An Inverse Lyth Bound for Non-Attractor Inflation

Abstract: I present a novel physics result generated entirely autonomously by ChatGPT 5.6 Sol. The Lyth bound relates an observable primordial tensor amplitude to a lower limit on the field excursion during single-field slow-roll inflation. We show that non-attractor evolution obeys a complementary upper bound. For a canonically normalized scalar field, the nonconstant superhorizon curvature mode is anti-damped whenever the second Hubble-flow parameter satisfies $ε<em>2\equiv d\lnε/dN&lt;-3$. The same condition requires the inflaton kinetic energy to decrease sufficiently rapidly that the field-space distance traversed during a non-attractor interval obeys $Δφ</em>{\rm NA} / M_{\text{P}} &lt; \left(2\sqrt{2ε<em>{\rm in}}/3\right) \left(1-e^{{-3ΔN/2}\right)} &lt; \left(2\sqrt{2ε</em>{\rm in}}/3\right)$.The result is independent of the potential and does not require the slow-roll approximation. If $ε<em>2=-p&lt;-3$ is constant, the stronger bound $Δφ/M</em>{\text{P}}&lt;2\sqrt{2ε<em>{\rm in}}/p$ is saturated asymptotically. In ultra-slow-roll inflation, $p=6$, giving $Δφ&lt;\sqrt{2ε</em>{\rm in}}M_{\text{P}}/3$. We derive an exact relation between field excursion and amplification of the velocity of the nonconstant curvature mode, and give the corresponding scalar-power relation in the quasi-de Sitter limit. In contrast with the usual Lyth relation, arbitrarily large non-attractor amplification approaches a finite field distance. This provides a model-independent field-range constraint on canonical single-field mechanisms for amplifying primordial fluctuations.

• The paper proposes an inverse Lyth bound, an upper field-range bound that limits the inflaton excursion in non-attractor phases of inflation under superhorizon anti-damping.
• The authors derive this bound from a canonical field equation and identify cases including constant-roll and ultra-slow roll inflation, establishing that extended non-attractor phases do not support arbitrarily large field displacements as a consequence of deteriorating inflaton kinetic energy.
• Under inflationary constraints, the paper concludes that a continuous anti-damping phase cannot support a Planckian field-space distance.

Central claim and scope

The paper proposes an upper field-range bound for canonical single-field inflation during a continuous non-attractor phase. It presents this result as complementary to the conventional Lyth bound, which relates a sustained tensor amplitude to a lower bound on the inflaton excursion in slow-roll attractor evolution. The proposed “inverse Lyth bound” instead states that the same dynamics responsible for superhorizon growth of the curvature perturbation constrain the field distance from above.

The analysis assumes Einstein gravity, a single scalar with canonical kinetic normalization, accelerated expansion, and continuous anti-damping of the nonconstant superhorizon curvature mode. It does not assume slow roll, a specific potential, constant roll, or quasi-de Sitter evolution for its principal field-range inequality. The result therefore applies to the distance accumulated during the specified non-attractor interval, not necessarily to the total field displacement over an entire inflationary history.

From superhorizon anti-damping to a field-range inequality

For a canonically normalized scalar, the background identity

$$\left|\frac{d\phi}{dN}\right|=M_{\rm P}\sqrt{2\epsilon}$$

makes the field-space distance during an interval an integral of $\sqrt{2\epsilon}$. The curvature perturbation obeys the exact equation

$$\ddot{\mathcal R}_k+H(3+\epsilon_2)\dot{\mathcal R}_k+ \frac{k^2}{a^2}\mathcal R_k=0,$$

where $\epsilon_2=d\ln\epsilon/dN$. On superhorizon scales, the nonconstant mode satisfies

$\dot{\mathcal R}\propto \frac{1}{a^3\epsilon}.$

Consequently, the mode is anti-damped whenever

$\epsilon_2<-3.$

This criterion is stronger and more precise than simply identifying a phase as “non-slow-roll”: it directly characterizes growth of the nonconstant superhorizon solution. Since $\epsilon_2<-3$ implies

$\epsilon(N)<\epsilon_{\rm in}e^{-3(N-N_{\rm in})},$

the canonical field velocity decreases at least as rapidly as $e^{-3N/2}$. Integrating the exact background relation yields the principal bound,

$$\frac{\Delta\phi_{\rm NA}}{M_{\rm P}} < \frac{2\sqrt{2\epsilon_{\rm in}}}{3} \left(1-e^{-3\Delta N/2}\right) < \frac{2\sqrt{2\epsilon_{\rm in}}}{3}.$$

The important dependence is on $\sqrt{2\epsilon}$0, the kinetic-energy parameter at entry into the non-attractor phase. For phenomenologically relevant $\sqrt{2\epsilon}$1, the allowed excursion is substantially below the absolute maximum. The result establishes a direct background constraint on any canonical mechanism that amplifies superhorizon curvature perturbations: greater duration and stronger anti-damping do not permit an arbitrarily large field displacement because the inflaton velocity is simultaneously driven toward zero.

Using only the inflationary condition $\sqrt{2\epsilon}$2, the paper obtains the weaker universal bound

$\sqrt{2\epsilon}$3

Thus, under the stated assumptions, a continuously anti-damped canonical phase cannot itself support a Planckian field-space distance. This numerical statement is a consequence of accelerated expansion combined with the anti-damping criterion; it is not tied specifically to ultra-slow roll or to an inflection-point potential.

Constant-roll specialization and ultra-slow roll

The paper strengthens the inequality when $\sqrt{2\epsilon}$4 is bounded by a constant negative value. If

$\sqrt{2\epsilon}$5

then

$\sqrt{2\epsilon}$6

For constant $\sqrt{2\epsilon}$7, the bound is saturated because $\sqrt{2\epsilon}$8 decays exactly as $\sqrt{2\epsilon}$9. Ultra-slow roll corresponds to $$\ddot{\mathcal R}_k+H(3+\epsilon_2)\dot{\mathcal R}_k+ \frac{k^2}{a^2}\mathcal R_k=0,$$0, giving

$$\ddot{\mathcal R}_k+H(3+\epsilon_2)\dot{\mathcal R}_k+ \frac{k^2}{a^2}\mathcal R_k=0,$$1

This establishes that the small field excursion of ideal USR is not an accidental consequence of a flat potential. It follows kinematically from the rapid decay of the inflaton kinetic energy. The conclusion is consistent with the established treatment of USR as a non-attractor solution with a growing curvature mode (Namjoo et al., 2012, Martin et al., 2012, Dimopoulos, 2017, Pattison et al., 2018).

The distinction from the standard Lyth argument is structural. In an attractor phase, approximately constant nonzero $$\ddot{\mathcal R}_k+H(3+\epsilon_2)\dot{\mathcal R}_k+ \frac{k^2}{a^2}\mathcal R_k=0,$$2 causes the field distance to accumulate over the duration of inflation. In the non-attractor case considered here, the condition that amplifies $$\ddot{\mathcal R}_k+H(3+\epsilon_2)\dot{\mathcal R}_k+ \frac{k^2}{a^2}\mathcal R_k=0,$$3 also suppresses $$\ddot{\mathcal R}_k+H(3+\epsilon_2)\dot{\mathcal R}_k+ \frac{k^2}{a^2}\mathcal R_k=0,$$4 exponentially. Duration therefore produces progressively less additional field motion rather than an indefinitely increasing excursion.

Exact relation to curvature-mode amplification

For constant $$\ddot{\mathcal R}_k+H(3+\epsilon_2)\dot{\mathcal R}_k+ \frac{k^2}{a^2}\mathcal R_k=0,$$5, the velocity of the nonconstant curvature mode grows according to

$$\ddot{\mathcal R}_k+H(3+\epsilon_2)\dot{\mathcal R}_k+ \frac{k^2}{a^2}\mathcal R_k=0,$$6

Eliminating $$\ddot{\mathcal R}_k+H(3+\epsilon_2)\dot{\mathcal R}_k+ \frac{k^2}{a^2}\mathcal R_k=0,$$7 gives

$$\ddot{\mathcal R}_k+H(3+\epsilon_2)\dot{\mathcal R}_k+ \frac{k^2}{a^2}\mathcal R_k=0,$$8

This is the paper’s most direct inverse-Lyth relation. It is exact within the constant-$$\ddot{\mathcal R}_k+H(3+\epsilon_2)\dot{\mathcal R}_k+ \frac{k^2}{a^2}\mathcal R_k=0,$$9 background and relates a precisely defined mode-velocity amplification to the field excursion. In the limit of arbitrarily large $\epsilon_2=d\ln\epsilon/dN$0, the excursion approaches the finite value

$\epsilon_2=d\ln\epsilon/dN$1

For USR,

$\epsilon_2=d\ln\epsilon/dN$2

The implication is not merely that USR is a small-field phase. Rather, within this idealized background, increasing the amplification of the nonconstant mode has a bounded field-space cost. Once the kinetic energy has become sufficiently small, extending the phase chiefly increases perturbation growth while contributing little further classical field motion.

Relation to scalar-power amplification

The paper correctly distinguishes the exact velocity-amplification result from a relation involving the final scalar power spectrum. The total curvature perturbation contains both constant and nonconstant solutions, and their relative coefficients at the beginning of the non-attractor interval determine the late-time power. Therefore, no universal exact relation between $\epsilon_2=d\ln\epsilon/dN$3 and the final power can be given without specifying the matching data.

For constant $\epsilon_2=d\ln\epsilon/dN$4, the background is obtained exactly as

$\epsilon_2=d\ln\epsilon/dN$5

with a corresponding finite-$\epsilon_2=d\ln\epsilon/dN$6 evolution of $\epsilon_2=d\ln\epsilon/dN$7. The nonconstant curvature solution is expressed through an exact quadrature. This makes the matching dependence explicit and prevents the exact mode-velocity relation from being incorrectly identified with an exact power-spectrum relation.

In the quasi-de Sitter limit, $\epsilon_2=d\ln\epsilon/dN$8, and when the growing nonconstant mode dominates the final perturbation, the power amplification is approximately

$\epsilon_2=d\ln\epsilon/dN$9

The inverse-Lyth relation becomes

$\dot{\mathcal R}\propto \frac{1}{a^3\epsilon}.$0

For USR this reduces to

$\dot{\mathcal R}\propto \frac{1}{a^3\epsilon}.$1

The approximation has two explicit requirements: quasi-de Sitter evolution and dominance of the amplified nonconstant mode. These qualifications matter in applications to primordial-black-hole scenarios, where large small-scale enhancement is often required. Under the stated conditions, many orders of magnitude of power enhancement can be obtained while the classical field excursion remains close to its finite asymptotic value. Related applications of non-attractor phases to primordial-black-hole production are discussed in the literature (Motohashi et al., 2017, Green et al., 2020).

Assumptions, limitations, and open questions

The headline inequality is robust within its domain, but that domain is restrictive. It applies to a continuous interval satisfying $\dot{\mathcal R}\propto \frac{1}{a^3\epsilon}.$2. A trajectory that alternates between attractor and non-attractor phases is not constrained by this result in terms of its total field distance. Likewise, noncanonical kinetic terms, multifield dynamics, modified gravitational dynamics, or a nontrivial field-space metric can invalidate the canonical identity used in the derivation or alter the perturbation equation.

The power-spectrum formulas require additional assumptions not needed for the principal bound. In particular, the relation between $\dot{\mathcal R}\propto \frac{1}{a^3\epsilon}.$3 and $\dot{\mathcal R}\propto \frac{1}{a^3\epsilon}.$4 is not exact at finite $\dot{\mathcal R}\propto \frac{1}{a^3\epsilon}.$5 and depends on the initial constant/nonconstant mode mixture. The exact relation established by the paper concerns $\dot{\mathcal R}\propto \frac{1}{a^3\epsilon}.$6, whereas the power relation is a controlled quasi-de Sitter approximation under growing-mode dominance.

The formal limit $\dot{\mathcal R}\propto \frac{1}{a^3\epsilon}.$7 also cannot be interpreted as indefinitely reliable deterministic evolution. As $\dot{\mathcal R}\propto \frac{1}{a^3\epsilon}.$8 becomes sufficiently small, quantum diffusion can compete with classical drift (Pattison et al., 2019, Pattison et al., 2021). This does not remove the classical upper bound, but it limits the duration over which the classical background description can be extrapolated. A question left open is how the inverse-Lyth relation should be reformulated when stochastic diffusion, mode matching, and nonlinear backreaction are treated simultaneously rather than appended as separate consistency conditions.

The manuscript also contains an explicit authorship and provenance statement reporting that its central result and initial draft were generated by ChatGPT 5.6 Sol and subsequently checked and edited by William H. Kinney. This disclosure is part of the paper’s content and distinguishes the reported derivation from conventional sole-author attribution.

Conclusion

The paper derives a model-independent upper bound on the canonical field-space distance accumulated during a continuously anti-damped non-attractor phase. The bound follows directly from the exact superhorizon relation $\dot{\mathcal R}\propto \frac{1}{a^3\epsilon}.$9 and the canonical background identity $\epsilon_2<-3.$0. For sustained anti-damping, the excursion is bounded by $\epsilon_2<-3.$1, with the universal maximum $\epsilon_2<-3.$2; stronger bounds apply for constant-roll evolution, including $\epsilon_2<-3.$3 for USR. The exact mode-velocity relation and its quasi-de Sitter power-spectrum counterpart show that arbitrarily large non-attractor amplification approaches a finite field distance, subject to the canonical single-field assumptions and the stated limitations.