Quadratic, Higgs and hilltop potentials in the Palatini gravity

Nilay Bostan

REVIEW 3 major objections 4 minor 1 cited by

Quadratic, Higgs and hilltop potentials in the Palatini gravity

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper maps which values of the non-minimal coupling and vacuum expectation value keep quadratic, Higgs, and hilltop inflation inside the Planck/BICEP2/Keck contours in Palatini gravity.

desk verdict A useful but currently conditional Palatini inflation parameter scan: the Higgs-domain claims are not closed until the F>0 condition is imposed and Eq. (4.3) is corrected. read the letter →

arxiv 1908.09674 v1 pith:O7LHBAGX submitted 2019-08-26 astro-ph.CO hep-ph

Nilay Bostan This is my paper · ORCID

classification astro-ph.COhep-ph PACS 98.80.Cq04.50.Kd

keywords Palatinigravitynon-minimalcouplingcosmicinflationquadraticpotentialHiggshilltoppotentialsspectralindextensor-to-scalarratio

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

This paper examines three inflation models---quadratic, Standard Model Higgs, and hilltop potentials---when the inflaton is non-minimally coupled to gravity through a $\xi\varphi^{2}R$ term and gravity is treated in the Palatini formalism. The author computes the spectral index $n_s$, tensor-to-scalar ratio $r$, running $\alpha$, and the required potential scale for each model under an instant-reheating (high-$N$) and a low-reheating-temperature (low-$N$) scenario, then compares them with the Planck/BICEP2/Keck contours. The central result is a narrow viability map: Palatini quadratic inflation fits the data only for $10^{-4}\lesssim\xi\lesssim10^{-3}$ with high $N$; Palatini Higgs inflation survives with $r$ heavily suppressed when $\xi\gg1$; and generalized hilltop potentials agree only for $\xi\lesssim0.005$, $\phi

What carries the argument

The central object is the non-minimal coupling function $F(\phi)=1+\xi(\phi^{2}-v^{2})$ (with $F=1+\xi\phi^{2}$ in the quadratic case), which multiplies the Ricci scalar in the Jordan-frame action. The machinery is the Weyl rescaling $g_{E,\mu\nu}=g_{\mu\nu}/F$ together with the Palatini result that the canonical kinetic term has prefactor $Z=1/F$, followed by the field redefinition $d\chi=d\phi/\sqrt{Z}$ that makes the Einstein-frame action canonical. The paper evaluates the slow-roll parameters in $\phi$-space using the $Z$-dependent formulas it quotes, and sets the number of e-folds $N$ by three reheating scenarios (high, middle, low). This converts each Jordan-frame potential into the $n_s$--$r$ predictions that are compared with the observed contours.

What would settle it

For each displayed positive-$\xi$ point in the $\phi<v$ Higgs and hilltop scans, evaluate $F(0)=1-\xi v^{2}$ and check that $F$ stays positive along the field trajectory from the start of inflation to the end; if any viable plotted point has $\xi v^{2}\ge1$, that point is not a valid Palatini model. Scanning the figures for such points would settle whether the claimed viable regions are fully physical.

Watch

Extended reading notes

Core claim

The paper claims that in Palatini gravity the observational viability of non-minimally coupled inflation depends sharply on the non-minimal coupling $\xi$ and the vacuum expectation value $v$, and it charts the allowed regions for the three potential families. For the quadratic potential with $F=1+\xi\phi^{2}$, only $10^{-4}\lesssim\xi\lesssim10^{-3}$ with a high number of e-folds brings $n_s$ inside the observed region, while for the low-$N$ scenario no value of $\xi$ works. For the Higgs potential $V=A[1-(\phi/v)^{2}]^{2}$, the paper finds that $\xi\gg1$ suppresses $r$ dramatically regardless of $v$ when $\phi>v$, that $\xi=10^{-3}$ is ruled out when $\phi<v$, and that negative $\xi$ can be compatible for large $v$; in the induced-gravity limit $\xi v^{2}=1$ all tested $\xi$ land inside the 68% contour. For hilltop potentials $V=A[1-(\phi/v)^{\mu}]^{2}$ with $\mu>2$, compatibility requires $\phi<v$, $\xi,v\ll1$, and $\xi\lesssim0.005$, and $\mu=4$ is excluded. The running $\alpha$ is too small to be observed in all models considered.

Load-bearing premise

The argument requires $F(\phi)=1+\xi(\phi^{2}-v^{2})$ to stay positive over the whole inflationary trajectory so the Weyl rescaling and $Z=1/F$ remain real, yet when $\xi v^{2}>1$ the function is negative at $\phi=0$, so some $\phi<v$ parameter points plotted may not define a valid Palatini theory.

Editorial extensions

If this is right

If the central claim is right, Palatini quadratic inflation is viable only inside a narrow strip $10^{-4}\lesssim\xi\lesssim10^{-3}$ for the instant-reheating scenario, so tighter future measurements of $n_s$ will either confirm or close this window.
For Palatini Higgs inflation with large $\xi$, $r$ is predicted far below current limits, so a future B-mode detection of $r$ around $10^{-3}$ would rule out the large-$\xi$ branch.
For the hilltop family, $\mu=4$ is excluded for every $\xi$ considered, while $\mu=6,8,10$ survive only for $\xi\lesssim0.005$ with $\phi<v$ and $v\ll1$; increasing $\xi$ pushes $n_s$ outside the observed region.
The predicted running $\alpha$ is of order $10^{-4}$--$10^{-3}$ in all these models, below what near-future 21-cm observations are expected to reach, so $\alpha$ will not discriminate among them.

Reading between the lines

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

A natural check the paper does not state: for $\xi>0$ the domain condition $F(\phi)>0$ over $\phi\in[0,v]$ requires $\xi v^{2}<1$, and some positive-$\xi$ points in the $\phi<v$ Higgs and hilltop scans may violate this, so the true viable regions could be smaller than plotted.
The same $(\xi,v)$ scan could be run in the metric formulation for these symmetry-breaking potentials; the comparison would show how strongly the choice of Palatini vs. metric gravity changes the observational predictions beyond the already-known difference in $r$.
Because large-$\xi$ Palatini Higgs inflation gives $r$ at or below $10^{-14}$, the cleanest experimental discriminator between Palatini and metric non-minimal inflation is a null or positive detection of primordial gravitational waves: a detection at $r\simeq10^{-3}$ would disfavour the Palatini large-$\xi$ branch.
The low-$N$ exclusion of quadratic inflation assumes a reheating temperature of 100 GeV with $w=0$; if reheating were more efficient, the low-$N$ band would shift, so the robust statement is the high-$N$ window rather than the low-$N$ exclusion.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a systematic, honest scan that genuinely extends earlier Palatini work, but the advertised Higgs parameter regions are not established as written. The reader's conditional verdict is about right, and the stress-test concern holds up.

What is new: earlier Palatini papers mostly used F = 1 + xi phi^2 with fixed N = 50 or 60. This paper uses F = 1 + xi(phi^2 - v^2), includes a nonzero VEV, scans negative xi, separates high-N and low-N reheating histories, and adds the induced gravity limit. That is a real extension. The quadratic section adds m and alpha as functions of xi on top of Tenkanen's results, and the hilltop analysis, with xi, v << 1 and the approximate xi <~ 0.005 compatibility region, is plausible and unaffected by the main problem. The citation pattern is honest; the relevant earlier work is cited, including the author's own [33], which is a legitimate source for the F form.

Soft spots, in proportion. First and most important: the Palatini Weyl rescaling requires F > 0 along the whole inflationary trajectory. For phi < v and xi > 0, F(0) = 1 - xi v^2, so F goes negative whenever v > 1/sqrt(xi). Figures 10-13 plot exactly those regions, with xi = 1e-3 and 1e-4 and v up to 1000. The negative-xi phi > v scans can also cross F <= 0. The paper never imposes this domain condition, and no tables or code are provided, so the claim that xi = 1e-4 or negative xi can be inside the 95% contour for phi < v may rest on unphysical trajectories. The hilltop conclusion is safe because it uses xi, v << 1, where F ~ 1. But the Higgs viability map is not reliable until this is fixed.

Second, Eq. (4.3) is presented as the analytic N* for nonminimally coupled Palatini Higgs inflation, but it contains no xi and is just the minimal-coupling result. It does not follow from Eq. (2.17) with F = 1 + xi(phi^2 - v^2). That is a concrete error, though it is contained in one formula and is easy to correct. Third, the absence of numerical tables or code makes the scan hard to audit; a few tables of representative (xi, v, N*, ns, r) values would greatly increase the paper's value.

Who this is for: people comparing inflation models in Palatini gravity, especially those interested in reheating-dependent N* and nonzero-VEV potentials. I would not desk-reject it. It deserves a serious referee, and the identified problems are fixable. If the author imposes F > 0 and reruns the affected scans, and corrects Eq. (4.3), I would be comfortable citing the paper. Until then, I would not build on the Higgs-specific parameter regions.

Referee Report

3 major / 4 minor

Summary. The paper studies single-field inflation in Palatini gravity with a non-minimal coupling of the form F(phi) = 1 + xi(phi^2 - v^2), applied to quadratic, Standard-Model-like Higgs, and hilltop potentials. It computes slow-roll predictions for ns, r, alpha, and the potential amplitude A or mass m, for high-, middle-, and low-N reheating scenarios, and compares the ns-r predictions with 68% and 95% CL contours from the Keck Array/BICEP2 and Planck collaborations. The central claims are a set of viability regions in the (xi, v) plane: small xi for the quadratic potential in the high-N case, specific xi and v ranges for the Palatini Higgs potential for phi>v and phi<v, and xi, v << 1 for hilltop potentials.

Significance. If the claimed parameter maps are correct, the paper provides a useful phenomenological survey of an interesting set of Palatini inflation models, complementing earlier work on Palatini quadratic and Higgs inflation. The paper carefully distinguishes high-N and low-N scenarios and includes both positive and negative xi, which is broader than much of the existing literature. However, two load-bearing issues prevent me from accepting the results as they stand: the analytic e-fold formula in Eq. (4.3) is not the non-minimally coupled result, and the paper never enforces the condition F(phi) > 0 that is required for the Einstein-frame action to be real. The quadratic and hilltop conclusions are likely unaffected, but the claimed Higgs viability regions are not established without imposing this domain restriction.

major comments (3)

[§4, Eq. (4.3)] Equation (4.3) is presented as the analytic N* for the non-minimally coupled Palatini Higgs potential, but it is independent of xi. From Eq. (2.17), N* = ∫ dphi F/(Z sqrt(2 epsilon_phi)) = ∫ dphi F V/V' for positive V'/V, and with Z = 1/F this is N* = ∫ dphi F V/V'. For the Higgs potential V = A[1 - (phi/v)^2]^2 one has V/V' = (phi^2 - v^2)/(4 phi), so with F = 1 + xi(phi^2 - v^2) the integrand contains the factor (1 + xi(phi^2 - v^2)). Equation (4.3) is the xi = 0 result, namely (phi*^2 - phi_e^2)/8 - (v^2/4) ln(phi*/phi_e), and it omits the xi-dependent terms. If the numerical scans use the full integral, this equation should be corrected or labelled as the minimal-coupling limit; otherwise the analytic foundations of the Higgs section are incorrect.
[§2, Eqs. (2.5)-(2.6), and Figs. 10-13] The Einstein-frame description requires F(phi) > 0 over the whole inflationary trajectory, because the Weyl rescaling g_E = g/F and Z = 1/F become imaginary or singular for F <= 0. The paper never imposes this condition. With F = 1 + xi(phi^2 - v^2) and F(v) = 1, for xi > 0 and phi < v one has F(0) = 1 - xi v^2, which is negative for xi v^2 > 1. The phi < v scans in Figs. 10-13 include xi = 10^-3 with v up to 10^3 and xi = 10^-4 with v up to 10^3, so for v > sqrt(1/xi) most of the interval phi in [0, v] has F < 0. For xi < 0 and phi > v, F crosses zero when |xi|(phi^2 - v^2) = 1, so parts of the phi > v scans in Figs. 6 and 8 may also be unphysical. Since no numerical tables or code are provided, I cannot verify whether the points claimed to lie inside the 95% CL contour correspond to valid F > 0 trajectories. The paper should impose F(phi) > 0 explicitly and rerun or exclude the unphysical parameter regions.
[§4, text after Eq. (4.5)] The statement that for phi << v the potential of Eq. (4.5) gives ns ≈ 1 - 8/v^2 is not an adequate prediction: it contains no dependence on N*, and for a fixed potential the spectral index at horizon crossing generally depends on the e-folds elapsed since that crossing. This is also inconsistent with the hilltop formulas in Section 5, Eq. (5.3), which give N*-dependent expressions. Please clarify the derivation and the regime of validity of this estimate.

minor comments (4)

[Throughout] The manuscript contains many typographical and grammatical errors, including 'inﬂaton', 'prehating', 'afterwards inﬂation', and missing spaces. These should be corrected in a careful proofreading pass.
[§5, Eq. (5.3)] Equation (5.3) contains a bracket typo: '[4µ - 2)N*]' should presumably be '[4(µ - 2)N*]' or similar. Please correct the expression so the reader can verify the formula.
[Figure captions, Figs. 15 and 17] The captions say 'The pink (red) line corresponds to the 95% (68%) CL contour', but these are two-dimensional contours, not lines; please reword.
[§2, Eq. (2.7) and notation] The field-redefinition equation d chi = d phi / sqrt(Z) with Z = 1/F is used repeatedly; it would help to state explicitly that this requires F > 0 and to note the sign of d chi/d phi when F is not positive on the whole trajectory.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the ns-r claims are independent model-curve comparisons to external CMB data; amplitude normalization is standard and not presented as a prediction.

full rationale

The paper's derivation chain is explicit: define the Jordan-frame action with F(phi)=1+xi(phi^2-v^2), Weyl-rescale to the Einstein frame, compute slow-roll parameters and N* from assumed reheating scenarios, normalize the amplitude to the observed curvature perturbation, and compare the resulting ns-r curves to the BICEP2/Keck/Planck contours. No step defines a target prediction in terms of the data or in terms of a fitted parameter by construction. The masses and amplitudes A and m are fixed using the observed Delta_R^2, which is standard normalization and not a claimed prediction; the paper never presents the curvature perturbation amplitude as an output. The non-minimal coupling form F(phi)=1+xi(phi^2-v^2) is attributed to the author's earlier work [33], but it is an explicit model ansatz stated in the paper rather than a result invoked to forbid alternatives or to force the ns-r values. The analytic approximations, such as eqs. (4.4) and (5.4), are checked against numerical integration of the Jordan-frame potential, which is internal consistency checking and not circular reasoning. The claimed viable parameter regions are comparisons of model curves to external CMB contours, and no equation in the paper is identical by construction to the central output. Therefore no circularity is identified.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central scan depends on free model parameters: xi, v, the amplitudes A/m fixed by the curvature perturbation, and the hilltop index mu. It also relies on background assumptions: slow roll, positive F for the Einstein frame, and the chosen reheating scenarios. The model choice F = 1 + xi(phi^2 - v^2) is adopted from [33] rather than derived. No new particles or forces are introduced.

free parameters (5)

xi = scanned from -10^-4 to 10^2 in the figures
Nonminimal coupling strength; allowed ranges are selected by matching Keck/BICEP2+Planck contours rather than derived from a theory.
v = scanned over log10 v from -2 to 4 in the figures
Inflaton VEV after symmetry breaking; free scale that controls the nonminimal function F and the potential shape.
A = fixed by the observed curvature amplitude, values depend on xi and v
Amplitude of the symmetry-breaking potentials; set via Eq. (2.14)/(2.18) to approximately the observed Delta_R.
m = about 6 x 10^-6 in the viable high-N quadratic region
Mass of the quadratic potential; fixed to the observed curvature amplitude and shown as a function of xi.
mu = benchmark values 4, 6, 8, 10
Hilltop exponent chosen by hand; the compatibility conclusion depends on mu, and mu = 4 is excluded in the scanned range.

assumptions (4)

domain assumption The slow-roll approximation for ns, r, alpha, and N* is valid on the scales considered.
All predictions use Eqs. (2.11)-(2.13) and (2.17); slow-roll is not verified post hoc.
domain assumption The Weyl rescaling to the Einstein frame with Z = 1/F is valid, requiring F > 0 on the inflationary trajectory.
Eqs. (2.5)-(2.7) assume positive F; the paper does not impose or check this in the xi > 0, phi < v scans.
ad hoc to paper The nonminimal coupling function is F(phi) = 1 + xi(phi^2 - v^2) with m^2 = 1 - xi v^2 so that F(v) = 1.
Chosen in Section 2 after Eq. (2.4) as the renormalizable form with a nonzero VEV; it is adopted from [33] rather than derived.
domain assumption The standard thermal history formula in Eq. (2.19) controls N*, with the high, middle, and low reheating scenarios bracketing the reheating uncertainty.
Several conclusions, such as the low-N exclusion of quadratic inflation, depend on these reheating assumptions.

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Pith. "Pith review of Quadratic, Higgs and hilltop potentials in the Palatini gravity." pith.science (2026). https://pith.science/paper/O7LHBAGX

@misc{pith2026190809674,
  author       = {Pith},
  title        = {Pith review of: Quadratic, Higgs and hilltop potentials in the Palatini gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O7LHBAGX}},
  note         = {Machine review of arXiv:1908.09674}
}

abstract

In this work, we study inflation with the non-minimally coupled quadratic, Standard Model (SM) Higgs and hilltop potentials through $\xi \phi^2R$ term in the Palatini gravity. We first analyze observational parameters of Palatini quadratic potential as functions of $\xi$ for high-$N$ scenario and low-$N$ scenario. In addition to this, taking into account inflaton $\phi$ has a non-zero vacuum expectation value $v$ afterwards inflation, we display observational parameters of well-known symmetry-breaking potentials type of Higgs potential and its generalizations which are hilltop potentials in the Palatini formalism for high-$N$ scenario and low-$N$ scenario. We calculate inflationary parameters of Palatini Higgs potential as functions of $v$ for different $\xi$ values where inflaton values both $\phi>v$ and $\phi<v$ during inflation as well as we show that observational parameters of Palatini Higgs potential in the induced gravity limit for high-$N$ scenario. On the other hand, we illustrate different from the Higgs potential the effect of $\xi$ on hilltop potentials which can agree with the observations for inflaton value solely $\phi<v$ and $\xi$, $v\ll1$ for both two scenarios, which we mentioned above. For each considered potentials, we also display $n_s-r$ values fit the current data given by the Keck Array/BICEP2 and Planck collaborations.

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