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The effective phase space and $e$-folding of the Starobinsky and extended Starobinsky model of inflation

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read With a conserved phase-space measure, most Starobinsky inflation trajectories produce far fewer than 60 e-folds, with an expectation value of only 3.5–4 in the observationally allowed window.

desk verdict First application of the Remmen-Carroll measure to Starobinsky inflation, but the central e-fold numbers rest on using slow-roll integrals at a kinetic-dominated Planck surface. read the letter →

arxiv 2501.04334 v2 pith:EPIJXXGZ submitted 2025-01-08 gr-qc hep-th

Theppawan Rukpakawong , Piyabut Burikham This is my paper · ORCID
classification gr-qchep-th PACS 98.80.Cq04.50.Kd
keywords Starobinskyinflationeffectivephasespaceconservedmeasuree-foldingnumberPlancksurfaceextendedR3slow-rollattractorhorizonproblem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks how likely the Starobinsky model of inflation is to produce the roughly 60 e-folds needed to solve the horizon problem, when initial conditions are weighted by a conserved measure on the zero-curvature cosmological phase space. It finds that most classical slow-roll trajectories crossing the Planck surface ($H = M^*_{\mathrm{Pl}}$) give far fewer than 60 e-folds: the expectation value of the total e-folding number is only about 3.5–4 when the field cutoff lies in the observationally allowed window $5.22 < \phi_{\mathrm{UV}}/M^*_{\mathrm{Pl}} < 5.50$, and even with an arbitrarily high cutoff the probability of exceeding 60 e-folds saturates at about 2.5 percent. For the extended Starobinsky model with a small $R^3$ term, the same calculation gives $\langle N\rangle \simeq 4.0$–4.3 and an even smaller probability of long inflation. If correct, this means slow-roll inflation is dynamically natural in these models but long-lasting inflation is not, a tension worth weighing against the observational success of the Starobinsky potential.

What carries the argument

The machinery is the conserved measure on the effective phase space $(x,y)$, where $x$ is proportional to $\sqrt{V(\phi)}$ and $y$ is proportional to $\dot\phi$. Conservation under the cosmological flow, $\partial_\mu(\omega v^\mu)=0$, yields a family of measures labeled by a real parameter $m$; the requirement that the probability density be finite and infinitely differentiable except on the apparent attractor selects $m=5$. On the Planck surface this becomes the probability distribution $P(\theta)\propto (1-\sin\theta)^2\sin^2\theta/|2-3\sin\theta|^{7/3}$, with $\theta$ the angle at which a trajectory intersects that surface. The e-folding counts $N_1,N_2$ from the leading and next-to-leading slow-roll formulas convert $\theta$ into a total e-folding number, and integrating $N(u)P(u)$ over the allowed $u=\cos\theta/\cos\theta_0$ range gives $\langle N\rangle$.

What would settle it

Sample the Planck surface with the paper's own measure by numerically integrating the full equations of motion for a dense grid of initial angles and cutoffs, and compare the empirical distribution of $N$ with the slow-roll counts $N_1,N_2$; if a significant fraction of trajectories with $\phi_{\mathrm{UV}}\le 5.5M^*_{\mathrm{Pl}}$ yield $N>60$, the claimed threshold and the $2.5\%$ saturation would fail. Alternatively, adopt the finite $m\neq 5$ solutions of the same conservation equation and recompute $P(N>60)$: a material change would show the central numbers are a consequence of the measure selection rather than of the Starobinsky dynamics.

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Extended reading notes

Core claim

The central claim is that, under the Remmen-Carroll conserved measure normalized on the Planck surface, the total e-folds $N$ achieved by a Starobinsky trajectory is typically small. For the observationally allowed cutoff $\phi_{\mathrm{UV}}\in[5.22,5.50]M^*_{\mathrm{Pl}}$, the expectation value is $\langle N\rangle\simeq 3.5$–$4$ (the leading-order count $N_1$ and the next-to-leading-order count $N_2$ give nearly identical answers), and $P(N>60)\simeq 0.5\%$ at the upper edge of the allowed window, saturating at $2.5\%$ as $\phi_{\mathrm{UV}}\to\infty$. Reaching $N>60$ requires $\phi_{\mathrm{UV}}>5.5M^*_{\mathrm{Pl}}$. In the extended model with an $R^3$ coupling $\alpha=10^{-4}$ or $6.5\times10^{-5}$, starting inflation at the top of the potential gives $\langle N\rangle=4.025$ or $4.336$, respectively. The paper also claims that the saturated energy density remains sub-Planckian, $V(\phi_{\mathrm{UV}}=50M^*_{\mathrm{Pl}}) = 1.1\times10^{-10} M^{*4}_{\mathrm{Pl}}$, so the semiclassical treatment is not invalidated by super-Planckian field excursions.

Load-bearing premise

The numbers depend on taking the $m=5$ solution of the conserved-measure equation, selected by infinite differentiability, as the true probability distribution over zero-curvature FLRW initial conditions; choose a different measure and the expectation and tail probabilities move substantially.

Editorial extensions

If this is right

  • In the Starobinsky model, the horizon problem is not generically solved: a randomly selected zero-curvature trajectory crossing the Planck surface has $\langle N\rangle\approx 3.5$–$4$ and only about one half of one percent chance of reaching 60 e-folds in the observationally allowed cutoff window.
  • At arbitrarily high UV cutoff, the probability $P(N>60)$ saturates at about $2.5\%$, so even allowing super-Planckian field values cannot make long inflation the rule.
  • In the extended Starobinsky model with a small $R^3$ coupling in the allowed range, starting from the top of the potential gives $\langle N\rangle$ between $4.025$ and $4.336$, and $P(N>60)$ is at most about $0.8\%$.
  • The saturated potential energy at $\phi_{\mathrm{UV}}=50M^*_{\mathrm{Pl}}$ is $V\simeq 1.1\times10^{-10} M^{*4}_{\mathrm{Pl}}$, so the inflationary dynamics stays sub-Planckian and semiclassical even though the field value is super-Planckian.
  • Slow-roll behavior is an attractor in the phase space, so inflation begins naturally; the paper's point is that its duration is the rare part.

Reading between the lines

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

  • Editorial extension: the $m=5$ choice is a prior over initial conditions. If future work treats other finite, differentiable solutions of the same conservation equation as equally physical, both $\langle N\rangle$ and the $2.5\%$ cap would move; the paper's numbers should be read as one well-motivated measure choice rather than a no-prior statement.
  • The same measure construction could be applied to other plateau-shaped inflation potentials, such as $α$-attractors, to test whether a small expected e-folding number is generic; if it is, the horizon problem becomes a question of why our own trajectory was atypically long.
  • A natural testable extension is to repeat the counting with an observer-selection weighting, for example weighting trajectories by the number of galaxies or by the total entropy produced; such weighting could raise the probability of seeing 60 e-folds without changing the unweighted measure.
  • The near-equality of the $N_1$ and $N_2$ e-folding counts suggests that slow-roll corrections are not the source of the small expectation, so a challenge to the conclusion would have to target the measure or the Planck-surface normalization rather than the e-folding formula.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper constructs the Remmen-Carroll conserved measure on the effective phase space of flat FLRW universes for the Starobinsky model and for an R^3-extended Starobinsky model. After selecting the m=5 mode of the measure by the infinite-differentiability condition, the authors define a probability distribution over trajectories intersecting the Planck surface H=M_Pl* and compute the expectation value of the total e-folding number and the probability of obtaining more than 50 or 60 e-folds. The main quantitative results are <N> ~ 3.5-4 for phi_UV in [5.22, 5.50] M_Pl* in the Starobinsky model, with P(N>50) and P(N>60) saturating near 3% and 2.5% for arbitrarily large cutoff, and analogous values for the extended model with alpha in the observationally allowed range.

Significance. If correct, the paper would provide a concrete application of the Remmen-Carroll measure to realistic inflationary models, yielding falsifiable probability statements about the total e-folds and about the role of the UV cutoff. The analytic construction of the conserved measure, the normalization integrals, and the closed-form expressions for <N> are useful and nontrivial. However, the quantitative conclusions are currently not supported because (i) the printed e-folding formula in Eq. (44) is inconsistent with the tables, and (ii) the identification of the Planck-surface crossing point with the onset of slow roll is not justified for the physical dynamics, so the reported expectations and probabilities are not yet established. The paper is a reasonable candidate for a major revision rather than a rejection, provided the authors can correct or justify the e-fold counting.

major comments (3)
  1. [III.B, Eq. (44)] The printed formula for N1 is inconsistent with its definition as an integral from sigma to sigma_end. Using sigma_end=1+2/sqrt(3), the bracket in Eq. (44) evaluates to about 3.40, so with A0=-1.616 one obtains N1(sigma_end) ~ 0.94, not 0. Evaluating the printed formula at phi=5.221 gives N1 ~ 54.9, whereas Table III lists 49.02; the table value follows from the standard slow-roll integral N1 = 0.75(sigma - ln sigma) - 1.040. Thus Eq. (44) and the derived expression Eq. (70) do not reproduce the numerical results, and Eq. (45) and Eq. (71) should be checked in the same way.
  2. [III.B-C and IV.B-C] The total e-folds are computed by inserting the field value phi(theta) at the Planck surface into the slow-roll integrals N1 and N2, but at H=M_Pl* the slow-roll conditions (19) fail badly. From Eq. (34), V0 ~ 1.1e-10 M_Pl^4, and Eq. (7) then forces phi_dot^2 ~ 6 M_Pl^4 at every Planck-surface point with x <= sqrt(3)/2, so the kinetic energy exceeds the potential by about ten orders of magnitude. During the subsequent kinetic phase the field changes by O(sqrt(6) M_Pl) per e-fold before the slow-roll attractor is reached, as follows from dphi/dN ~ sqrt(6) M_Pl in kinetic domination. Therefore the map sigma(theta) -> N_slow-roll(phi) does not give the total e-folds from the Planck surface; it gives the e-folds that would be obtained if the field were already slow-rolling. Since the headline numbers <N>, P(N>50), and P(N>60) in Tables III-V and Figs. 5-6, 11-12 are built on this identification, the central quantitative claims are not yet supported.
  3. [III.C, Eqs. (65)-(66)] The upper and lower half-plane branches are combined into a single probability distribution P(u) using the identity cos(theta)=cos(2pi-theta), but the dynamics are not symmetric under theta -> 2pi-theta. The sign of y, hence the sign of phi_dot, differs between the two branches, so with the same |cos theta| one trajectory initially climbs the potential while the other descends; their kinetic transients and total e-folds are different. Assigning the same N(phi(theta)) to both branches in Eq. (66) is therefore not justified. Because the measure is defined over these two branches, the reported expectation values and tail probabilities are sensitive to this treatment.
minor comments (6)
  1. [Abstract and title] The phrase 'ande-folding' in the title line should be 'and e-folding'.
  2. [Conclusions] The text near the end of Section V contains the typo 'Starosbinky'; it should read 'Starobinsky'.
  3. [Section II.B, Eq. (34)] The coupling beta is dimensionful (M_Pl^-2 in the units used); state this explicitly after Eq. (34) to avoid confusion in later dimensionless combinations.
  4. [Section III.C, after Eq. (67)] The statement that the discrepancy between the approximate and full probability distributions is 'on the order of 10^-15' should specify whether this is absolute or relative, and for which quantity.
  5. [Section III.C and Fig. 5] The saturation value of P(N>60) is quoted as 0.0250 in the figure discussion and as 0.0251 in the concluding paragraph; use one consistent value, or explain the difference.
  6. [Fig. 5 caption] The sentence beginning 'At phi_UV ~ 7.925 M_Pl for N0=50 and phi_UV ~ 7.346 M_Pl for N0=60' is confusing because the threshold field should increase with N0; please rephrase or correct the values.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the conserved measure and e-fold counts are derived from dynamics and external observables; only a minor, non-load-bearing self-citation to Ref. [32] is present.

full rationale

The central derivation chain is self-contained. The Starobinsky potential (32) and the coupling beta are fixed by the COBE normalization (33)-(34), which is an external amplitude constraint; beta is not fitted to the target average <N>. The conserved measure is obtained from the Liouville flow (56)-(64), with m=5 selected by the infinite-differentiability condition taken from Remmen-Carroll [9], an external source, not from the desired e-fold statistics. The e-fold functions N1 and N2 (44)-(45) follow from slow-roll integrals with end points fixed by epsilon=1, and the probability distribution P(u) (66) is built from the measure flux through the Planck surface via |dot H| (63), then integrated against N(u). The spectral-index and tensor constraints (51)-(54) are computed from the same potential and compared with Planck/BK18 data, providing an external benchmark; no parameter is adjusted to reproduce <N> or P(N>60). The only author-overlapping citation is Ref. [32], used for the standard action and the COBE estimate ('For beta value, we impose COBE normalization ... [29,32]'), but the same COBE constraint is also attributed to independent references [29,46,47], so this self-citation is not load-bearing. A physical concern that trajectories crossing the Planck surface may not yet be slow-rolling (kinetic-dominated crossing) is a correctness or robustness issue about the slow-roll identification, not a circular reduction of the argument. Therefore no specific circular step is identified; the paper earns a low score reflecting only a minor non-load-bearing self-citation.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central calculation rests on the Remmen-Carroll measure and the choice of the m=5 mode. The model parameters beta and alpha are fixed by standard external constraints. No new particles or forces are introduced.

free parameters (3)
  • beta (R^2 coupling) = N_e^2/(3(0.027)4 M_Pl^2) with N_e=60, from COBE normalization
    Sets the Starobinsky mass scale. It cancels in the Starobinsky probability distribution but enters the extended-model subleading measure term.
  • alpha (R^3 coupling) = 6.5e-5 to 1.04e-4 (allowed by ns and r constraints)
    Perturbative parameter in the extended model; the paper scans its observationally allowed range.
  • m (measure mode) = 5
    Selected by the physicality condition of infinite differentiability. Not fitted to data but a modeling choice that determines the probability distribution.
assumptions (6)
  • domain assumption The Remmen-Carroll conserved measure on the k=0 effective phase space yields the probability distribution over FLRW trajectories.
    Adopted from references [8,9]; the entire probability interpretation rests on this.
  • ad hoc to paper The physical measure is the m=5 mode, selected by requiring infinite differentiability except at the apparent attractor.
    Other modes are discarded; this choice determines P(theta) and hence the average N. Stated in Section III C.
  • domain assumption The large-z (early universe) form of the flow is used to solve for the measure, then evaluated at the Planck surface z=sqrt(beta) M_Pl.
    Assumes the asymptotic large-z solution remains valid at the Planck surface.
  • domain assumption Total e-folds are computed with slow-roll integrals from the Planck surface to the point where epsilon equals unity.
    Assumes trajectories are on the slow-roll attractor when crossing the Planck surface.
  • domain assumption COBE normalization fixes beta with N_e=60.
    Standard normalization; the paper states 'we set N_e=60' after Eq. (34).
  • domain assumption For the extended model, only O(alpha) terms in the potential and measure are kept, requiring alpha(sigma-1) much less than 1.
    Justified by alpha <= 1e-4; higher-order terms are dropped.

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Cite this review

Pith. "Pith review of The effective phase space and $e$-folding of the Starobinsky and extended Starobinsky model of inflation." pith.science (2026). https://pith.science/paper/EPIJXXGZ

@misc{pith2026250104334,
  author       = {Pith},
  title        = {Pith review of: The effective phase space and $e$-folding of the Starobinsky and extended Starobinsky model of inflation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EPIJXXGZ}},
  note         = {Machine review of arXiv:2501.04334}
}
abstract

For zero spatial curvature, cosmological phase space of Starobinsky and extended Starobinsky inflationary model show three apparent attractors; the fixed angle attractor in the large field limit, the final attractor representing reheating phase in the small field region, and the apparent attractor corresponding to the slow-roll condition connecting between the large-field and small-field region. To consider the total $e$-folding likelihood of the model, Remmen-Carroll conserved measure is constructed and normalized. Using the measure, the total e-folding number $N$ and its expectation value $\left\langle N \right\rangle$ are calculated. Our results show that most classical slow-roll trajectories which intersect the Planck surface have $N<60$, and $\phi_{\rm UV}>5.5 M^{*}_{\rm Pl}$ is required for $N>60$. It is found that for $\phi_{\rm UV}\in [5.22,5.50]M^{*}_{\rm Pl}$ which satisfies the constraint on the spectral index, $n_s = 0.9658 \pm 0.0040\,\,\,(68\%\,\,{\rm CL})$, the expectation value $\langle N \rangle \simeq 3.5 - 4$ for trajectories intersecting the Planck surface in the Starobinsky model. For extended Starobinsky model with additional $R^3$ term parametrized by a coupling parameter $\alpha$, the expectation value when inflation starts from the top of the potential shifts to $\left\langle N \right\rangle = 4.025,4.336$ for $\alpha = 10^{-4},6.5\times 10^{-5}$ respectively. In the Starobinsky model even at very large inflaton cutoff $\phi_{\rm UV}$ where the field value is super-Planckian, the energy density from the (saturating) inflaton potential and the Hubble parameter are still sub-Planckian and therefore the inflation occurs within the semi-classical regime.

Figures

Figures reproduced from arXiv: 2501.04334 by the authors.

Figure 1
Figure 1. The flow vector fields in the effective phase space given by Eqn. ( [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. The e-folding number achieved by the initial infla￾ton ϕ. As shown in the zoom-in figure, N2, evaluated using Eqn.(27), results in a greater number of e-folds compared to N1, which is calculated using Eqn. (21). N2 includes contri￾butions from both ϵV and ηV , whereas N1 considers only ϵV . definition of cos θ0 in Eqn. (39) to obtain σ(θ) = 1 1 − cos θ cos θ0 = 1 1 − u , (48) where u ≡ cos θ/ cos θ0. The slow-roll r… view at source ↗
Figure 3
Figure 3. Hubble parameter versus inflaton value under slow [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (9 more)
Figure 5
Figure 5. Figure 5: The probability to obtain at least 50 and 60 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 4
Figure 4. Figure 4: Expectation value of e-folds ⟨N⟩ from the leading￾order expression (44) and the next-to-leading-order expres￾sion (45), using the conserved measure (66) for the Starobin￾sky model versus ϕUV cutoff. For a specific inflaton cutoff of ϕUV = 5.5M∗ Pl, the probability dist…
Figure 6
Figure 6. Figure 6: The probability distribution on the space of trajec [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: Potentials of the extended Starobinsky model for [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: The stream plot of vector fields in the effective phase space illustrated by Eqn. ( [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: Constraints from the spectral index [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: Lastly, we evaluate how likely all possible initial condi￾tions on the Planck surface with e-folding number given by Eqn. (89) would have N > 50, 60, assuming inflation starts at ϕmax. Namely, we compute the fraction of the trajectories within the subregion (θmax, θN1…
Figure 11
Figure 11. Figure 11: The probability of obtaining at least 50 and 60 [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]
Figure 12
Figure 12. Figure 12: The probability distribution [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]

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Reviewed August 10, 2026 · model on record in the stance chip above.