.94	0.10	0.21	hedged possibility 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.86.

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sources read

arXiv listing abstract v2

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References

Works cited (18)

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H. Murayama, H, Suzuki, T. Yanagida, and J. Yokoyama, Phys. Rev. D50, R2356 (1994)

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G. ’t Hooft, in Recent developments in gauge theories , edited by G. ’t Hooft et al. (Plenum Press, Carg` ese, 1980)

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M. Yu. Khlopov and A.D. Linde, Phys. Lett. 138B, 265 (1984); J. Ellis, G.B. Gelmini, J.L. Lopez, D.V. Nanopoulos and S. Sarker, Nucl. Phys. B373, 399 (1992); M. Kawasaki and T. Moroi, Prog. Theor. Phys. 93, 879 (1995). 8

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astro-ph 2007-11 conditional novelty 5.0 of 10

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$\alpha $-Attractors: Planck, LHC and Dark Energy
hep-th 2015-06 conditional novelty 4.0 of 10

A family of supergravity models reproduces known inflation and dark-energy observables while leaving the gravitino mass and the tensor-to-scalar ratio free, so future B-mode and collider data can pin them down.
Chaotic Inflation in Supergravity after Planck and BICEP2
hep-th 2014-05 conditional novelty 4.0 of 10

A cubic term added to the superpotential of quadratic chaotic inflation in supergravity produces a two-parameter family of potentials whose tilt and tensor ratio can be fitted to Planck 2013 and BICEP2 data.
Multi-Natural Inflation in Supergravity and BICEP2
hep-ph 2014-03 conditional novelty 4.0 of 10

A two-sine-wave inflaton potential can span a broad region of (n_s, r) space, including n_s near or above unity with r around 0.1 to 0.2, so the model is not excluded by BICEP2.
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hep-th 2014-03 conditional novelty 4.0 of 10

Identifying the inflaton with the imaginary part of the supergravity scalaron multiplet, plus matter couplings that stabilize the real part, yields quadratic chaotic inflation with r around 0.16, consistent with BICEP2.
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hep-ph 2013-03 conditional novelty 4.0 of 10

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hep-th 2013-03 accept novelty 1.0 of 10

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astro-ph.CO 2011-01 accept novelty 1.0 of 10

Supergravity can accommodate new, hybrid, chaotic, and topological inflation, and Jordan-frame supergravity can host Higgs inflation, provided the Kähler potential or D-term structure is chosen to avoid the eta problem.

How later papers used it

143 later papers cite it

Evidence: substantial. 122 of the citing papers share no author with it, and 10 of those carry a confident label from Pith's citation classifier.

Pith read 19 of the 236 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

1 later paper by other authors is listed below: 1 compared its own results with it.

Compares with it 1 of 5 papers with this label

Said that its simple chaotic inflation model with a polynomial potential almost coincides with the model of chaotic inflation in supergravity proposed in this paper.

New models of chaotic inflation in supergravity (2010) · 0.84 · paraphrased: the licence does not allow quoting

How the literature uses it

132 arXiv papers cite this in 221 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 155 sentences without a confident label

Builds on it 4 papers

There, it was shown that one can significantly generalize the model of Ref. [ref] by studying more general potentials of the functional form $K((Φ-Φ̅)^2,SS̅)$ and by introducing models with a superpotential $W = S f(Φ)$, where $f(Φ)$ is an arbitrary holomorphic function.

Chaotic inflation and supersymmetry breaking · Introduction · 0.89
One can notice that this Kähler potential is invariant under the following transformation; [equation] This shift symmetry rather generalizes the shift symmetries proposed in [ref] , in the context of supergravity realization of the chaotic inflation and running kinetic inflation.

A New Class of Non-canonical Conformal Attractors for Multifield Inflation · Basic setup · 0.83
We will start with a simple polynomial potential that is a generalization of the potential used in Ref. [ref] : [equation] Note that the stabilizing terms $- ζ (SS̅)^2 + γ2 SS̅ (Φ-Φ̅)^2$ was added to the potential of the model of [ref] .

General inflaton potentials in supergravity · Examples · 1.00

Uses its method or tool 2 papers

In this review, following Refs. [ref] , we introduce the Nambu-Goldstone-like shift symmetry of the inflaton superfield $Φ$ in order to naturally realize chaotic inflation in supergravity.

Supergravity based inflation models: a review · Chaotic inflation · 0.81
We use the approach in [ref] and introduce two “orthogonal" chiral superfields [ref] , the goldstino, $S$, and inflaton superfield, $Φ$, where we denote the scalar components of these superfields with the same letter.

Sharp turns in axion monodromy: primordial black holes and gravitational waves · Multifield axion monodromy in supergravity · 0.96

Compares with it 5 papers

In [ref] the mass of the gravitino was vanishing.

$\alpha $-Attractors: Planck, LHC and Dark Energy · Shift Symmetry and Z, T, and $Φ$ variables · 0.90
This problem seems generic for the canonical scalar kinetic terms in the Jordan frame supergravity[One can compare this case to the generalized chaotic inflation for the minimal Kähler potential and the superpotential $W=XS^n$ with $n$ being a natural number in Refs. [ref] where the accompanying singlet has a vanishing mass at the origin.].

Chaotic inflation in Jordan frame supergravity · The tachyonic mass problem in Jordan-frame supergravity inflation · 0.97
Also, to bring this model closer to the notations of [ref] , we use the term $- 1/2(Φ-Φ̅)^2$ instead of the term $1/2(Φ+Φ̅)^2$ in the potential of the model of Ref. [ref] .

New models of chaotic inflation in supergravity · Chaotic inflation with a quadratic potential · 0.95

Cites it as background 48 papers

Among many (F-term) inflation models, chaotic inflation [ref] is one of the most interesting models from this viewpoint.

Phase transition and monopole production in supergravity inflation · Introduction · 0.90
Such a shift-symmetry has been for the first time introduced in the context of chaotic inflation in supergravity (SUGRA) [ref] .

Dynamical breaking of shift-symmetry in supergravity-based inflation · Introduction · 0.86
Sgoldstino-less models of inflation [ref] are characterized by the presence of a nilpotent chiral superfield $S^2=0$ [ref] , i.e. the goldstino superfield, responsible for spontaneous supersymmetry breaking and another chiral superfield $Φ$ whose imaginary component (in our conventions) is the inflaton field, which exhibits a shift symmetry in order to make its potential viable for inflation [ref] .

Sgoldstino-less inflation and low energy SUSY breaking · Effective field theory for sgoldstino-less inflation · 0.98

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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