Running cosmological constant in light of recent CMB-BAO data

A. Racioppi; M. Vijar

REVIEW

Running cosmological constant in light of recent CMB-BAO data

A. Racioppi, M. Vijar

Cosmology and Nongalactic Astrophysics · Submitted 28 Oct 2026

Pith's verdict: A clean slow-roll analysis of a known potential family whose real news is a sign choice — positive δ tracks the BAO-inclusive data, regularized negative δ prefers CMB-only — but the 1σ/2σ claims are read by eye from someone else's contours, not computed.

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arXiv ID2610.38973 (version 2)

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CitationA. Racioppi, M. Vijar. "Running cosmological constant in light of recent CMB-BAO data." arXiv:2610.38973 (2026). https://pith.science/paper/PAMQGPFY

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Editor's note

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What you should know: this is a tidy slow-roll analysis of a familiar potential — a log-corrected cosmological constant, V = Λ⁴(1 + δ ln(φ/M)) — with δ promoted from a SUSY-fixed small positive number to a free parameter of either sign. The point of interest is the sign complementarity. δ > 0 interpolates between a logarithmic regime (n_s → 1 − 1/N_e, r → 0) and linear inflation, and sits comfortably inside the SPA+BK+DESI region; δ < 0, once the mass term inside the log is restored, becomes a hilltop that prefers CMB-only data. One family, one sign choice, both sides of the current BAO-CMB tension. That is a map rather than a new mechanism, and it is the paper's real contribution.

The math holds up. I checked the slow-roll ingredients, the exact φ₀ reparametrization, and the central move: the "regularized" hilltop potential (3.9) is not an ad hoc guess but a genuine rewrite of the one-loop CW potential with an explicit mass term, via (A.4)–(A.9). The asymptotic limits are right, and the exclusions (unregularized δ < 0, r ≳ 0.07) are robust because 0.07 sits far above the BK18 bound. The authors state their own limits: effective treatment only, UV completion deferred, one-log truncation acknowledged.

The soft spot is the one the stress-test names. Every significance claim in the abstract and conclusions — positive δ "fitting easily," the hilltop at "1σ for β = 0.9, N_e = 60," CMB-BAO at "only 2σ" — is a visual reading of the contours of ref. [11], a preprint, and no chi-squared or likelihood value appears anywhere. The δ < 0 one-sigma fit sits at β = 0.9, right at the β < 1 boundary, at N_e = 60, the top of the standard range; the δ > 0 CMB-only fit is a stripe near the 2σ edge. Contour-reading is common in this literature, and the qualitative complementarity would probably survive a proper likelihood calculation, but the significance language overstates what was computed: a shift of a few hundredths in n_s, or in the r normalization, would move each claim across a sigma boundary. Fixable, not fatal.

Minor: the one-log truncation is acknowledged but unquantified. A scan over plausible higher-order corrections would say how much the tracks move.

Who this is for: inflation model-builders watching the ACT/DESI tension and people working with CW potentials. It deserves a serious referee, with the condition that the comparison against [11] be redone with a real likelihood and the β ≈ 0.9 window traced carefully. I would send it out.

Referee report

Significance

If the claims hold, the paper's value is that a single single-field potential with one sign choice covers both sides of the current BAO-CMB tension, and it does so transparently: the slow-roll expressions (2.3)-(2.10), the exact φ0 reparametrization (3.1)-(3.4), the reduction of the one-loop Coleman-Weinberg form (A.4) to the regularized potential (3.9), and the hilltop observables (3.11)-(3.14) are all analytic and, as far as I have checked, mutually consistent. The resulting mass scales Λ̄∼10^15-10^16 GeV are sensible, and the r∼10^-2 predictions are falsifiable by Simons Observatory and LiteBIRD. The main weakness is that every quantitative compatibility statement (the 'wide range' for δ>0, the 1σ point at β=0.9, the 'only 2σ' status with BAO) is read visually from the contours of the arXiv preprint [11], no goodness-of-fit statistic is reported, and the sharpest claim sits at a boundary point of the assumed effective potential (β close to the β<1 bound, N_e at the top of the standard range). The qualitative two-branch structure is likely robust, but the claimed confidence levels are not yet supported by the analysis as presented.

Major objections (3)

[§3.1, §3.3, abstract, conclusions] All σ-level compatibility statements — 'fits easily within the SPA+BK+DESI constraints in a wide range of parameters' (abstract, [b75]), 'compatible with SPA+BK at 1σ for β=0.9 and N_e=60' (just after eq. (3.14), [b110]), and 'agreement with the SPA+BK+DESI data is achievable only at 2σ' — are read by eye from the contour plots of ref. [11], an arXiv preprint, and no χ² or likelihood value is reported anywhere in the paper. The fragility of this procedure is visible in where the claims sit: the δ<0 1σ point uses β=0.9, close to the hard bound β<1 required by V̄(0)>0 (eqs. (3.10)/(A.6)), and N_e=60, the upper end of the range fixed at [b36]; the δ>0 CMB-only compatibility is, by the authors' own words, a 'narrow stripe near the 2σ border' at N_e=50, achievable only for δ≳0.1. A shift of a few hundredths in n_s in either the model tracks or the likelihood contours would move each claim across a σ boundary. I request a quantitative evaluation against the likelihood underlying refs. [10]/[11] — e.g., a Δχ² or posterior probability for representative parameter points, with N_e marginalized over [50,60] — and that the abstract and conclusions be restated with those numbers.
[§3.3 and Appendix A] The headline negative-δ result rests on the one-log effective potential (A.4)/(3.9) with the identifications (A.8), and the claimed 1σ point is a shallow hilltop: for β=0.9, V̄(0)=-(1/2)Λ̄^4 ln β≈0.05Λ̄^4, so the tree-level and radiative contributions nearly cancel, and the fit sits at the top of the N_e range. The authors acknowledge the truncation of the effective potential — the neglect of further φ-dependent terms from the field-dependent masses in (A.2) is stated at [b145], and the validity range of (2.2) at [b92] — but they do not quantify how much the (n_s, r) tracks, and hence the claimed 1σ/2σ status, move under plausible corrections (for example, the next-order term in the loop expansion, or a regularization of (A.2) that differs from (3.9) away from the β→0 limit). Footnote 3 justifies (3.9) only as 'the most economical' choice. Since the hilltop compatibility for δ<0 is the paper's main new phenomenological result, a quantitative robustness estimate is needed before the 1σ claim can be regarded as established.
[§3.1, eqs. (3.5)-(3.7), Fig. 2] The claim that the δ>0 model 'fits easily within the SPA+BK+DESI constraints in a wide range of parameters' is not delimited, and the model family includes a regime that is certainly excluded: for φ0≫M_P the potential approaches the linear form (3.5) and the asymptotics of eqs. (3.3)-(3.4) give r→4/N_e≈0.067-0.08 for N_e=50-60, above the BK18 upper limit r<0.036 (95% CL) that enters the SPA+BK combination. The paper should specify the allowed region in (δ, M, N_e) — in particular, confirm that the wide allowed region is the logarithmic/intermediate regime and that the large-M, large-δ (linear-inflation) end of the tracks in Fig. 2 is excluded — rather than leaving the impression that the whole one-parameter family is compatible.

Minor objections (5)

[Abstract] The abstract's statement that for negative δ the regularized potential 'achiev[es] compatibility with CMB-only data' omits the conditions under which this holds (β=0.9, N_e=60); the body shows the 1σ result holds at a single point near the β<1 boundary, so the abstract should carry this qualifier.
[§3.3, paragraph after Fig. 5] The sentence 'at β=0.9 the same is achieved with a bit more than 50 e-folds' is terse; please state explicitly that 'the same' refers to 2σ compatibility with SPA+BK, and that the 1σ result requires N_e=60.
[Figs. 2 and 5] Because δ (and φ0, β) ranges over several decades and the tracks in the (n_s, r) plane are one-parameter curves, marking a few representative parameter values on each track would let the reader locate the claimed 'wide range' and the 1σ/2σ boundary points.
[Ref. [11]] All σ-level claims are read from the contours of ref. [11], cited as an arXiv preprint; please check for a published version, cite it if available, and note in the text that the numerical contours may shift between versions.
[Footnote 3 and Appendix A] The phrase 'most economical way' in footnote 3 could mislead: Appendix A shows that (3.9) is exactly equivalent to the one-loop form (A.4) under the identifications (A.8); consider saying so explicitly in the main text.

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The paper

The paper claims

The paper claims a cosmological constant with a logarithmic radiative correction of free sign can match either the CMB-only or the CMB-BAO constraints, predicting a tensor-to-scalar ratio near one percent.

The reading

This paper asks whether a single minimal modification—treating the cosmological constant as running, i.e. receiving a logarithmic radiative correction $V(\phi) = \Lambda^4(1 + \delta \ln(\phi/M))$—can account for the latest inflationary data when the sign and size of the correction $\delta$ are left free. It establishes that positive $\delta$ produces predictions interpolating between a logarithmic regime and linear inflation that fit the CMB-BAO constraints (SPA+BK+DESI) easily over a wide parameter range, while negative $\delta$ only works once the effective mass term inside the logarithm is restored, which turns the potential into a hilltop model fitting the CMB-only data at $1\sigma$ ($\beta = 0.9$, $N_e = 60$) and the CMB-BAO data only at $2\sigma$. If correct, a single-field family with a sign choice covers both sides of the current BAO-CMB spectral-index tension, with a tensor-to-scalar ratio $r \sim 10^{-2}$ that upcoming CMB experiments can test.

What carries the argument

The load-bearing object is the one-loop effective potential in its running-cosmological-constant form, eq. (2.2), which the paper reparametrizes through $\phi_0 \equiv M e^{-1/\delta}$ as $V(\phi) = \delta \Lambda^4 \ln(\phi/\phi_0)$; in the $\phi_0$ variable the slow-roll observables $N_e$, $r$, and $n_s$ depend on the single parameter $\phi_0$, which is why curves with different $M$ values overlap in the $(n_s, r)$ plane. The second mechanism is the regularization of eq. (3.9), $\bar{V}(\phi) = -\tfrac{1}{2}\bar{\Lambda}^4 \ln(\beta + \phi^2/\phi_0^2)$, which restores the subleading effective mass term inside the logarithm and converts the negative-$\delta$ potential into a hilltop with local maximum at $\phi = 0$; the condition $0 < \beta < 1$ keeps that maximum positive. Standard slow-roll machinery—$\epsilon$, $\eta$, the e-fold integral, and the amplitude normalization $A_s \simeq 2.1 \times 10^{-9}$—converts each potential shape into the $(n_s, r)$ tracks compared with the observational contours.

Extended reading notes

Core claim

The central claim is that the running-cosmological-constant potential of eq. (2.2), $V(\phi) = \Lambda^4(1 + \delta \ln(\phi/M))$, remains a viable single-field inflaton when $\delta$ is promoted to a free parameter of either sign, free of assumptions about a specific underlying construction. For $\delta > 0$, the observables interpolate between a logarithmic regime ($n_s \to 1 - 1/N_e$, $r \to 0$) and linear inflation, and the predictions fit the CMB-BAO combination (SPA+BK+DESI) in a wide parameter range, while compatibility with the CMB-only data set SPA+BK (Planck, ACT, SPT with BICEP/Keck) is confined to a narrow stripe near the $2\sigma$ border needing $N_e \simeq 50$ and $\delta \gtrsim 0.1$. For $\delta < 0$ the unregularized potential is excluded by data, with $r \gtrsim 0.07$ everywhere and the slow-roll approximation breaking down for $\phi_0 \lesssim 15 M_P$; including the effective mass term inside the logarithm gives the regularized potential $\bar{V}(\phi) = -\tfrac{1}{2}\bar{\Lambda}^4 \ln(\beta + \phi^2/\phi_0^2)$, which develops a local maximum at $\phi = 0$ and behaves like a hilltop model, compatible with the CMB-only constraints at $1\sigma$ for $\beta = 0.9$ and $N_e = 60$ and with the CMB-BAO constraints only at $2\sigma$. The two signs therefore show complementary behavior, with the inflationary scale at $\bar{\Lambda} \sim 10^{15}$–$10^{16}$ GeV and $r \sim 10^{-2}$ in both surviving branches.

Load-bearing premise

The load-bearing premise is that the single-logarithm potential of eq. (2.2), with the effective mass term inside the logarithm negligible except where deliberately restored in eq. (3.9), describes the inflaton potential exactly over the whole scanned field range, so a different ultraviolet completion, higher-loop corrections, or a different regularization could shift the $(n_s, r)$ tracks and move the positive-$\delta$ branch out of the CMB-BAO band or erase the negative-$\delta$ hilltop fit.

Editorial extensions

If this is right

If the positive-$\delta$ branch is right, inflation from a running cosmological constant is consistent with the DESI-inclusive CMB-BAO constraints over a wide parameter range, and its compatibility with CMB-only data is marginal, sitting in a narrow $2\sigma$ stripe with $N_e \simeq 50$ and $\delta \gtrsim 0.1$.
If the regularized negative-$\delta$ branch is right, hilltop inflation emerges from a radiative correction alone, fitting the CMB-only data at $1\sigma$ for $\beta = 0.9$, $N_e = 60$, while the CMB-BAO agreement remains at $2\sigma$.
Both surviving branches predict $r \sim 10^{-2}$ and an inflationary scale $\Lambda \sim 10^{15}$–$10^{16}$ GeV, putting the scenario within reach of next-generation CMB polarization experiments.
A resolution of the BAO-CMB tension would select the sign of $\delta$, i.e. whether bosonic or fermionic degrees of freedom dominate the radiative correction to the cosmological constant.
Within the positive-$\delta$ branch the predictions form a one-parameter family in the $(n_s, r)$ plane bounded by the logarithmic limit ($n_s \to 1 - 1/N_e$, $r \to 0$) and the linear-inflation limit, so the model's fate is decided along a single curve.

Reading between the lines

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

Editorial inference: the one-parameter track structure offers a diagnostic beyond this paper—if future data pin $(n_s, r)$ precisely, the running-cosmological-constant shape forces the pair onto a specific locus, so a measurement clearly off that locus would rule out the log-shaped potential regardless of the sign of $\delta$.
Editorial inference: keeping the explicit mass term in a one-loop potential is generic rather than special, so the regularized negative-$\delta$ branch suggests that any logarithmically corrected cosmological constant with a subleading mass acquires a hilltop completion at small field values, a pattern that could be tested against other one-loop inflationary potentials.
Editorial inference: because the inflationary observables are independent of the coupling $y$ (Appendix A), the model is degenerate across underlying couplings; quantities sensitive to that coupling, such as the reheating temperature or non-Gaussianity, would break the degeneracy and are a natural next computation.
Editorial inference: if the DESI-inclusive data continue to tighten, the negative-$\delta$ branch, already only $2\sigma$ compatible, would be pushed toward exclusion, effectively converting the fit into an upper bound on $\beta$ at fixed $N_e$.

Original abstract

We revisit single-field slow-roll inflation driven by a running cosmological constant, i.e. a cosmological constant receiving a logarithmic radiative correction, in light of the latest CMB-BAO data. The relative radiative correction $\delta$ is treated as a free parameter of the model and allowed to be positive or negative. For positive $\delta$, the predictions interpolate between a logarithmic regime and a linear inflation one. Compatibility with CMB-only data is confined to a narrow stripe near the $2\sigma$ border where $\delta$ is relatively large. On the other hand, the predictions fit easily within the constraints including BAO data in a wide range of parameters. For negative $\delta$, the slow-roll approximation breaks down at small field values and the predictions are excluded by data. However, once the effective mass contribution inside the logarithmic correction is taken into account, the negative $\delta$ potential develops a local maximum and behaves like a hilltop model, achieving compatibility with CMB-only data, while agreement with CMB-BAO data is possible only at $2\sigma$.

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.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-30: 1 request, 5,626 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.80 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.75 · compares with prior results 0.32 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We revisit single-field slow-roll inflation driven by a running cosmological constant, i.e. a cosmological constant receiving a logarithmic radiative correction, in light of the latest CMB-BAO data. The relative radiative correction $\delta$ is treated as a free parameter of the model and allowed to be positive or negative. For positive $\delta$, the predictions interpolate between a logarithmic regime and a linear inflation one. Compatibility with CMB-only data is confined to a narrow stripe near the $2\sigma$ border where $\delta$ is relatively large. On the other hand, the predictions fit easily within the constraints including BAO data in a wide range of parameters. For negative $\delta$, the slow-roll approximation breaks down at small field values and the predictions are excluded by data. However, once the effective mass contribution inside the logarithmic correction is taken into account, the negative $\delta$ potential develops a local maximum and behaves like a hilltop model, achieving compatibility with CMB-only data, while agreement with CMB-BAO data is possible only at $2\sigma$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We revisit single-field slow-roll inflation driven by a running cosmological constant, i.e. a cosmological constant receiving a logarithmic radiative correction, in light of the latest CMB-BAO data.	contribution 0.96	0.83	0.06	asserted 0.67
s1	The relative radiative correction $\delta$ is treated as a free parameter of the model and allowed to be positive or negative.	method 0.94	0.77	0.03	asserted 0.97
s2	For positive $\delta$, the predictions interpolate between a logarithmic regime and a linear inflation one.	result 0.84	0.65	0.05	asserted 1.00
s3	Compatibility with CMB-only data is confined to a narrow stripe near the $2\sigma$ border where $\delta$ is relatively large.	result 1.00	0.87	0.14	asserted 1.00
s4	On the other hand, the predictions fit easily within the constraints including BAO data in a wide range of parameters.	result 1.00	0.87	0.27	asserted 0.93
s5	For negative $\delta$, the slow-roll approximation breaks down at small field values and the predictions are excluded by data.	result 0.99	0.83	0.10	asserted 1.00
s6	However, once the effective mass contribution inside the logarithmic correction is taken into account, the negative $\delta$ potential develops a local maximum and behaves like a hilltop model, achieving compatibility with CMB-only data, while agreement with CMB-BAO data is possible only at $2\sigma$.	result 0.98	0.84	0.25	asserted 0.94

What kind of paper the classifier reports this is: theoretical result 0.97 · evidence from analysis of existing data 0.99.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** The inflationary potential and the inflaton roll direction for positive (green) and negative (red) values of δ. In the following we consider the standard range Ne ∈ [50, 60]. The tensor-to-scalar ratio and the scalar spectral index are respectively given by r = 16 ϵ, ns = 1 − 6ϵ + 2η, (2.7) and applying eqs. (2.3) and (2.4), they become r = 8 M2 P δ 2 ϕ 2 N [1 + δ ln(ϕN /M)]2 , (2.8) ns = 1 − 3 M2 P δ 2 ϕ 2 N [1 + δ ln(ϕN /M)]2 − 2 M2 P δ ϕ 2 N [1 + δ ln(ϕN /M)]. (2.9) Finally, the amplitude of the scalar perturbations is As = 1 24π 2M4 P V (ϕN ) ϵ(ϕN ) = Λ 4 ϕ 2 N [1 + δ ln(ϕN…

**Figure 2.** r vs. ns (top left), r vs. δ (top right), δ vs. ns (bottom left) and λ vs. δ (bottom right) with δ > 0 and M/MP = 10−2 , 10−1 , 1, 10, 103 (blue, cyan, yellow, orange, red) for Ne = 50 (continuous) and Ne = 60 (dashed). For reference, we also show the 1σ and 2σ constraints from SPA+BK (gray) and SPA+BK+DESI (purple) [11] and the predictions of linear inflation for Ne ∈ [50, 60] (brown). between 0 and M (see eq. (2.12)). In fact, for δ → ∞ the predictions approach asymptotic values corresponding to ϕ0 = M. Finally, we comment on the two different asymptotic limits, respectively for δ…

**Figure 3.** r vs. ns (top left), r vs. ϕ0/MP (top right), ϕ0/MP vs. ns (bottom left) and λ¯ vs. ϕ0/MP (bottom right) with δ < 0 for Ne = 50 (black, continuous) and Ne = 60 (black, dashed). For reference, we also show the 1σ and 2σ constraints from SPA+BK (gray) and SPA+BK+DESI (purple) [11] and the predictions of linear inflation for Ne ∈ [50, 60] (brown). this maximum number being exactly 50 (60) e-folds. This means that the system is in a regime where eq. (2.2) cannot be trusted. In fact, eq. (2.2) is derived on the assumption that ϕ is big enough so that an effective mass term inside the…

**Figure 4.** The inflationary potential with δ < 0 for the regularized (blue) and unregularized (red, dashed) cases. local maximum of the potential in eq. (3.9) is located at ϕ = 0 and corresponds to V¯ (0) = − 1 2 Λ¯ 4 ln β . (3.10) In order to ensure V¯ (0) > 0, we immediately see that β is constrained to be 0 < β < 1. The number of e-folds for the potential in eq. (3.9) is Ne = " βϕ2 0 + ϕ 2 4M2 P ln β + ϕ 2 ϕ 2 0 + β ϕ 2 0 4M2 P ln β ln ϕ 2 ϕ 2 0 − Li2 − ϕ 2 βϕ2 0 − ϕ 2 4M2 P #ϕN ϕend , (3.11) where Li2 is the dilogarithm function, and the other inflationary observables are…

**Figure 5.** r vs. ns (top left), r vs. ϕ0/MP (top right), ϕ0/MP vs. ns (bottom left) and λ¯ vs. ϕ0/MP (bottom right) for the potential in eq. (3.9) with β = 0.01, 0.05, 0.1, 0.5, 0.9 (blue, cyan, yellow, orange, red) for Ne = 50 (continuous) and Ne = 60 (dashed). For reference, we also show the 1σ and 2σ constraints from SPA+BK (gray) and SPA+BK+DESI (purple) [11], the predictions of linear inflation for Ne ∈ [50, 60] (brown) and the ones of the potential in eq. (3.1) (black). inflation limit. For small β, when reducing ϕ0, r increases and the predictions depart from the unregularized ones,…

Figures reproduced from arXiv:2610.38973 by the authors.

Discussion

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