Cosmic microwave background anomalies in an open universe

Andrew R Liddle; Marina Cort\^es

REVIEW

Cosmic microwave background anomalies in an open universe

Andrew R Liddle, Marina Cortês

Cosmology and Nongalactic Astrophysics · Submitted 24 Jun 2013 · 31 references

Pith's verdict: The constraint arithmetic is sound and the derived window on 1−Ω is a genuinely useful target, but the 'specific implementation' is a template — no tunnelling potential is ever written down, and the curvaton super-curvature claim is borrowed from an inflaton calculation.

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arXiv ID1306.5698 (version 1)

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CitationAndrew R Liddle, Marina Cort\^es. "Cosmic microwave background anomalies in an open universe." arXiv:1306.5698 (2013). https://pith.science/paper/XKAT5Y6Y

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Worth your time, but read it as a feasibility study, not as the model its abstract advertises. The real content is Eqs. (5)–(10): take the Erickcek/Lyth gradient formula, plug in the Garcia-Bellido et al. Grishchuk–Zel'dovich quadrupole bound, impose k_curv,L < 1 and P_R,L < 1, and you get |A| < 0.03|f_NL| with a viable window 8×10^-5 < 1−Ω_0 < 10^-3. That arithmetic is clean and I checked the saturation logic; it holds. The conversion to a bound on m_F/H_T in Eq. (10) is a nice touch, and the observation that H_F cancels is the kind of thing that makes a scenario paper useful to someone else. The idea of using open inflation's super-curvature mode as the modulating perturbation is also the right instinct — it is the one place in the literature where such a mode comes out of a first-principles calculation rather than a hand-wave about the onset of inflation. Credit for that, and credit for the honesty: they say outright that considerable fine-tuning may be needed and cite Linde–Mezhlumian for how bad it is in these models. That is not a hidden flaw, it is a stated one, and I would not hold it against them as if they had concealed it. The soft spots are elsewhere. First, the 'specific implementation' in Eq. (11) is a Lagrangian with an unspecified W(ψ); the sentence 'suitable values of each can be obtained by choice of the tunnelling potential shape' is doing all the work and is never cashed out. Second, and this is the one I would actually press on: they assert the curvaton's super-curvature mode 'has the same form as computed in Ref. [24],' but Ref. [24] computed the inflaton's adiabatic perturbations. A super-curvature isocurvature mode converting into curvature after curvaton decay is not shown, and it is not obviously the same calculation. Third, Eq. (5) is a flat-space formula with an underived replacement k_L x_ls → 3|1−Ω|^{1/2} k_curv,L; an order-unity error there shifts the whole window. None of this sinks the paper, but it means 'most naturally explained' and 'specific implementation' in the abstract are ahead of the body. Send it out. It is short, it is honest, and the window it identifies is a concrete target for anyone doing open inflation or anomaly phenomenology.

Referee report

Significance

If the construction can be realized, it would be a genuine step beyond the existing modulation literature: the modulating super-horizon mode would no longer be an unexplained postulate but a calculable feature of an open inflationary sector, with a specific curvature window and a specific relation to f_NL. The constraint arithmetic in Eqs. (5)–(10) is clean and internally consistent, and the physics inputs are drawn from published sources (Erickcek et al.; Lyth; Garcia-Bellido et al.; Yamamoto–Sasaki–Tanaka). The paper is also honest about the main open problem, stating that considerable fine-tuning may be needed and citing Ref. [19] for the difficulty of making open inflation work at all. The value of the paper therefore rests on whether the 'specific implementation' advertised in the abstract is actually delivered, and that is where the manuscript falls short of its own framing.

Major objections (3)

[Abstract, sentence 3; §'A specific implementation', Eq. (11)] The abstract states 'We provide a specific implementation of the scenario which appears compatible with all existing constraints' (abstract:s2), but Eq. (11) is a template rather than an implementation: the tunnelling potential W(ψ) is never written down, and the two properties that make the scenario work — the enhancement H_F^2/H_T^2 ~ 10^8 and the placement of the super-curvature spike at k_curv,L^2 ≃ 2(m_σ^2 + g^2ψ_F^2)/(3H_F^2) < 1 — are asserted ('suitable values of each can be obtained by choice of the tunnelling potential shape') rather than demonstrated. The consistency conditions quoted after Eq. (10), (2/3)m_F^2 < H_F^2 < 4×10^8 H_T^2, are necessary but not shown sufficient, and no explicit W(ψ) is exhibited that satisfies them while leaving an acceptable sub-curvature spectrum. Either a worked W(ψ) should be provided, or the abstract and the concluding sentence should be restated to say that the paper identifies the required parametric corner and a plausible field content, not that it supplies an implementation.
[§'A specific implementation', paragraph following Eq. (11)] The paper argues that because σ enters Eq. (11) in the same way as φ enters the first Linde–Mezhlumian model, 'the field perturbations of σ have the same form as computed in Ref. [24]'. Ref. [24] computed the adiabatic perturbations of the inflaton in an open inflationary background; σ, by contrast, is an isocurvature field during inflation whose fluctuations must be converted into curvature at curvaton decay. The mode-function statement may well be correct, but the transfer step is not shown, and it is not obvious that the standard local curvaton relation ζ = (r/3) δσ/σ applies unchanged to modes with k_curv^2 < 1, which are never sub-horizon. The manuscript should either demonstrate that the isocurvature-to-adiabatic transfer holds mode-by-mode for super-curvature modes, or state explicitly that this is an assumption inherited from the flat-space curvaton calculation. As written, this step is load-bearing for the claim that the spike survives in the curvature perturbation.
[§'Modulation in an open universe', Eqs. (7)–(10)] The headline quantitative result, 8×10^-5 < 1-Ω_total < 10^-3, follows from simultaneously saturating three independent inequalities: the quadrupole bound Eq. (7), the super-curvature condition k_curv,L < 1, and the perturbativity bound P_R,L^{1/2} < 1. The paper presents this window as the viable range without quantifying how much tuning it represents, even though the introduction invokes the string landscape as the motivation for a marginally open universe. It would strengthen the paper to state explicitly whether the window is a prediction of the framework or a tuning of it, and, if the latter, to estimate what fraction of the landscape-inspired parameter space lands in it. Relatedly, at saturation P_R,L^{1/2} is of order 0.5, so the observed A ≃ 0.07 requires |f_NL| of order a few; the paper should state this number explicitly rather than leaving it implicit in Eq. (9).

Minor objections (8)

[Eq. (1)] The displayed formula for the comoving curvature scale appears inverted: it should be r_curv = a^{-1}H^{-1}/√|1-Ω_total|, not a^{-1}H^{-1}√|1-Ω_total|. The sentence immediately following ('at least ten times the Hubble radius' for a universe within a percent of flatness) is consistent only with the inverse form, and the later relation k_curv,L = k_L x_ls/(3|1-Ω_total|^{1/2}) also assumes it. This should be corrected.
[Eq. (5)] Eq. (5) is a flat-space relation taken from Refs. [9,11], while the scenario operates at 1-Ω_total ~ 10^-4–10^-3. The paper should state that the O(1-Ω_total) corrections to this relation are negligible over the viable window, or cite an open-universe derivation. Eq. (6) is explicitly labelled as a flat-geometry limit, so the same remark applies there.
[§'A specific implementation', paragraph after Eq. (11)] The statement that the continuum spectrum gives n_s ≃ 0.98 should specify the assumed number of e-folds and whether the curvaton changes the standard m^2φ^2 prediction. Planck 2013 (Ref. [28]) gives n_s = 0.9603 ± 0.0073, so 0.98 sits about 2.7σ high; the paper should either justify the value or note the tension.
[Reference list, Ref. [5]] The third entry of Ref. [5] (Hansen, Banday, and Gorski) ends with the truncated identifier 'astro-ph/040420'; the full arXiv number should be supplied.
[Eq. (11)] Eq. (11) is labelled L but is written as a sum of potential terms; it should be labelled as the potential (or Lagrangian density) consistently with the surrounding text.
[§'A specific implementation', Eq. (11) and following] The paper defines the true vacuum at ψ = 0 and the false vacuum at ψ_F, which is the opposite of the convention in Ref. [19]. This is stated, but a sentence explaining why the flip is convenient (the coupling g^2ψ^2σ^2 vanishing in the true vacuum) would help the reader.
[Introduction and Abstract] The abstract refers to 'anomalies' in the plural and the introduction states that a compelling explanation ought to explain several, but the mechanism addresses only the dipole modulation. The paper should say explicitly which of the other anomalies (low-multipole deficit, cold spot, alignments) it does and does not claim to address.
[Conclusions] The sentence 'Ours is the first proposal of a model which permits a complete first-principles calculation of a perturbation spectrum including a large-scale modulation effect of the observed amplitude' should be softened in light of the two major comments above; 'first proposal of a model with the required field content and parametric window' would be defensible, 'complete first-principles calculation' is not yet supported.

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The paper

The paper claims

Open inflation in a marginally open universe can supply the single very-large-scale perturbation needed to produce the observed dipole power asymmetry in the cosmic microwave background, within existing bounds on curvature and the quadrup

The reading

The cosmic microwave background carries a set of large-scale oddities that survived the move from WMAP to Planck: aligned low multipoles, a cold spot, a deficit of power at the largest angles, and a dipole modulation in which one half of the sky has roughly 7 percent more power than the other. This paper argues that these are not statistical accidents but the fingerprint of a universe that is very slightly open, so that the curvature scale sits only a few times beyond the observable horizon. The argument has two moving parts: open inflation, in which our bubble universe nucleates out of a false vacuum, naturally excites a 'super-curvature' perturbation mode whose wavelength exceeds our horizon; and a curvaton field converts that single long-wavelength mode into a gradient across our observable patch, which modulates the small-scale power in a dipolar pattern. If the picture is right, the size of the anomaly and the curvature of the universe are tied together by a small set of measurable numbers, so that a modest improvement in the limits on spatial curvature or on primordial non-Gaussianity would confirm or exclude it.

What carries the argument

The load-bearing object is the super-curvature mode: an eigenmode of the spatial Laplacian with wavenumber in the range $0 < k^2 < 1$ in curvature units, which is not needed to expand a general radial function but is required to describe a general random field, and which open inflation typically excites in a single narrow spike. Matching quantum fluctuations across the bubble wall gives that spike an amplitude enhanced over the ordinary continuum spectrum by the factor $H_F^2/H_T^2$, the ratio of the Hubble rate before tunnelling to that after. The second ingredient is the curvaton, a light scalar whose fluctuations are converted into curvature after inflation; adding it to an open inflation model lets the very-large-scale perturbation be carried by a field that does not source the CMB quadrupole the way the inflaton would, which is the obstacle that blocks a purely adiabatic version of the same idea. Together the spike and the curvaton give a spectrum with a sharp super-horizon feature plus a nearly scale-invariant sub-curvature tail, which is exactly the shape the modulation scenario needs.

Extended reading notes

Core claim

The paper's claim is that the observed dipole power asymmetry, modelled as $1 + A$ times a modulation along a preferred direction with $|A|$ of order $0.07$, can be generated by a single super-curvature perturbation in a marginally open universe, and that the required amplitude fits inside the existing quadrupole and curvature bounds. Combining the large-scale gradient relation for the asymmetry with the Grishchuk–Zel'dovich quadrupole limit gives $k^{curv}_L P^{1/2}_{R,L}|1-Ω_{total}| < 8×10^{-5}$ and hence $|A| < 3×10^{-4}|f_{NL}||1-Ω_{total}|^{-1/2}$; saturating the super-curvature condition $k^{curv}_L < 1$ and the perturbativity bound at the same time yields a ceiling $|A| < 0.03|f_{NL}|$. A viable window is $8×10^{-5} < 1-Ω_{total} < 10^{-3}$ for $|f_{NL}|$ below about 10, with typical parameters $1-Ω_0$ around $10^{-4}$ and $m_F$ around $10^4 H_T$ giving $|A|$ around $0.01|f_{NL}|$. The authors then assemble a two-field open inflation model with a curvaton, in which a tunnelling field $ψ$ ends inflation while a $φ$ field and a $σ$ field carry the spectra, and a coupling $g^2ψ^2σ^2$ lets the curvaton be heavy before tunnelling, so that it inherits the super-curvature spike, and light afterwards, so that it does not drive its own inflation.

Load-bearing premise

The scenario needs a tunnelling potential that simultaneously places the super-curvature spike at the right wavenumber, boosts its amplitude by the large factor $H_F^2/H_T^2$, and leaves an acceptable ordinary spectrum, and the paper never writes such a potential down, conceding that considerable fine-tuning may be needed.

Editorial extensions

If this is right

Spatial curvature becomes a diagnostic rather than a nuisance parameter: the scenario works only in the band roughly $8×10^{-5} < 1-Ω_{total} < 10^{-3}$, close to but above Planck's current sensitivity.
The asymmetry amplitude and primordial non-Gaussianity are locked together, so a measurement of $f_{NL}$ at the level of a few would either supply the required amplitude or close the window.
Because the modulating perturbation is a single discrete spike rather than a scale-invariant tail, the induced modulation should be nearly scale-independent across the sky, with departures appearing only where extra physics such as isocurvature or a scale-dependent $f_{NL}$ enters.
Since the same tunnelling event sets both the curvature and the spike, a detection of curvature at the $10^{-4}$ level would be a direct probe of the bubble-nucleation picture that motivates the whole construction.

Reading between the lines

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

The narrowness of the allowed window cuts both ways: it makes the model falsifiable, but it also means a modest tightening of the curvature constraint without a detection would kill the explanation by excluding the only region where it works, rather than by refuting its mechanism.
The same super-curvature machinery breaks isotropy in one definite direction, so it should also leave a mark on the quadrupole–octupole alignment and the cold spot; the paper concentrates on the dipole, but the alignment anomalies are a second, independent handle on the same perturbation.
If the asymmetry really is scale-dependent, as quasar and small-angle CMB data hint, a single spike is too rigid; a broadened or two-component super-curvature spectrum is the natural extension to test.
The coupling that shifts the curvaton mass across the tunnelling is a reusable device, so the same trick could be transplanted to other post-tunnelling mass hierarchies in landscape-motivated models.

Original abstract

We argue that the observed large-scale cosmic microwave anomalies, discovered by WMAP and confirmed by the Planck satellite, are most naturally explained in the context of a marginally-open universe. Particular focus is placed on the dipole power asymmetry, via an open universe implementation of the large-scale gradient mechanism of Erickcek et al. Open inflation models, which are motivated by the string landscape and which can excite `super-curvature' perturbation modes, can explain the presence of a very-large-scale perturbation that leads to a dipole modulation of the power spectrum measured by a typical observer. We provide a specific implementation of the scenario which appears compatible with all existing constraints.

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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 0.03 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.36 · 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 argue that the observed large-scale cosmic microwave anomalies, discovered by WMAP and confirmed by the Planck satellite, are most naturally explained in the context of a marginally-open universe. Particular focus is placed on the dipole power asymmetry, via an open universe implementation of the large-scale gradient mechanism of Erickcek et al. Open inflation models, which are motivated by the string landscape and which can excite `super-curvature' perturbation modes, can explain the presence of a very-large-scale perturbation that leads to a dipole modulation of the power spectrum measured by a typical observer. We provide a specific implementation of the scenario which appears compatible with all existing constraints.

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 argue that the observed large-scale cosmic microwave anomalies, discovered by WMAP and confirmed by the Planck satellite, are most naturally explained in the context of a marginally-open universe.	interpretation 0.58	0.25	0.92	hedged possibility 0.65
s1	Particular focus is placed on the dipole power asymmetry, via an open universe implementation of the large-scale gradient mechanism of Erickcek et al. Open inflation models, which are motivated by the string landscape and which can excite `super-curvature' perturbation modes, can explain the presence of a very-large-scale perturbation that leads to a dipole modulation of the power spectrum measured by a typical observer.	method 0.50	0.26	0.18	hedged possibility 0.58
s2	We provide a specific implementation of the scenario which appears compatible with all existing constraints.	contribution 0.84	0.89	0.09	hedged possibility 0.80

What kind of paper the classifier reports this is: theoretical result 0.83 · evidence from argument 0.82.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
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relations to other papers

levels run

P abstract

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S sections
T body sentences
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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

References

Works cited (31)

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Forward citations

Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Planck 2015 results. XX. Constraints on inflation
astro-ph.CO 2015-02 accept novelty 6.0 of 10

Full-mission Planck data give a scalar spectral index ns = 0.9655 +/- 0.0062 and a tensor-to-scalar bound r < 0.11, disfavouring phi-squared and natural inflation relative to R-squared inflation.
Inflation after False Vacuum Decay: Observational Prospects after Planck
hep-th 2013-09 conditional novelty 6.0 of 10

A steepening inflaton potential following false vacuum decay suppresses CMB power at large angles without necessarily producing detectable curvature, and two toy models of this effect fit the Planck low-l data better ...
Curvature constraints from Large Scale Structure
astro-ph.CO 2016-03 conditional novelty 5.0 of 10

Extending CLASS to curved geometry, the authors find that neglecting cosmic magnification in a tomographic galaxy survey can bias the inferred spatial curvature by more than one standard deviation.
Inflation After False Vacuum Decay: New Evidence from BICEP2
astro-ph.CO 2014-04 conditional novelty 4.0 of 10

With BICEP2's r of about 0.2 assumed primordial, Planck's low-multipole temperature deficit becomes a 3.5 to 4 sigma tension with standard cosmology, which a hand-tuned steepening of the inflaton potential absorbs.

How the literature uses it

32 arXiv papers cite this in 39 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 16 sentences without a confident label

Builds on it 2 papers

Extending Ref. [ref] , there are also potentially interesting consequences of zero modes for other CMB anomalies such as the hemispherical power asymmetry.

Cosmological Zero Modes · Conclusion, Discussion, Future Prospects · 0.98
After this work was completed [ref] appeared, which extends the model of [ref] to open universes.

Large-scale anomalies from primordial dissipation · Previous work · 0.94

Cites it as background 18 papers

The proposal in Ref. [ref] advocates the use of the super-curvature perturbation [ref] produced in the open inflation scenario

A viable explanation of the CMB dipolar statistical anisotropy · Introduction · 0.83
After Planck Collaboration reported this anomaly in the data release, a few papers appeared which attempted to provide a physical mechanism, namely see [ref] .

CMB Power Asymmetry from Primordial Sound Speed Parameter · Introduction · 0.91
The idea of generating power asymmetry from a long-mode modulation [ref] has been studied extensively in the literature [ref] .

Inflationary power asymmetry from primordial domain walls · Introduction · 0.99

What it cites · 25 sentences without a confident label

Uses the method or tool of 2 papers

The asymmetry $A$ is related to the non-gaussianity parameter $f_NL$, given a single modulating mode with wavenumber $k_L$, using [ref] [equation]

The CMB asymmetry from inflation · Modulation in an open universe · 0.81
The asymmetry $A$ is related to the non-gaussianity parameter $f_NL$, given a single modulating mode with wavenumber $k_L$, using [ref] [equation]

Hemispherical Asymmetry and Local non-Gaussianity: a Consistency Condition · Modulation in an open universe · 0.81

Uses data or values from 2 papers

The modulation amplitude $A$ is measured in the CMB to be $0.07 ±0.02$ in maps smoothed at 5 degrees [ref] .

Planck 2013 results. XXIII. Isotropy and statistics of the CMB · Modulation by large-scale perturbations · 1.00
We adopt the same limit on such a contribution to the quadrupole as Ref. [ref] , $C_2^GZ < 3.6×10^-11$, obtaining [equation] This implies [equation]

A Hemispherical Power Asymmetry from Inflation · Modulation in an open universe · 0.97

Discussion

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