Mixed inflaton and curvaton perturbations

David Langlois; Filippo Vernizzi

REVIEW

Mixed inflaton and curvaton perturbations

David Langlois, Filippo Vernizzi

Astrophysics · Submitted 11 Mar 2004 · 22 references

Pith's verdict: A careful, correct calculation of the mixed inflaton-curvaton regime; the modified consistency relation is the real payoff, and the stress-test's S-conservation worry does not fully land.

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arXiv IDastro-ph/0403258 (version 2)

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CitationDavid Langlois, Filippo Vernizzi. "Mixed inflaton and curvaton perturbations." arXiv:astro-ph/0403258 (2004). https://pith.science/paper/FECTM6RD

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Referee report

Significance

The paper addresses a well-posed and physically motivated question: what happens when inflaton and curvaton perturbations are comparable rather than one being negligible. The main deliverables are a clean linear decomposition with a numerically computed interpolation function, analytic limits that correctly reproduce the known pure-curvaton and double-inflation results, and a genuinely falsifiable prediction in the form of a modified tensor consistency relation (Eq. 68) that depends only on sigma_* and not on the inflaton potential. The alpha = 9/10 coefficient rests on an independent physical argument (conservation of total zeta for adiabatic initial conditions) and is cross-checked numerically. The paper is explicit that the quadratic curvaton potential and instantaneous decay are simplifications, and it cites the published justification for the latter. The observational discussion is appropriately hedged about the current inability to measure n_T.

Major objections (3)

[Sec. 3.3.2, Eqs. (40)-(45)] The secondary-inflaton estimate evaluates the entropy perturbation S at the radiation-to-curvaton transition using Eq. (41), which the authors themselves state is rigorously valid only while the curvaton is negligible, and then substitutes it into Eq. (40), which presupposes S is conserved. The text argues that S becomes constant once the curvaton dominates or oscillates, but the value carried forward is the one frozen in at the transition, not the value at decay. For sigma_* >~ m_P the curvaton drives a secondary inflationary phase, and the super-Hubble evolution of the coupled system during that phase is precisely where the linear decomposition of Eq. (26) could fail. The numerical support in Fig. 3 covers only two values of Phi_* at sigma_* = m_P. This is load-bearing for the large-sigma_* branch of f(sigma_*) and hence for the claimed interpolation. A concrete test would be to show, for several sigma_* in the range 1-5 m_P and several Phi_*, that the numerically evolved Phi_f agrees with Eq. (45) to better than the accuracy claimed, and to state that accuracy explicitly.
[Sec. 3.3.3, Eq. (47) and Fig. 4] The central result is the function f(sigma_*), but it is presented only graphically in Fig. 4 with no fitting formula, table of values, or stated numerical error. Since the observational predictions in Sec. 4 (Eqs. 55, 61, 66, 68) are quoted in terms of f, a reader cannot reproduce the numbers in Fig. 5 without digitizing the plot. Providing an explicit fit or a short table of f(sigma_*) over the relevant range, together with the numerical tolerance of the evolution code, would make the main result reproducible and would let the reader judge how sharply the minimum near sigma_* ~ 1.2 m_P is determined.
[Sec. 3.3, Eq. (26) and Eq. (32)] The claim that alpha = 9/10 holds 'for all types of initial conditions' is justified by the adiabaticity argument plus a numerical check, but the argument as written applies to the total zeta only when the initial entropy vanishes. The text does note that delta_sigma_* = 0 corresponds to S^(i) = 0, so the logic is defensible, yet the statement in the sentence following Eq. (32) would be clearer if it distinguished the adiabatic case (where the argument is analytic) from the general case (where the result is numerical). As written, a reader may read the sentence as claiming an analytic proof for arbitrary initial conditions.

Minor objections (8)

[Sec. 3.2, Eq. (19)] The step from Eq. (19) to Eq. (20) drops the curvaton term by invoking subdominance; it would help to state explicitly the order of the neglected term relative to the retained one.
[Sec. 3.3.1, Eq. (37)] The averaging over oscillations that yields delta^(i)_m = 2 delta_sigma_*/sigma_* - (3/2)Phi_* deserves one sentence of justification, since it is the step that produces the 9/10 coefficient in Eq. (39).
[Sec. 4.1, Eq. (50)] The constant C is defined as -2 + ln 2 + gamma; it would be useful to note explicitly that this is the standard Stewart-Lyth normalization constant and to cross-reference the convention in [15].
[Sec. 4.1, Eq. (57)] The definition R = f-tilde^2 epsilon/(1 - 2(1+3C)epsilon + 2C eta) is introduced as the relative importance of curvaton over inflaton perturbations, but the denominator is the slow-roll correction to the inflaton amplitude rather than the amplitude itself; a brief clarification of what R measures would avoid confusion.
[Sec. 4.2, Eq. (76)] The correction to N_* from a curvaton domination phase is estimated as -(1/12) ln(rho_CD/rho_dec); the derivation of the prefactor 1/12 from Eq. (75) is not shown and would benefit from one line of algebra.
[Fig. 5] The figure caption refers to 'f = 1' and 'f = 3' while the text uses f-tilde; the notation should be made consistent between caption and text.
[Sec. 5, Eq. (83)] The final expression for zeta is stated to be valid only for large f-tilde, with second-order slow-roll corrections otherwise; it would be helpful to state the condition on f-tilde quantitatively (e.g. f-tilde^2 epsilon >> epsilon) rather than qualitatively.
[General] There are several typographical issues in the extracted text (spacing in 'pri mordial', 'ﬂ uctuatio ns', 't he', 'rˆ oles'), which presumably reflect the arXiv source rather than the journal submission; these should be checked at proof stage.

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The paper

The paper claims

A single curve weighs inflaton against curvaton seeds of structure

The reading

Inflation usually gets all the credit for the primordial ripples that grew into galaxies, but a second light field, the curvaton, can generate them too. This paper asks what happens when the two contributions are comparable rather than one being negligible, and follows the perturbations from the first radiation era through the epoch when the curvaton starts to oscillate up to its decay. The authors show that the final curvature perturbation is a linear combination of the inflaton's contribution and the curvaton's, with coefficients fixed entirely by the curvaton's initial amplitude $f(1/2)(1/2)$ and by the background history. The curvaton coefficient is not a constant: it falls off as $1/1/2$ for small amplitudes, rises linearly for large ones, and has a minimum near $f(1/2) = 1.2, m_P$, so the same function controls the amplitude, the spectral index and the tensor-to-scalar ratio at once. The practical consequence is a degeneracy: a quartic inflaton model that the data seem to exclude can be pulled back inside the 95% contour by adding a curvaton, so the microwave background alone no longer pins down the inflaton potential.

What carries the argument

The load-bearing object is the interpolation function $f(1/2_*)$, defined through $(1/2)_{RD} = (1/2)_* - (f(1/2_*)/m_P),(1/2)_*$. It packages the entire post-inflationary history of the curvaton: whether it oscillates before or after it dominates, how its energy fraction grows, and how much of the initial isocurvature perturbation survives to be converted into adiabatic curvature. The authors compute it in two analytic limits using the entropy perturbation $S = 3((1/2)_r - (1/2)_1/2)$ and the relation $(1/2)_f = (9/10)(1/2)_* + (1/5) S$ that holds once the entropy perturbation is conserved, and they extract the intermediate range numerically. Because $f$ enters the amplitude, the spectral index and the tensor-to-scalar ratio identically, it is the single knob that a mixed model turns.

Extended reading notes

Core claim

The paper's central claim is that when inflaton-generated and curvaton-generated curvature perturbations are of the same order, the primordial curvature perturbation after curvaton decay can be written as a two-term linear combination, $$(1/2)_{RD} = (1/2)_* - (1/2) f(1/2)_* , (1/2)_*,$$ where the first term is what inflation imprinted and the second is the curvaton's contribution. The coefficient of the inflaton term is fixed to $9/10$ by conservation of the total curvature perturbation on super-Hubble scales for purely adiabatic initial conditions, independent of the details of the transition. The function $f(1/2)_*$ carries all the model dependence: it reduces to $f = (4/9)(m_P/1/2_*)$ in the standard curvaton regime where the field oscillates long before it dominates, and to $f = 1/2_*/(3 m_P)$ in the secondary-inflaton regime where it stays frozen until it dominates and drives a second burst of inflation. In between, the authors solve the coupled background and perturbation equations numerically and find that $f$ dips to about $0.55$ near $1/2_* = 1.2, m_P$. Feeding this into the standard slow-roll normalization gives the scalar amplitude, the spectral index, the tensor-to-scalar ratio and a modified consistency relation $r = -8 n_T/(1 - (1/2)^2 n_T/2)$, in which the deviation from the single-field relation depends only on the curvaton's initial value.

Load-bearing premise

The whole interpolation rests on the curvaton having an exactly quadratic potential and decaying instantaneously, so that the function $f(1/2_*)$ can be computed once and used everywhere.

Editorial extensions

If this is right

The final curvature perturbation is a linear combination of the inflaton and curvaton contributions, with the inflaton coefficient fixed at $9/10$ and the curvaton coefficient given by the numerically computed $f(1/2_*)$.
Both known limits are recovered as special cases: the standard curvaton result for $1/2_*$ well below the Planck mass, and the double-inflation result for $1/2_*$ above it.
The tensor-to-scalar ratio is always reduced by adding a curvaton, and the standard single-field consistency relation $r = -8 n_T$ is replaced by $r = -8 n_T/(1 - (1/2)^2 n_T/2)$, which offers a way to measure the curvaton's initial value.
For a quartic inflaton potential, a mixed model with $1/2_*$ below about $0.5, m_P$ brings the $N = 64$ prediction inside the 95% confidence contour of the WMAP data, and below about $0.1, m_P$ inside the 68% contour.
The scalar spectral index and its running interpolate smoothly between the pure-inflation expressions and the pure-curvaton expression $n_s - 1 = -2(1/2)$.

Reading between the lines

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

Because a curvaton admixture and a change in the inflaton potential's slope both shift the same point in the $(n_s, r)$ plane, breaking the degeneracy with data alone will require either a measurement of the tensor spectral index or a tight measurement of the running of the scalar index.
The minimum of $f$ near $1/2_* = 1.2, m_P$ marks a region where the curvaton is least efficient at seeding structure, so mixed models there are hardest to distinguish from pure inflation even though the curvaton is dynamically important.
The interpolation relies on a quadratic curvaton potential and instantaneous decay; a non-quadratic potential or gradual decay would move the curve $f(1/2_*)$ and hence the predicted $(n_s, r)$ trajectory, so the qualitative interpolation would survive but the quoted numbers would not.
If curvaton self-interactions suppress the perturbation amplitude at large field values, the rising secondary-inflation branch of $f$ could be cut off, which would restore the pure-curvaton branch as the only viable large-amplitude option.

Original abstract

A recent variant of the inflationary paradigm is that the ``primordial'' curvature perturbations come from quantum fluctuations of a scalar field, subdominant and effectively massless during inflation, called the ``curvaton'', instead of the fluctuations of the inflaton field. We consider the situation where the primordial curvature perturbations generated by the quantum fluctuations of an inflaton and of a curvaton field are of the same order of magnitude. We compute the curvature perturbation and its spectrum in this case and we discuss the observational consequences.

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The abstract, sentence by sentence

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A recent variant of the inflationary paradigm is that the ``primordial'' curvature perturbations come from quantum fluctuations of a scalar field, subdominant and effectively massless during inflation, called the ``curvaton'', instead of the fluctuations of the inflaton field. We consider the situation where the primordial curvature perturbations generated by the quantum fluctuations of an inflaton and of a curvaton field are of the same order of magnitude. We compute the curvature perturbation and its spectrum in this case and we discuss the observational consequences.

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	A recent variant of the inflationary paradigm is that the ``primordial'' curvature perturbations come from quantum fluctuations of a scalar field, subdominant and effectively massless during inflation, called the ``curvaton'', instead of the fluctuations of the inflaton field.	background 1.00	0.04	0.05	asserted 0.92
s1	We consider the situation where the primordial curvature perturbations generated by the quantum fluctuations of an inflaton and of a curvaton field are of the same order of magnitude.	contribution 0.87	0.25	0.09	asserted 0.86
s2	We compute the curvature perturbation and its spectrum in this case and we discuss the observational consequences.	contribution 0.97	0.94	0.12	asserted 0.91

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.88.

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sources read

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not examined

full text
figures and tables
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Figures

**Figure 1.** The standard curvaton limit, for σ∗ = 0.3mP. Evolution of the curvaton, its energy density fraction Ωσ ≡ ρσ/(ρσ + ρr) (upper part), and of the curvature perturbation Φ (lower part). The initial conditions for the perturbations are δσ∗ = mP and Φ∗ = −2. 10

**Figure 2.** The secondary inflaton limit, for σ∗ = 3mP. Evolution of σ, its energy density fraction (upper part), and of the curvature perturbation (lower part). The initial conditions for the perturbations are δσ∗ = mP and Φ∗ = −2. 13

**Figure 3.** The intermediate case, for σ∗ = mP. Evolution of σ, its energy density fraction (upper part), and of the curvature perturbation (lower part) for two different initial conditions: Φ∗ = 0 and for Φ∗ = −2 (both cases with δσ∗ = mP). 15

**Figure 4.** The function f(σ∗), which characterizes the amplitude of the contribution of σ to the curvature perturbation. For σ∗ ≪ mP one recognizes the ∝ 1/σ∗ contribution of the pure curvaton model. For σ∗ ≫ mP, one recognizes the contribution of a secondary inflaton, proportional to σ∗ . 16

**Figure 5.** Two-dimensional likelihood contours at 68% and 95% confide

Figures reproduced from arXiv:astro-ph/0403258 by the authors.

References

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Forward citations

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astro-ph.CO 2009-11 accept novelty 6.0 of 10

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astro-ph.CO 2009-07 conditional novelty 6.0 of 10

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How later papers used it

64 later papers cite it

Evidence: substantial. 54 of the citing papers share no author with it, and 7 of those carry a confident label from Pith's citation classifier.

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1 later paper by other authors is listed below: 1 used its method or tool.

Uses its method or tool 1 of 2 papers with this label

Used the method in this paper to solve the Bessel equation with a source term.

Non-Gaussianity, Isocurvature Perturbation, Gravitational Waves and a No-Go Theorem for Isocurvaton (2008) · 0.95 · paraphrased: the licence does not allow quoting

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63 arXiv papers cite this in 102 sentences

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Papers citing it · 64 sentences without a confident label

Builds on it 2 papers

More specifically, in the present work, we extend the general formalism given in [ref] to third order, then apply it to the mixed curvaton and inflaton scenario [ref] .

Primordial Trispectrum from Isocurvature Fluctuations · Introduction · 1.00
The first example is a more refined treatment of the isocurvature perturbations and their non-Gaussianity in the mixed curvaton-inflation scenario [ref] .

General treatment of isocurvature perturbations and non-Gaussianities · Introduction · 0.93

Uses its method or tool 2 papers

In the presence of the late-decaying $ϕ$ field, the following expressions are obtained using perturb2 [ref] : [equation]

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It can be checked that this result is exactly the same as in the mixed inflaton-curvation scenario [ref] .

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Beauty is Distractive: Particle production during multifield inflation · The power-spectrum · 0.86
In the limit $q=1$, which is equivalent to $p≫1$, $Q_σ$ becomes [equation] This agrees with the result in [ref] where asymptotic form of $Q_σ$ for $σ_* ≪M_pl$ is provided under the assumption that curvaton once dominates the universe before the curvaton decays.

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Ricci Reheating · Scalar Perturbations · 0.98

Cites it as background 25 papers

This leaves us with 10 possible cases (including situations where $σ$ drives a secondary phase of inflation [ref] ), depending on the vev of $σ$ at the end of inflation $σ_$.

Constraining Curvatonic Reheating · Curvaton and Reheating · 0.81
The cases where the inflaton contribution is not negligible correspond to the mixed inflaton-curvaton scenario [ref] .

General treatment of isocurvature perturbations and non-Gaussianities · Primordial adiabatic and isocurvature perturbations · 0.93
Such a mixed scenario in the framework of the modulated reheating [ref] and the curvaton [ref] has been discussed by several authors.

Primordial Non-Gaussianity in Models with Dark Matter Isocurvature Fluctuations · Introduction · 0.87

Discussion

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