The large-N running of the spectral index of inflation

Diederik Roest; Juan Garcia-Bellido

REVIEW

The large-N running of the spectral index of inflation

Juan Garcia-Bellido, Diederik Roest

Cosmology and Nongalactic Astrophysics · Submitted 10 Feb 2014 · 24 references

Pith's verdict: A tidy, honest extension of the 1/N classification, but the headline 'clustering' of alpha_s is an algebraic identity dressed as a discovery.

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Pith ID6R3A6B6L, Pith's permanent record of this paper and its review

CitationDiederik Roest, Juan Garcia-Bellido. "The large-N running of the spectral index of inflation." arXiv:1402.2059 (2014). https://pith.science/paper/6R3A6B6L

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Editor's note

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The useful part of this paper is the classification, not the headline. Garcia-Bellido and Roest take the perturbative 1/N story of Mukhanov and Roest and add three more large-N behaviours: constant, non-perturbative, and logarithmic. The N-formalism write-up (Eqs. 8 and 13) is clean, the table is a handy reference, and the model-by-model mapping onto classes is done carefully. I checked the leading-order expressions for the constant, p=1, p>1, p=2, non-perturbative and logarithmic cases and they come out right. They are also honest where the expansion fails: Section III.C says plainly that for new and natural inflation the leading 1/N approximation is poor and the full expression is needed. The k-inflation appendix is a sensible addendum. That is real, if incremental, work, and it is the kind of thing I would point a student to when they ask which potentials land in which large-N bin. Now the soft spot, and it is the one the abstract leans on. For the perturbative and logarithmic classes, alpha_s = (n_s - 1)/N is an exact leading-order identity, not a consequence of the class structure: p=1 gives -(2eps_1+1)/N^2 against n_s-1 = -(2eps_1+1)/N, p>1 gives -p/N^2 against -p/N, and the logarithmic p=2 case is the same. The constant class has alpha_s = 0 and the non-perturbative class has alpha_s = -c r/4, smaller still. So |alpha_s| is bounded by |n_s-1|/N by construction, and with n_s = 0.96 and N around 55 you get log10|alpha_s| around -3.2 whether or not any universality class exists. The authors say as much in Section IV ('purely phenomenological observation') and give the Delta n_s argument, so they know. The abstract's 'Remarkably, we find' does not reflect that, and the closing claim that permil-level measurements 'could allow us to differentiate these models among themselves' is weaker than stated: if alpha_s is pinned by n_s and N, a better measurement mostly re-measures those two. Minor point: the BICEP2 framing has not aged well, though the classification stands without it. Recommendation: worth a referee report, with a request to soften the abstract and the discrimination claim. The physics is sound; the packaging oversells one equation.

Referee report

Major objections (3)

[Abstract, sentence 4; Section IV] The abstract's 'Remarkably, we find that the tilt's runnings essentially cluster around the permil level' presents as an empirical discovery what is, at leading order, an identity. For the perturbative class, Eqs. (19)–(20) give α_s = (n_s − 1)/N exactly (p = 1: n_s − 1 = −(2ϵ₁ + 1)/N, α_s = −(2ϵ₁ + 1)/N²; p > 1: n_s − 1 = −p/N, α_s = −p/N²), and the logarithmic class, Eq. (35), gives the same relation. The constant class has α_s = 0 (Eq. 16) and the non-perturbative class has α_s = −c r/4 (Eq. 30), both even smaller. Hence |α_s| ≲ |n_s − 1|/N by construction for every class, and with n_s ≈ 0.96 and N ≈ 55 the value −7×10⁻⁴ follows without any model input beyond the fit. The authors acknowledge this in Section IV ('This is a purely phenomenological observation'), but the abstract and the Introduction do not. I recommend stating the identity α_s ≈ (n_s − 1)/N explicitly in Section IV and tempering the abstract's framing accordingly; the result remains worth reporting, but as a consistency relation rather than a surprise.
[Section IV, final paragraph] The claim that a permil-level measurement of α_s 'could allow us to differentiate these models among themselves' is not supported by the class structure. Since all four classes share α_s ≈ (n_s − 1)/N at leading order, a precise α_s measurement at fixed n_s primarily determines N rather than discriminating between classes; the spread visible in Fig. 3b arises from subleading terms and from scanning N over [50, 60], not from the leading class behaviour. The authors should either qualify this sentence or show explicitly which classes separate at permil precision and by how much, for example by quoting α_s at fixed N = 55 for each class.
[Section IV and Fig. 3b] The text states that the clustering has 'a range of only half a decade', but the vertical axis of Fig. 3b spans log₁₀|α_s| from −4.0 to −2.6, i.e. about 1.4 decades, and the plotted models (including hybrid near the top) appear to cover more than half a decade. The quoted spread should be reconciled with the figure: state the actual range among models that survive the Planck n_s constraint, say whether hybrid inflation is included in that range, and give the numerical endpoints rather than a qualitative 'half a decade'.

Minor objections (8)

[Section III.C, paragraph on natural inflation] The sentence 'It is different from that of New inflation since α_s = −cr/4 is not satisfied in this case ∀N, since (ϵ′/ϵ)′ ≠ 0' is hard to parse. Eq. (30) presents α_s = −4cϵ₀e^{−2cN} = −cr/4 as the class prediction, so the intended point seems to be that natural inflation deviates from the pure exponential form at subleading order. Please rewrite this sentence to say so directly.
[Section III.C, new inflation] New inflation is described as 'almost ruled out' and then as lying 'just inside the 2σ contour allowed by Planck'. These two statements pull in opposite directions; please choose one characterization and state the margin explicitly.
[Table I, non-perturbative row] The non-perturbative row lists only the potential 1 − φ², but natural inflation with V ∝ 1 + cos(φ/μ) is also presented in Section III.C as a member of this class. Either add it to the table or note that the row shows a single representative.
[Section IV] The paper fixes each parameter to n_s = 0.96 but does not propagate the Planck uncertainty, n_s = 0.9603 ± 0.0073 (Eq. 1), into α_s. A one-line estimate, δα_s ≈ δn_s/N ≈ 1.3×10⁻⁴, would show that the clustering is robust against the observational error and would strengthen the section.
[Eq. (13)] The tensor running α_t = 2ϵ′ is defined but never discussed or evaluated for any class. Either comment briefly on its magnitude or drop it from the equation to avoid an unused definition.
[Abstract, sentence 2] The claim that the four classes cover 'the large majority of inflationary models' is asserted rather than demonstrated. The paper gives a representative list, not a census; please soften the wording or justify it with a count of models in a standard compilation such as the Encyclopaedia Inflationaris cited as Ref. [4].
[Eq. (10)] The displayed expression for the conformal time τ is difficult to parse as typeset, with nested integrals and exponents that do not read cleanly. Please check the typesetting and, if the expression is only meant to be schematic, say so.
[Note added] The appended note changes the emphasis of the paper (from Starobinsky-like to chaotic models) without altering the body text, which still describes chaotic p = 1 as 'almost ruled out'. For a journal version, either integrate the BICEP2 discussion into Sections I and IV or state clearly in the body which statements the note supersedes.

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The paper

The paper claims

Most inflation models that fit the measured spectral tilt predict a running of that tilt near one part in a thousand.

The reading

Inflation's observables are usually written as functions of $N$, the number of e-foldings between when a mode left the horizon and when inflation ended. Since $N$ is at least about 50, the authors argue that one should sort models by how the equation-of-state parameter $N$ behaves at large $N$, and they extend the earlier 1/N classification with three further classes: constant, exponentially falling (non-perturbative), and logarithmic. They then compute the running of the spectral tilt, $alpha_s$, for every class, after fixing each model's free parameter so that it reproduces the observed tilt $n_s = 0.96$ over $N$ between 50 and 60. The result they emphasise is that the running lands near $-0.001$ in every class, a spread of only about half a decade in $log_{10}|alpha_s|$, even though the tensor-to-scalar ratio $r$ ranges over several orders of magnitude across the same models. If that holds, no near-future CMB experiment short of a dedicated survey can separate these classes by their running, and the running is not a useful discriminator of inflation models.

What carries the argument

The carrying device is the 'N-formalism': the whole inflationary background is specified by the equation-of-state parameter $epsilon(N)$, with the Hubble slow-roll parameters recovered as $delta = epsilon + epsilon'/(2epsilon)$ and $xi = 3epsilon'/2 + epsilon^2 + (epsilon'/epsilon)'/2$, and the observables as $n_s - 1 = -2epsilon + epsilon'/epsilon$, $r = 16epsilon$, and $alpha_s = 2epsilon' - (epsilon'/epsilon)'$. Because everything reduces to one function of $N$, models can be sorted by the large-$N$ asymptotics of that function into four classes: constant ($epsilon = epsilon_0$), perturbative ($epsilon = epsilon_p/N^p$, with $p=1$ giving chaotic inflation and $p=2$ giving Starobinsky/T-models), non-perturbative ($epsilon = epsilon_0 e^{-2cN}$, covering new and natural inflation), and logarithmic ($epsilon = epsilon_p ln^q N / N^p$, covering Kähler moduli inflation). The classification is the paper's organising structure; the running calculation is then a short derivative away in each case.

Extended reading notes

Core claim

The paper's central claim is a clustering: take the models that survive the Planck constraint on the scalar tilt, fix their parameters so that $n_s = 0.96$ at $N$ between 50 and 60, and the running of the tilt comes out around $alpha_s = -0.001$ ($log_{10}|alpha_s|$ near $-3.2$) regardless of which large-$N$ universality class the model belongs to. This is stated in the abstract as the tilt's runnings clustering 'around the permil level' and displayed in the paper's Fig. 3b, which plots $n_s$ against $log_{10}(-alpha_s)$ for chaotic, Starobinsky, natural, brane, log, Kähler, T-model, moduli, hybrid and power-law cases. The contrast the authors draw is with the tensor-to-scalar ratio: in Fig. 3a the same models spread over several orders of magnitude in $r$, so the running is far more uniform than the amplitude of gravitational waves. The authors are careful to say this is a phenomenological observation rather than a theorem, and they explain it by noting that models inside the Planck contour vary the tilt by less than about $0.01$ across $Delta N$ near 10, which forces $alpha_s$ to be of order $(n_s-1)/N$.

Load-bearing premise

The clustering is only as meaningful as the sample that produces it: the authors fix each model's free parameter to match the observed tilt over $N$ between 50 and 60, and they themselves call the resulting uniformity 'a purely phenomenological observation' rather than a consequence of the class structure.

Editorial extensions

If this is right

Because $alpha_s$ sits near $-0.001$ for every class, measuring the running cannot by itself tell the universality classes apart; the tensor-to-scalar ratio remains the sharper discriminator.
Detecting a running at the permil level requires a probe with a much wider lever arm in scale than the CMB, which is why the authors point to 21 cm surveys such as the Square Kilometer Array.
Models in the non-perturbative class (new and natural inflation) are the exception where the leading large-$N$ approximation fails: the effective expansion parameter is $1/(cN)$, not $1/N$, so the full expression is needed even inside the Planck region.
Supergravity hybrid inflation, whose $epsilon(N)$ oscillates as $sin(kappa N/pi)$, falls outside the classification altogether, so the scheme covers most but not all models.

Reading between the lines

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

The clustering can be read as a consistency check on the fitting procedure rather than a prediction of the class structure: if every model is pinned to the same tilt over the same narrow range of $N$, a running of order $(n_s-1)/N$ is close to guaranteed, so the half-decade spread is the informative part, not the central value.
The quoted Planck value $alpha_s = -0.0134 \pm 0.0090$ is roughly an order of magnitude larger in magnitude than $-0.001$ while still consistent within about one and a half sigma; a future measurement settling near $-0.013$ would put the clustering under real pressure rather than merely refining it.
If the running really is near the permil level across classes, then a detection of a substantially larger running would be evidence for a model outside the four classes, which makes the running a background discriminator rather than a model discriminator.

Original abstract

We extend previous classifications of inflationary models by means of their behaviour at large-N, where N is the number of e-foldings. In addition to the perturbative 1/N case, whose slow-roll parameters fall off as powers of 1/N, we introduce the constant, non-perturbative and logarithmic classes. This covers the large majority of inflationary models. Furthermore, we calculate the running of the spectral tilt for all these classes. Remarkably, we find that the tilt's runnings essentially cluster around the permil level. We comment on the implications for future experiments.

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.

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What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.15 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.92 · compares with prior results 0.45 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We extend previous classifications of inflationary models by means of their behaviour at large-N, where N is the number of e-foldings. In addition to the perturbative 1/N case, whose slow-roll parameters fall off as powers of 1/N, we introduce the constant, non-perturbative and logarithmic classes. This covers the large majority of inflationary models. Furthermore, we calculate the running of the spectral tilt for all these classes. Remarkably, we find that the tilt's runnings essentially cluster around the permil level. We comment on the implications for future experiments.

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 extend previous classifications of inflationary models by means of their behaviour at large-N, where N is the number of e-foldings.	contribution 0.98	0.92	0.04	asserted 0.98
s1	In addition to the perturbative 1/N case, whose slow-roll parameters fall off as powers of 1/N, we introduce the constant, non-perturbative and logarithmic classes.	contribution 0.98	0.91	0.07	asserted 0.99
s2	This covers the large majority of inflationary models.	contribution 0.44	0.40	0.05	asserted 0.98
s3	Furthermore, we calculate the running of the spectral tilt for all these classes.	contribution 0.91	0.96	0.08	asserted 0.99
s4	Remarkably, we find that the tilt's runnings essentially cluster around the permil level.	result 1.00	0.96	0.73	asserted 1.00
s5	We comment on the implications for future experiments.	interpretation 0.79	0.64	0.03	unclear 0.96

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 1.00.

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full text
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Figures

**Figure 2.** • p = 2: The Whitt potential corresponding to the Starobinsky model [6, 15] V = V0 1 − e − √ 2/3φ 2 , (24) is characterized by an equation of state parameter (N) = 3 4N2 , (25) at lowest order in 1/N. This model is not ruled out but precisely agrees with the scalar spectral index ns = 0.96, and one finds that the model is well inside the 2σ contour allowed by Planck. An interesting generalisation in the same universality class is provided by T-models of inflation with a potential [16] V = V0 tanh2n (φ/√ 6α), with α arbitrary, which corresponds to an equation of state parameter…

**Figure 3.** a the spectral index and the tensor-to-scalar ratio are plotted for all models that we have discussed. In case these models have free parameters we have fixed these in order to comply with the Planck data. Moreover, we have varied N from 50 to 60 to give an indication of the N-dependence of these models. Secondly, in Fig. 3b we show the spectral index and its running, for the same parameter values. While the ratio of tensor to scalar amplitudes in all models discussed above, which fall in various universality classes, can vary significantly over several orders of magnitude, see Fig

Figures reproduced from arXiv:1402.2059 by the authors.

References

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Computed the scalar spectral index and its running, as well as the tensor-to-scalar ratio, using this paper's exact expressions for epsilon(N).

Primordial black holes from single field models of inflation (2017) (same authors) · 0.85 · paraphrased: the licence does not allow quoting

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This parameterization has subsequently been developed in [ref] , where different terms in the $1/N$ expansion have been introduced.

Classification of inflationary models and constraints on fundamental physics · Reasons to classify inflationary models. · 0.81
for the equation of state during inflation[For an extension of the above ansatz, see [ref] .].

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To this end we consider the classification of single field slow roll inflationary models as demonstrated in [ref] .

Inflationary field excursion in broad classes of scalar field models · Introduction · 0.83
In the present article we study tachyon inflation in the so called large-$N$ formalism [ref] , where relevant quantities are functions of the number of $e$-folds $N$, taken as an evolution variable instead of the usual inflaton field $ϕ$, or cosmic time.

Tachyon inflation in the Large-$N$ formalism · Introduction · 0.94
The general features of inflationary dynamics at this scale can be discussed in a model-independent way in terms of the large-$N$ formalism [ref] .

Single Field Double Inflation and Primordial Black Holes · The first phase of inflation · 0.86

Cites it as background 9 papers

In popular single field slow-roll inflation models, the spectral index $n_s$ of primordial scalar perturbation follows [ref] , n_s-1 = -O(1)N_* in large-$N_*$ limit, where $N_* = ∫Hdt$ is the efolds number before the end of inflation.

Single field slow-roll inflation with step uplift to $n_s=1$ · Introduction · 0.89
In the first section we discuss some issues related with the standard method to define inflationary models by specifying potentials and we give a brief review of the classifications proposed by Mukhanov in [ref] and Roest [ref] .

Classification of inflationary models and constraints on fundamental physics · Primordial GW and direct GW detectors. · 0.97
In addition to that there are constant, non-perturbative and logarithmic classes [ref] .

Inflationary field excursion in broad classes of scalar field models · Comparison of field excursion in different classes of inflaton · 0.86

What it cites · 16 sentences without a confident label

Builds on 1 paper

An interesting generalisation in the same universality class is provided by T-models of inflation with a potential [ref] [equation] with $α$ arbitrary, which corresponds to an equation of state parameter ϵ(N) = 3αn/4n N^2+2N√(3α(3α+4n^2))+3αn .

Universality Class in Conformal Inflation · Perturbative class · 0.94

Uses data or values from 3 papers

In particular, the spectral index has been measured to be [ref] [equation] although subsequent analyses have argued for a slightly higher value [ref] .

Planck 2013 results. XXII. Constraints on inflation · Introduction · 0.96
In particular, the spectral index has been measured to be [ref] [equation] although subsequent analyses have argued for a slightly higher value [ref] .

Planck Data Reconsidered · Introduction · 0.96
Similarly, the running of the spectral index has been constrained with some precision: [equation] On the other hand, the BICEP2 collaboration [ref] has recently announced the detection of the primordial gravitational wave background produced during inflation, with an amplitude responsible for a tensor-to-scalar ratio [equation] with $r=0$ ruled out at more than 5$σ$.

BICEP2 I: Detection Of B-mode Polarization at Degree Angular Scales · Introduction · 0.81

Cites as background 1 paper

$∙$ $p>2$: In contrast to the previous models, hilltop inflation takes place near the origin [ref] .

Hilltop Inflation · Perturbative class · 0.91

Discussion

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