Probing Primordial Features with the Stochastic Gravitational Wave Background

Dhiraj Kumar Hazra; Matteo Braglia; Xingang Chen

REVIEW

Probing Primordial Features with the Stochastic Gravitational Wave Background

Matteo Braglia, Xingang Chen, Dhiraj Kumar Hazra

Cosmology and Nongalactic Astrophysics · Submitted 10 Dec 2020 · 149 references

Pith's verdict: A careful numerical study that delivers a usable template bank for feature-induced SGWB spectra, but the claim that feature classes can be told apart is asserted rather than tested.

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CitationDhiraj Kumar Hazra, Matteo Braglia, Xingang Chen. "Probing Primordial Features with the Stochastic Gravitational Wave Background." arXiv:2012.05821 (2020). https://pith.science/paper/4ENIBE4I

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Editor's note

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The thing worth knowing: this paper takes the standard-clock two-field model (Chen–Namjoo and successors) and pushes it into the LISA band, then hands the GW community a concrete product — the multi-domain template in Eq. (4.9) with parameter sets in Appendix B, plus a binned-PLS reconstruction forecast in Table 1. The numerical machinery is not new (it is the pipeline from Braglia et al. 2020, ref. [47]), and the authors say so. What is new is the template bank and the reconstruction analysis, and both are done with care. Appendix A gives the two-field perturbation equations, the Bunch-Davies conditions, and the two-pass integration that kills curvature-isocurvature correlations; the induced-GW integral is cited correctly. The concurrent paper [145] is disclosed in a note added. That is honest practice and I'd like to see more of it. The paper also does the unglamorous thing of showing where its own models fail: Model II gives r of order 0.1, Model III gives ns around 0.93–0.94, both in tension with Planck, and the authors defer the fix rather than hiding it. Fine-tuning of the peak into the LISA band is stated up front [abstract:s3]. So the paper is not overselling the physics. Where it does oversell is [abstract:s4]. The templates in Eqs. (4.6)–(4.8) share a skeleton — power-law rise, a log-normal-ish first peak, a second peak built from up to three log-normal components, then a modulation g(x) — and differ mainly in whether g oscillates in f or in ln f, with l, ω, φ free in both. With 15–20 free parameters per class and one numerical spectrum per class in Figs. 11–14, a good fit is weak evidence for anything. Nothing in §4.2 or §4.3 fits a class-A spectrum with a class-B template, and no second example per class is shown. The reader's worry about single-example fits is aimed slightly off — the templates are flexible fitting functions, not fixed predictors, so the Appendix B values don't need to generalize; the functional form does. The real gap is discrimination, and the paper never quantifies at what SNR l, ω, φ become measurable, which is exactly the information carrying the class label. Table 1 hints the answer is uncomfortable: 55–90 bins at SNR 10 per bin. §4.3's comparison with phase transitions, cosmic strings, and preheating is qualitative and useful but doesn't close this. None of this sinks the paper; it bounds what the template claim can support. Worth refereeing, and I'd cite it for the template.

Referee report

Significance

The paper addresses a well-motivated question — whether the SGWB can serve as a probe of inflationary features at scales inaccessible to the CMB and LSS — and it does so with a concrete, non-trivial model rather than a toy parametrization. The numerical machinery is documented in enough detail to be reproduced: the perturbation equations are given explicitly in App. A, the initial conditions and the two-pass procedure are stated, and the PBH constraint P_R,max < 0.01 is imposed throughout. The paper is also candid about its own weak points: it states that fine-tuning in scale and amplitude is required (abstract:s3), that Model II overproduces tensors with r ~ O(0.1) (§3.2), and that Model III yields n_s ≈ 0.93–0.94, redder than Planck (§3.3). The template bank, if validated, would be a genuinely useful deliverable for the GW community, and the qualitative comparison with broken power-laws, log-normals, phase transitions, cosmic strings, and preheating in §4.3 is a sensible first step toward a falsifiable classification. The value of the paper therefore turns on whether the templates are demonstrated to be class estimators rather than per-spectrum curve fits.

Major objections (4)

[§4.2, Eqs. (4.6)–(4.9), Figs. 11–14, App. B] The templates are validated only in-sample: each of Figs. 11–14 shows one numerical spectrum and one fit to that same spectrum, with 15–20 free parameters per class listed in App. B. The claim in abstract:s8 that the templates 'accurately capture the spectral features of several classes' and in abstract:s4 that the classes 'may be distinguished' requires two things that are not shown: that a template fitted to one representative example generalizes to other parameter points in the same class, and that a template from one class fits the other classes measurably worse. The concern is sharpened by the near-nested structure of Eqs. (4.6)–(4.8): all share the same skeleton (power-law rise below f1, a log-normal-like first peak, a second peak built from up to three log-normal components, then a modulation g(x)), and they differ essentially only in whether g is 1 + l sin(ω(f−f_p)+φ) or 1 + l sin(ω ln x + φ), with l, ω, and φ free in both. With that much freedom a good fit is weak evidence for the underlying class. A concrete test would be to fit each of the four numerical spectra with all four templates and report Δχ² or a Bayes factor, and to fit a second, independently chosen parameter point per class (the A/B/C spectra already plotted in Figs. 3, 5, and 8 are available for this). Without such a test the templates read as curve fits to individual spectra rather than as class estimators.
[§4.1 vs §4.2, Table 1] The paper presents the template approach as 'the second more sensitive approach' (abstract:s7) but never quantifies its sensitivity. Table 1 shows that reconstructing the sinusoidal and both resonant spectra requires 55–90 bins at SNR = 10 per bin, i.e. a total SNR of order several hundred, which is far beyond a realistic LISA mission. If the template approach is the practical route to detection, the paper should provide at least a Fisher-matrix or matched-filter estimate of the SNR that a LISA observation would accumulate for each template at a representative amplitude, and state the minimum Ω_GW h² at which each class becomes distinguishable from the others. As written, the reader is told that the reconstruction route is expensive and that the template route is cheaper, but is given no number for the latter.
[§3.2, §3.3, §5] The templates for the bump, sinusoidal, and resonant-2 classes are drawn from models that are already in tension with Planck: Model II gives r ~ O(0.1) (§3.2) and Model III gives n_s = 0.9397, 0.9356, 0.9351, 0.9369, 0.9327 (§3.3), with the authors stating that a different potential for σ is needed and deferring this to future work. The authors acknowledge both points, but the implication for the template bank is not drawn out: it should be stated explicitly which of the four classes survive in a model that satisfies current CMB constraints, and whether the template shapes are robust to the modifications (e.g. an inflection-point potential for σ) needed to restore agreement with Planck. If the distinctive oscillatory structure is generic to the feature mechanism while the envelope is model-dependent, that should be argued; if not, the observational target is narrower than the abstract suggests.
[§4.1, Eq. (4.1), §5 final remark] The induced SGWB is computed under the assumption of Gaussian scalar perturbations, and the Conclusions correctly flag that non-Gaussianities 'can significantly change the spectral shape of the SGWB' and may enhance or wash out the oscillatory signal. Given that the models here involve large slow-roll violation, turns in field space, and a transient tachyonic growth of isocurvature modes, the non-Gaussianity is plausibly substantial rather than a small correction. The acknowledgment is not quantified anywhere, and since the templates are derived from the Gaussian calculation, an order-of-magnitude estimate of the expected f_NL (or of the fractional distortion of Ω_GW) in at least one representative case would let the reader judge whether the template shapes are robust. This is a limitation the authors state, so the request is for an estimate, not for a full non-Gaussian treatment.

Minor objections (10)

[Abstract] The abstract states that the spectrum 'can significantly exceed the ns = 5 limit in single-field inflation and grow as fast as ns = 9.1', while §3.3 phrases the single-field bound as n_s − 1 ≤ 4. The two statements are equivalent but the notation should be made consistent, and 'grow as fast as n_s = 9.1' should be clarified as the local tilt d ln P_R/d ln k rather than a growth rate in time.
[Eq. (4.5) and App. B, bump feature] The parameter list gives 'd1 = d4 = −d2 = −0.36', which is ambiguous as written; each coefficient should be listed separately. The term d4 f/f_p inside an otherwise log-normal exponent is dimensionally unusual and deserves a sentence of justification.
[App. B, sinusoidal and resonant feature 1] 'A2 = 1.07−8' appears twice and is presumably 1.07 × 10⁻⁸; the exponent is missing. Please check all numerical values in App. B for similar typographical losses.
[Eq. (4.8) and App. B, resonant feature 2] The parameter list for resonant feature 2 includes f5 = 0.261, but f5 does not appear anywhere in Eq. (4.8), which uses f1 through f4. Either the template or the parameter list is inconsistent and should be corrected.
[Eq. (4.9) and g(x) definition] For bump features g(x) is set to zero, which removes the A4 term entirely; the template then carries a redundant parameter. It would be clearer to state that the A4 term is absent for the bump class rather than to define it and multiply by zero.
[Fig. 1 caption vs §3.1] Fig. 1 uses Cσ = 0.02, whereas the Model I examples A, B, C in §3.1 fix Cσ = 10. The caption should state explicitly that Fig. 1 is illustrative and is not one of the A/B/C parameter sets, to avoid confusion about which regime is being shown.
[Figs. 3, 5, 8, top-left panels] The CMB-scale panels use axis labels of the form '1e 9' and plot the Planck curve without a legend entry in some panels. Please label the Planck comparison consistently and use standard scientific-notation axis formatting.
[§4.1, Eq. (4.4)] The SNR expression and the choice SNR_thr = 10 with T = 4 years should be stated together with the precise PSD and channel (TDI X) assumed, and with the version of schNell used, so that the numbers in Table 1 can be reproduced.
[Note added] The overlap with Ref. [145] is properly disclosed. It would help the reader if the note stated in one sentence which results overlap (e.g. the oscillatory SGWB shapes from sharp features) and whether the templates here differ from those in that work.
[§4.3] The comparison with other SGWB sources is qualitative. A short table listing, for each alternative mechanism, which of the four template features (broad peak, narrow peak, linear-in-f modulation, log-periodic modulation) it can and cannot reproduce would make the discrimination argument in §4.3 much easier to assess.

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The paper

The paper claims

A two-field inflation model can stamp four distinguishable shapes onto the stochastic gravitational-wave background, and each shape carries a template that identifies which kind of primordial feature produced it.

The reading

The paper asks whether the stochastic gravitational-wave background (SGWB) — the random hum of gravitational waves filling space — could be used to see 'primordial features': brief disturbances in the rolling of the inflaton field during inflation. The authors work through a two-field model in which a 'clock' field falls tachyonsically into a potential well, oscillates, and then hands inflation over to a second field, and they scan its parameter space to produce four families of scalar power spectra: a single broad bump, a sinusoidal (sharp-feature) running, a resonant (high-frequency) running, and the 'standard clock' combination of the last two. Those large scalar perturbations later source gravitational waves when they re-enter the horizon, and the authors compute the resulting $h^2Omega_{GW}(f)$ numerically, tuning the feature so the peak lands in LISA's band around $k sim 10^{10}$–$10^{13}$, Mpc$^{-1}$. The payoff, if it holds, is a channel that reads inflationary dynamics at scales twenty-odd e-folds before the end of inflation — scales no CMB, galaxy survey, or 21-cm experiment can reach. The paper then offers two ways to use this in data analysis: a binned power-law sensitivity reconstruction with LISA, which needs a high signal-to-noise ratio and 15 to 90 frequency bins depending on the class, and a set of analytic templates meant to act as estimators even at modest signal-to-noise.

What carries the argument

The engine is a non-canonical two-field Lagrangian with $f(sigma) = 1 + xi sigma$ (or $exp(xi sigma)$ in Model III), in which a strong coupling makes the isocurvature mass squared $m^2_{iso}$ transiently negative as the clock field falls, so isocurvature modes grow tachyonsically and source curvature perturbations far above the slow-roll level. That growth is what puts $mathcal{P}_mathcal{R}$ at $mathcal{O}(10^{-3})$–$10^{-2}$ at LISA scales while keeping CMB scales consistent with Planck. The second piece of machinery is the standard induced-GW convolution, Eq. (4.1), which turns the scalar spectrum into $Omega_{GW}(f)$ through the radiation-era transfer function; the third is the binned power-law integrated sensitivity curve, which quantifies how finely LISA would have to slice its band to reconstruct each spectral shape.

Extended reading notes

Core claim

The central claim is that different underlying inflationary physics leaves recognisably different oscillatory structure on the SGWB, and that this structure can be summarised well enough to be used as a matched filter. Concretely: a bump feature shows only a single rounded broad peak; a sinusoidal feature adds secondary peaks with a linear-in-frequency oscillation inherited from a sharp feature; a resonant feature adds ripples with a logarithmic-in-frequency oscillation inherited from the oscillating massive field; and the standard clock case interpolates between the two. The authors compress all of these into one piecewise template, Eq. (4.9), built as 'broad peak + narrow peak $times$ (1 + oscillatory feature)', with separate domains for the rising power law, the broad peak, the narrow peak plus ripples, and the small-scale cutoff or plateau. They also report that the growth of the power spectrum can beat the single-field bound $n_s - 1 le 4$, reaching an effective $n_s$ as steep as 9.1 in one realisation, which is a multi-field effect worth noting on its own. The claim is not that these signals are natural — the authors say plainly that fine-tuning in both the scale and the amplitude is required for a feature to land in an observational window — but that once one does land there, its shape is diagnostic.

Load-bearing premise

Each template in Eq. (4.9) is fitted, with fifteen to twenty free parameters listed in Appendix B, to a single numerical spectrum per feature class, so the claim that it captures the class rests on the untested assumption that a fit to one representative example generalises to other parameter choices within that class.

Editorial extensions

If this is right

A LISA detection of a peaked SGWB with the right oscillatory envelope would identify which class of primordial feature generated it, at scales around $k sim 10^{10}$–$10^{13}$, Mpc$^{-1}$ that no electromagnetic probe reaches.
The linear-in-frequency versus logarithmic-in-frequency ripple distinction gives an operational test between sharp-feature and resonant-feature physics, since the two oscillatory patterns are qualitatively different functions of $f$.
A measured spectral index above the single-field bound $n_s - 1 = 4$ in the rising part of the spectrum would itself be evidence for multi-field dynamics or a temporary non-inflationary epoch.
The bump template's nearly flat top makes it a candidate fit for a nHz-band common-spectrum signal if the clock field oscillated earlier in inflation, shifting the feature into the pulsar-timing window.
The templates are written to be usable directly by the gravitational-wave community as estimators, without re-running the inflation code.

Reading between the lines

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

The paper's real deliverable is a template bank of one, and the obvious next step it invites is a populated bank: one template per class, cross-validated against many parameter draws, would turn these fits into genuine class discriminators rather than representative curve fits.
Because the oscillatory ripples sit on top of a peaked envelope, a matched-filter search would likely be limited by how well the envelope is constrained first; a staged analysis that fits the smooth part before looking for ripples may be more sensitive than a joint fit.
The same induced-GW logic applies to any model that spikes the scalar spectrum at small scales, so the oscillatory signature is what distinguishes primordial features from inflection-point or non-canonical single-field models that produce similar broad peaks.
The authors flag non-Gaussianity as a possible modifier of the SGWB shape but do not compute it; if the feature-generating dynamics is strongly non-Gaussian, the templates derived here under Gaussian assumptions could misestimate the ripple amplitude.

Original abstract

The stochastic gravitational wave background (SGWB) offers a new opportunity to observe signals of primordial features from inflationary models. We study their detectability with future space-based gravitational waves experiments, focusing our analysis on the frequency range of the LISA mission. We compute gravitational wave spectra from primordial features by exploring the parameter space of a two-field inflation model capable of generating different classes of features. Fine-tuning in scales and amplitudes is necessary for these signals to fall in the observational windows. Once they show up, several classes of frequency-dependent oscillatory signals, characteristic of different underlying inflationary physics, may be distinguished and the SGWB provides a window on dynamics of the primordial universe independent of cosmic microwave background and large-scale structure. To connect with future experimental data, we discuss two approaches of how the results may be applied to data analyses. First, we discuss the possibility of reconstructing the signal with LISA, which requires a high signal-to-noise ratio. The second more sensitive approach is to apply templates representing the spectra as estimators. For the latter purpose, we derive templates that can accurately capture the spectral features of several classes of feature signals and compare them with the SGWB produced by other physical mechanisms.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.84 · human participants 0.01 · controlled comparison 0.14 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.59 · mentions code or data availability 0.02.

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

The stochastic gravitational wave background (SGWB) offers a new opportunity to observe signals of primordial features from inflationary models. We study their detectability with future space-based gravitational waves experiments, focusing our analysis on the frequency range of the LISA mission. We compute gravitational wave spectra from primordial features by exploring the parameter space of a two-field inflation model capable of generating different classes of features. Fine-tuning in scales and amplitudes is necessary for these signals to fall in the observational windows. Once they show up, several classes of frequency-dependent oscillatory signals, characteristic of different underlying inflationary physics, may be distinguished and the SGWB provides a window on dynamics of the primordial universe independent of cosmic microwave background and large-scale structure. To connect with future experimental data, we discuss two approaches of how the results may be applied to data analyses. First, we discuss the possibility of reconstructing the signal with LISA, which requires a high signal-to-noise ratio. The second more sensitive approach is to apply templates representing the spectra as estimators. For the latter purpose, we derive templates that can accurately capture the spectral features of several classes of feature signals and compare them with the SGWB produced by other physical mechanisms.

background contribution method result interpretation main claim

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s0	The stochastic gravitational wave background (SGWB) offers a new opportunity to observe signals of primordial features from inflationary models.	background 1.00	0.06	0.09	asserted 0.75
s1	We study their detectability with future space-based gravitational waves experiments, focusing our analysis on the frequency range of the LISA mission.	contribution 0.71	0.82	0.04	asserted 0.89
s2	We compute gravitational wave spectra from primordial features by exploring the parameter space of a two-field inflation model capable of generating different classes of features.	method 0.87	0.95	0.06	asserted 0.99
s3	Fine-tuning in scales and amplitudes is necessary for these signals to fall in the observational windows.	result 0.63	0.50	0.13	asserted 1.00
s4	Once they show up, several classes of frequency-dependent oscillatory signals, characteristic of different underlying inflationary physics, may be distinguished and the SGWB provides a window on dynamics of the primordial universe independent of cosmic microwave background and large-scale structure.	interpretation 0.84	0.18	0.50	hedged possibility 0.99
s5	To connect with future experimental data, we discuss two approaches of how the results may be applied to data analyses.	method 0.60	0.66	0.05	asserted 0.54
s6	First, we discuss the possibility of reconstructing the signal with LISA, which requires a high signal-to-noise ratio.	method 0.87	0.29	0.05	hedged possibility 0.96
s7	The second more sensitive approach is to apply templates representing the spectra as estimators.	method 0.94	0.53	0.07	asserted 0.98
s8	For the latter purpose, we derive templates that can accurately capture the spectral features of several classes of feature signals and compare them with the SGWB produced by other physical mechanisms.	method 0.75	0.95	0.09	asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.40 · evidence from simulation or computation 0.98.

What was examined, and what was not

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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.** Evolution of the clock field [left] and the first slow-roll parameter [right] for Model I in Eq. (3.1). The parameters used in this plot are Cσ = 0.02, σf = 0.0164, CΘ = 0.01, σi = 0.0556, and Θi = 3.16. The specific details of the second stage of inflation and consequences on the shape of the PPS and SGWB will be the subject of the following sections. In the following, we always solve for σi to get the maximum violation of the slow-roll around ∼ 20 e-folds before the end of inflation, so that the peak of the power spectrum occurs at scales where LISA has its maximum sensitivity…

**Figure 2.** Evolution of the two scalar fields [left] and of the parameter [right] for Model I. The parameters used in the three examples are A:{V0 = 3.99 × 10−13 , ξσf = 3.32, σf = 0.775, σi = 2.12, Θi = 9.84×10−3}, B:{V0 = 2.84×10−14 , ξσf = 1.45, σf = 0.258, σi = 0.817, Θi = 6.01×10−3} and C:{V0 = 8.62 × 10−15 , ξσf = 1.086, σf = 0.155, σi = 0.517, Θi = 3.96 × 10−3}. For all three cases we fix Cσ = 10 and CΘ = 0.0476. peaks decays and at very smaller scales we observe a plateau produced during the second stage of inflation by the field Θ and a superimposed oscillatory pattern with the…

**Figure 3.** PPS [bottom] zoomed at CMB [top-left] and LISA [top-right] scales for Model I. The parameters used in the three examples are the one reported in the caption of

**Figure 4.** Evolution of the two scalar fields [left] and of the parameter [right] for Model II. The parameters used in the three examples are A: {V0 = 1.86 × 10−9 , ξσf = 1.15, σf = 0.0258, σi = 10.6, Θi = 3.16}, B: {V0 = 6.51 × 10−10 , ξσf = 0.6, σf = 0.0258, σi = 10.47, Θi = 0.079} and C: {V0 = 9.78 × 10−10 , ξσf = 0.6, σf = 0.0111, σi = 9.86, Θi = 0.079}. For all three cases we fix Cσ = 0.1, m0 = 0.5 and CΘ = 0.01. of the PPS in

**Figure 5.** PPS [bottom] zoomed at CMB [top-left] and LISA [top-right] scales for Model II. The parameters used in the three examples are the one reported in the caption of

**Figure 6.** Evolution of σ [left] and Θ [right] for Model III. The parameters used in the three examples are {Cσ = 500 , σf = 0.18, σi = 3.27, Θi = 8} and we vary ξ ∈ [1, 30, 300]. 50.0 50.5 51.0 51.5 52.0 52.5 53.0 53.5 54.0 N -10 5 -10 3 -10 1 -10 -1 -10 -3 -10 -5 -10 -7 -10 -09 10 -9 10 -7 10 -5 10 -3 10 -1 10 1 10 3 10 5 10 7 m2 iso= H2 f(¾) = 1 + »¾ » = 1 » = 30 » = 300 20 18 16 14 12 Nend ¡ N -10 9 -10 7 -10 5 -10 3 -10 1 -10 -1 -10 -3 -10 -5 -10 -7 -10 -09 10 -9 10 -7 10 -5 10 -3 10 -1 10 1 10 3 10 5 10 7 m2 iso=H2 f(¾) = exp(»¾) ¾f = 0:078 ¾f = 0:18 ¾f = 0:55 ¾f = 1:83 ¾f = 5:48

**Figure 7.** Evolution of the mass squared of isocuvature perturbations defined in Eq. (A.8) around the time when slow-roll is violated for the coupling f(σ) = 1+ξσ [left] and for the coupling f(σ) = exp(ξσ) [right]. The parameters used in the former case are the ones used in

**Figure 8.** PPS [bottom] zoomed at CMB [top-left] and LISA [top-right] scales for Model III. We fix mΘ = 1 and use the following parameters: A: {V0 = 2.3 × 10−12, Cσ = 500, ξ = 7.5, σf = 5.477, σi = 9.311, Θi = 4.4}, B: {V0 = 1.34 × 10−13, Cσ = 500, ξ = 9.0, σf = 1.826, σi = 4.290, Θi = 6.7}, C: {V0 = 6.3 × 10−15, Cσ = 550, ξ = 12.2, σf = 0.574, σi = 1.649, Θi = 8.25}, D: {V0 = 5.0 × 10−16, Cσ = 500, ξ = 14.5, σf = 0.183, σi = 0.597, Θi = 8.0}, E: {V0 = 8.2 × 10−17, Cσ = 500, ξ = 18.9, σf = 0.078, σi = 0.277, Θi = 8.0}, . For a comparison, we also plot the Planck 2018 power-law spectrum with ns…

**Figure 9.** Reconstructed spectra for the bump [left] and sinusoidal [right] features. We also plot the numerical spectra in magenta lines and the LISA binned (unbinned) PLS in black (orange) lines. Note however, that features at larger scales, probed by Pulsar Timing Array experiments, or smaller scales, probed by the next generation of space-based GW interferometers such as BBO and DECIGO [79], are also possible if the massive field starts to oscillate earlier or later during inflation respectively. Moving the feature even closer to the end of inflation we enter in the regime of Einstein…

**Figure 10.** Reconstructed spectra for resonant feature 1 [left] and 2 [right]. We also plot the numerical spectra in magenta lines and the LISA binned (unbinned) PLS in black (orange) lines. shapes. In order to propose accurate templates5 , and to understand how they arise in our models, as well as to make contact with the types of feature produced in inflationary models reviewed in the Introduction and in Section 3, we now divide our results in four classes and discuss them in turn. • Bump feature. In certain situations, when the mass of the massive field (and the frequency of its…

**Figure 11.** Primordial scalar power spectrum [left] and SGWB energy density [right] for the bump feature produced in Model III (blue line in

**Figure 12.** Primordial scalar power spectrum [left] and SGWB energy density [right] for the sinusoidal feature produced in Model III (light green line in

**Figure 13.** Primordial scalar power spectrum [left] and SGWB energy density [right] for the resonant feature 1 produced in Model I (purple line in

**Figure 14.** Primordial scalar power spectrum [left] and SGWB energy density [right] for the resonant feature 2 produced in Model II (blue line in

**Figure 15.** We plot our templates explicitly showing the contribution of each of the domains in which they are divided. • resonant feature 1. The parameters that we use to reproduce

Figures reproduced from arXiv:2012.05821 by the authors.

References

Works cited (149)

[47]

M. Braglia, D. K. Hazra, F. Finelli, G. F. Smoot, L. Sriramkumar and A. A. Starobinsky, Generating PBHs and small-scale GWs in two-ﬁeld models of inﬂation , JCAP 08 (2020) 001 [2005.02895]

View on Pith Pith review arXiv 2020
[145]

J. Fumagalli, S. Renaux-Petel and L. T. Witkowski, Oscillations in the stochastic gravitational wave background from sharp features and particle production during inﬂation , 2012.02761

View on Pith Pith review arXiv 2012
[1]

A. A. Starobinsky, Spectrum of adiabatic perturbations in the universe when there are singularities in the inﬂation potential , JETP Lett. 55 (1992) 489

View on Pith 1992
[2]

X. Chen, R. Easther and E. A. Lim, Generation and Characterization of Large Non-Gaussianities in Single Field Inﬂation , JCAP 04 (2008) 010 [ 0801.3295]

View on Pith Pith review arXiv 2008
[3]

J. A. Adams, B. Cresswell and R. Easther, Inﬂationary perturbations from a potential with a step, Phys. Rev. D 64 (2001) 123514 [ astro-ph/0102236]

View on Pith Pith review arXiv 2001
[4]

X. Chen, R. Easther and E. A. Lim, Large Non-Gaussianities in Single Field Inﬂation , JCAP 06 (2007) 023 [ astro-ph/0611645]

View on Pith Pith review arXiv 2007
[5]

D. K. Hazra, M. Aich, R. K. Jain, L. Sriramkumar and T. Souradeep, Primordial features due to a step in the inﬂaton potential , JCAP 10 (2010) 008 [ 1005.2175]

View on Pith Pith review arXiv 2010
[6]

P. Adshead, C. Dvorkin, W. Hu and E. A. Lim, Non-Gaussianity from Step Features in the Inﬂationary Potential, Phys. Rev. D 85 (2012) 023531 [ 1110.3050]

View on Pith Pith review arXiv 2012
[7]

D. K. Hazra, A. Shaﬁeloo, G. F. Smoot and A. A. Starobinsky, Wiggly Whipped Inﬂation, JCAP 08 (2014) 048 [ 1405.2012]

View on Pith Pith review arXiv 2014
[8]

R. Bean, X. Chen, G. Hailu, S.-H. Tye and J. Xu, Duality Cascade in Brane Inﬂation , JCAP 03 (2008) 026 [ 0802.0491]

View on Pith Pith review arXiv 2008
[9]

V. Miranda, W. Hu and P. Adshead, Warp Features in DBI Inﬂation , Phys. Rev. D 86 (2012) 063529 [1207.2186]

View on Pith Pith review arXiv 2012
[10]

A. Achucarro, J.-O. Gong, S. Hardeman, G. A. Palma and S. P. Patil, Features of heavy physics in the CMB power spectrum , JCAP 01 (2011) 030 [ 1010.3693]

View on Pith Pith review arXiv 2011
[11]

X. Chen, Primordial Features as Evidence for Inﬂation , JCAP 01 (2012) 038 [ 1104.1323]

View on Pith Pith review arXiv 2012
[12]

X. Gao, D. Langlois and S. Mizuno, Inﬂuence of heavy modes on perturbations in multiple ﬁeld inﬂation, JCAP 10 (2012) 040 [ 1205.5275]

View on Pith Pith review arXiv 2012
[13]

X. Chen and M. H. Namjoo, Standard Clock in Primordial Density Perturbations and Cosmic Microwave Background, Phys. Lett. B 739 (2014) 285 [ 1404.1536]

View on Pith Pith review arXiv 2014
[14]

X. Chen, M. H. Namjoo and Y. Wang, Models of the Primordial Standard Clock , JCAP 1502 (2015) 027 [ 1411.2349]

View on Pith Pith review arXiv 2015
[15]

R. Flauger, L. McAllister, E. Pajer, A. Westphal and G. Xu, Oscillations in the CMB from Axion Monodromy Inﬂation, JCAP 06 (2010) 009 [ 0907.2916]

View on Pith Pith review arXiv 2010
[16]

R. Flauger and E. Pajer, Resonant Non-Gaussianity, JCAP 01 (2011) 017 [ 1002.0833]

View on Pith Pith review arXiv 2011
[17]

X. Chen, Folded Resonant Non-Gaussianity in General Single Field Inﬂation , JCAP 12 (2010) 003 [ 1008.2485]

View on Pith Pith review arXiv 2010
[18]

M. Aich, D. K. Hazra, L. Sriramkumar and T. Souradeep, Oscillations in the inﬂaton potential: Complete numerical treatment and comparison with the recent and forthcoming CMB datasets, Phys. Rev. D 87 (2013) 083526 [ 1106.2798]

View on Pith Pith review arXiv 2013
[19]

X. Chen, Fingerprints of Primordial Universe Paradigms as Features in Density Perturbations, Phys. Lett. B 706 (2011) 111 [ 1106.1635]

View on Pith Pith review arXiv 2011
[20]

X. Chen, Primordial Non-Gaussianities from Inﬂation Models , Adv. Astron. 2010 (2010) 638979 [1002.1416]

View on Pith Pith review arXiv 2010
[21]

J. Chluba, J. Hamann and S. P. Patil, Features and New Physical Scales in Primordial Observables: Theory and Observation , Int. J. Mod. Phys. D 24 (2015) 1530023 [ 1505.01834]

View on Pith Pith review arXiv 2015
[22]

A. Slosar et al., Scratches from the Past: Inﬂationary Archaeology through Features in the Power Spectrum of Primordial Fluctuations , 1903.09883. – 28 –

View on Pith Pith review arXiv 1903
[23]

Planck collaboration, Y. Akrami et al., Planck 2018 results. X. Constraints on inﬂation , 1807.06211

View on Pith Pith review arXiv 2018
[24]

D. K. Hazra, D. Paoletti, M. Ballardini, F. Finelli, A. Shaﬁeloo, G. F. Smoot et al., Probing features in inﬂaton potential and reionization history with future CMB space observations , JCAP 02 (2018) 017 [ 1710.01205]

View on Pith Pith review arXiv 2018
[25]

Z. Huang, L. Verde and F. Vernizzi, Constraining inﬂation with future galaxy redshift surveys , JCAP 04 (2012) 005 [ 1201.5955]

View on Pith Pith review arXiv 2012
[26]

D. K. Hazra, Changes in the halo formation rates due to features in the primordial spectrum , JCAP 03 (2013) 003 [ 1210.7170]

View on Pith Pith review arXiv 2013
[27]

X. Chen, C. Dvorkin, Z. Huang, M. H. Namjoo and L. Verde, The Future of Primordial Features with Large-Scale Structure Surveys, JCAP 11 (2016) 014 [ 1605.09365]

View on Pith Pith review arXiv 2016
[28]

M. Ballardini, F. Finelli, C. Fedeli and L. Moscardini, Probing primordial features with future galaxy surveys, JCAP 10 (2016) 041 [ 1606.03747]

View on Pith Pith review arXiv 2016
[29]

G. A. Palma, D. Sapone and S. Sypsas, Constraints on inﬂation with LSS surveys: features in the primordial power spectrum, JCAP 06 (2018) 004 [ 1710.02570]

View on Pith Pith review arXiv 2018
[30]

B. L’Huillier, A. Shaﬁeloo, D. K. Hazra, G. F. Smoot and A. A. Starobinsky, Probing features in the primordial perturbation spectrum with large-scale structure data , Mon. Not. Roy. Astron. Soc. 477 (2018) 2503 [ 1710.10987]

View on Pith Pith review arXiv 2018
[31]

M. Ballardini, F. Finelli, R. Maartens and L. Moscardini, Probing primordial features with next-generation photometric and radio surveys , JCAP 04 (2018) 044 [ 1712.07425]

View on Pith Pith review arXiv 2018
[32]

F. Beutler, M. Biagetti, D. Green, A. Slosar and B. Wallisch, Primordial Features from Linear to Nonlinear Scales, Phys. Rev. Res. 1 (2019) 033209 [ 1906.08758]

View on Pith Pith review arXiv 2019
[33]

M. Ballardini, R. Murgia, M. Baldi, F. Finelli and M. Viel, Non-linear damping of superimposed primordial oscillations on the matter power spectrum in galaxy surveys , JCAP 04 (2020) 030 [ 1912.12499]

View on Pith Pith review arXiv 2020
[34]

I. Debono, D. K. Hazra, A. Shaﬁeloo, G. F. Smoot and A. A. Starobinsky, Constraints on features in the inﬂationary potential from future Euclid data , Mon. Not. Roy. Astron. Soc. 496 (2020) 3448 [ 2003.05262]

View on Pith Pith review arXiv 2020
[35]

X. Chen, P. Meerburg and M. M¨ unchmeyer,The Future of Primordial Features with 21 cm Tomography, JCAP 09 (2016) 023 [ 1605.09364]

View on Pith Pith review arXiv 2016
[36]

Y. Xu, J. Hamann and X. Chen, Precise measurements of inﬂationary features with 21 cm observations, Phys. Rev. D 94 (2016) 123518 [ 1607.00817]

View on Pith Pith review arXiv 2016
[37]

S. Matarrese, O. Pantano and D. Saez, General relativistic dynamics of irrotational dust: Cosmological implications, Phys. Rev. Lett. 72 (1994) 320 [ astro-ph/9310036]

View on Pith Pith review arXiv 1994
[38]

S. Matarrese, S. Mollerach and M. Bruni, Second order perturbations of the Einstein-de Sitter universe, Phys. Rev. D58 (1998) 043504 [ astro-ph/9707278]

View on Pith Pith review arXiv 1998
[39]

V. Acquaviva, N. Bartolo, S. Matarrese and A. Riotto, Second order cosmological perturbations from inﬂation, Nucl. Phys. B667 (2003) 119 [ astro-ph/0209156]

View on Pith Pith review arXiv 2003
[40]

C. Carbone and S. Matarrese, A Uniﬁed treatment of cosmological perturbations from super-horizon to small scales , Phys. Rev. D71 (2005) 043508 [ astro-ph/0407611]

View on Pith Pith review arXiv 2005
[41]

K. N. Ananda, C. Clarkson and D. Wands, The Cosmological gravitational wave background from primordial density perturbations, Phys. Rev. D 75 (2007) 123518 [ gr-qc/0612013]

View on Pith Pith review arXiv 2007
[42]

D. Baumann, P. J. Steinhardt, K. Takahashi and K. Ichiki, Gravitational Wave Spectrum Induced by Primordial Scalar Perturbations, Phys. Rev. D76 (2007) 084019 [hep-th/0703290]. – 29 –

View on Pith Pith review arXiv 2007
[43]

H. Assadullahi and D. Wands, Gravitational waves from an early matter era , Phys. Rev. D 79 (2009) 083511 [ 0901.0989]

View on Pith Pith review arXiv 2009
[44]

D. Polarski and A. A. Starobinsky, Spectra of perturbations produced by double inﬂation with an intermediate matter dominated stage , Nucl. Phys. B385 (1992) 623

View on Pith 1992
[45]

D. Polarski and A. A. Starobinsky, Isocurvature perturbations in multiple inﬂationary models , Phys. Rev. D50 (1994) 6123 [ astro-ph/9404061]

View on Pith Pith review arXiv 1994
[46]

D. Polarski and A. A. Starobinsky, Structure of primordial gravitational waves spectrum in a double inﬂationary model , Phys. Lett. B356 (1995) 196 [ astro-ph/9505125]

View on Pith Pith review arXiv 1995
[48]

S. Renaux-Petel and K. Turzy´ nski,Geometrical Destabilization of Inﬂation , Phys. Rev. Lett. 117 (2016) 141301 [ 1510.01281]

View on Pith Pith review arXiv 2016
[49]

N. Bartolo, V. De Luca, G. Franciolini, A. Lewis, M. Peloso and A. Riotto, Primordial Black Hole Dark Matter: LISA Serendipity , Phys. Rev. Lett. 122 (2019) 211301 [ 1810.12218]

View on Pith Pith review arXiv 2019
[50]

B. Carr, K. Kohri, Y. Sendouda and J. Yokoyama, Constraints on Primordial Black Holes , 2002.12778

View on Pith Pith review arXiv 2002
[51]

D. K. Hazra, L. Sriramkumar and J. Martin, BINGO: A code for the eﬃcient computation of the scalar bi-spectrum, JCAP 1305 (2013) 026 [ 1201.0926]

View on Pith Pith review arXiv 2013
[52]

S. Tsujikawa, D. Parkinson and B. A. Bassett, Correlation - consistency cartography of the double inﬂation landscape, Phys. Rev. D 67 (2003) 083516 [ astro-ph/0210322]

View on Pith Pith review arXiv 2003
[53]

Z. Lalak, D. Langlois, S. Pokorski and K. Turzynski, Curvature and isocurvature perturbations in two-ﬁeld inﬂation , JCAP 0707 (2007) 014 [ 0704.0212]

View on Pith Pith review arXiv 2007
[54]

X. Chen, M. H. Namjoo and Y. Wang, On the equation-of-motion versus in-in approach in cosmological perturbation theory, JCAP 01 (2016) 022 [ 1505.03955]

View on Pith Pith review arXiv 2016
[55]

G. A. Palma, S. Sypsas and C. Zenteno, Seeding primordial black holes in multi-ﬁeld inﬂation , 2004.06106

View on Pith Pith review arXiv 2004
[56]

J. Fumagalli, S. Renaux-Petel, J. W. Ronayne and L. T. Witkowski, Turning in the landscape: a new mechanism for generating Primordial Black Holes , 2004.08369

View on Pith Pith review arXiv 2004
[57]

J. Garcia-Bellido and E. Ruiz Morales, Primordial black holes from single ﬁeld models of inﬂation, Phys. Dark Univ. 18 (2017) 47 [ 1702.03901]

View on Pith Pith review arXiv 2017
[58]

G. Ballesteros and M. Taoso, Primordial black hole dark matter from single ﬁeld inﬂation , Phys. Rev. D 97 (2018) 023501 [ 1709.05565]

View on Pith Pith review arXiv 2018
[59]

I. Dalianis, A. Kehagias and G. Tringas, Primordial black holes from α-attractors, JCAP 01 (2019) 037 [ 1805.09483]

View on Pith Pith review arXiv 2019
[60]

I. Dalianis and K. Kritos, Exploring the Spectral Shape of Gravitational Waves Induced by Primordial Scalar Perturbations and Connection with the Primordial Black Hole Scenarios , 2007.07915

View on Pith Pith review arXiv 2007
[61]

G. Ballesteros, J. Rey, M. Taoso and A. Urbano, Primordial black holes as dark matter and gravitational waves from single-ﬁeld polynomial inﬂation , JCAP 07 (2020) 025 [ 2001.08220]

View on Pith Pith review arXiv 2020
[62]

M. Braglia, D. K. Hazra, L. Sriramkumar and F. Finelli, Generating primordial features at large scales in two ﬁeld models of inﬂation , 2004.00672

View on Pith Pith review arXiv 2004
[63]

S. Pi, Y.-l. Zhang, Q.-G. Huang and M. Sasaki, Scalaron fromR2-gravity as a heavy ﬁeld , JCAP 1805 (2018) 042 [ 1712.09896]. – 30 –

View on Pith Pith review arXiv 2018
[64]

Y. Aldabergenov, A. Addazi and S. V. Ketov, Primordial black holes from modiﬁed supergravity, Eur. Phys. J. C 80 (2020) 917 [ 2006.16641]

View on Pith Pith review arXiv 2020
[65]

D. V. Nanopoulos, V. C. Spanos and I. D. Stamou, Primordial Black Holes from No-Scale Supergravity, Phys. Rev. D 102 (2020) 083536 [ 2008.01457]

View on Pith Pith review arXiv 2020
[66]

Y. Aldabergenov, A. Addazi and S. V. Ketov, Testing Primordial Black Holes as Dark Matter in Supergravity from Gravitational Waves , 2008.10476

View on Pith Pith review arXiv 2008
[67]

A. Gundhi, S. V. Ketov and C. F. Steinwachs, Primordial black hole dark matter in dilaton-extended two-ﬁeld Starobinsky inﬂation , 2011.05999

View on Pith Pith review arXiv 2011
[68]

A. Gundhi and C. F. Steinwachs, Scalaron-Higgs inﬂation reloaded: Higgs-dependent scalaron mass and primordial black hole dark matter , 2011.09485

View on Pith Pith review arXiv 2011
[69]

M. A. Garcia, M. A. Amin and D. Green, Curvature Perturbations From Stochastic Particle Production During Inﬂation, JCAP 06 (2020) 039 [ 2001.09158]

View on Pith Pith review arXiv 2020
[70]

C. T. Byrnes, P. S. Cole and S. P. Patil, Steepest growth of the power spectrum and primordial black holes, JCAP 06 (2019) 028 [ 1811.11158]

View on Pith Pith review arXiv 2019
[71]

P. Carrilho, K. A. Malik and D. J. Mulryne, Dissecting the growth of the power spectrum for primordial black holes , Phys. Rev. D 100 (2019) 103529 [ 1907.05237]

View on Pith Pith review arXiv 2019
[72]

A. D. Gow, C. T. Byrnes, P. S. Cole and S. Young, The power spectrum on small scales: Robust constraints and comparing PBH methodologies , 2008.03289

View on Pith Pith review arXiv 2008
[73]

C. Unal, E. D. Kovetz and S. P. Patil, Multi-messenger Probes of Inﬂationary Fluctuations and Primordial Black Holes , 2008.11184

View on Pith Pith review arXiv 2008
[74]

J. R. Espinosa, D. Racco and A. Riotto, A Cosmological Signature of the SM Higgs Instability: Gravitational Waves , JCAP 1809 (2018) 012 [ 1804.07732]

View on Pith Pith review arXiv 2018
[75]

K. Kohri and T. Terada, Semianalytic calculation of gravitational wave spectrum nonlinearly induced from primordial curvature perturbations, Phys. Rev. D 97 (2018) 123532 [1804.08577]

View on Pith Pith review arXiv 2018
[76]

G. Dom` enech,Induced gravitational waves in a general cosmological background , Int. J. Mod. Phys. D 29 (2020) 2050028 [ 1912.05583]

View on Pith Pith review arXiv 2020
[77]

G. Dom` enech, S. Pi and M. Sasaki, Induced gravitational waves as a probe of thermal history of the universe , JCAP 08 (2020) 017 [ 2005.12314]

View on Pith Pith review arXiv 2020
[78]

LISA collaboration, P. Amaro-Seoane et al., Laser Interferometer Space Antenna, 1702.00786

View on Pith Pith review arXiv
[79]

K. Yagi and N. Seto, Detector conﬁguration of DECIGO/BBO and identiﬁcation of cosmological neutron-star binaries, Phys. Rev. D83 (2011) 044011 [ 1101.3940]

View on Pith Pith review arXiv 2011
[80]

C. Caprini, D. G. Figueroa, R. Flauger, G. Nardini, M. Peloso, M. Pieroni et al., Reconstructing the spectral shape of a stochastic gravitational wave background with LISA , 1906.09244

View on Pith Pith review arXiv 1906
[81]

R. Flauger, N. Karnesis, G. Nardini, M. Pieroni, A. Ricciardone and J. Torrado, Improved reconstruction of a stochastic gravitational wave background with LISA , 2009.11845

View on Pith Pith review arXiv 2009
[82]

T. L. Smith and R. Caldwell, LISA for Cosmologists: Calculating the Signal-to-Noise Ratio for Stochastic and Deterministic Sources , Phys. Rev. D 100 (2019) 104055 [ 1908.00546]

View on Pith Pith review arXiv 2019
[83]

E. Thrane and J. D. Romano, Sensitivity curves for searches for gravitational-wave backgrounds, Phys. Rev. D 88 (2013) 124032 [ 1310.5300]

View on Pith Pith review arXiv 2013
[84]

D. Alonso, C. R. Contaldi, G. Cusin, P. G. Ferreira and A. I. Renzini, Noise angular power spectrum of gravitational wave background experiments , Phys. Rev. D 101 (2020) 124048 [2005.03001]. – 31 –

View on Pith Pith review arXiv 2020
[85]

S. Pi and M. Sasaki, Gravitational Waves Induced by Scalar Perturbations with a Lognormal Peak, JCAP 09 (2020) 037 [ 2005.12306]

View on Pith Pith review arXiv 2020
[86]

NANOGrav collaboration, Z. Arzoumanian et al., The NANOGrav 12.5-year Data Set: Search For An Isotropic Stochastic Gravitational-Wave Background , 2009.04496

View on Pith Pith review arXiv 2009
[87]

V. Vaskonen and H. Veerm¨ ae,Did NANOGrav see a signal from primordial black hole formation?, 2009.07832

View on Pith Pith review arXiv 2009
[88]

K. Kohri and T. Terada, Solar-Mass Primordial Black Holes Explain NANOGrav Hint of Gravitational Waves, 2009.11853

View on Pith Pith review arXiv 2009
[89]

V. De Luca, G. Franciolini and A. Riotto, NANOGrav Hints to Primordial Black Holes as Dark Matter, 2009.08268

View on Pith Pith review arXiv 2009
[90]

C. Caprini et al., Science with the space-based interferometer eLISA. II: Gravitational waves from cosmological phase transitions, JCAP 04 (2016) 001 [ 1512.06239]

View on Pith Pith review arXiv 2016
[91]

C. Caprini and D. G. Figueroa, Cosmological Backgrounds of Gravitational Waves , Class. Quant. Grav. 35 (2018) 163001 [ 1801.04268]

View on Pith Pith review arXiv 2018
[92]

N. Bhaumik and R. K. Jain, Primordial black holes dark matter from inﬂection point models of inﬂation and the eﬀects of reheating , 1907.04125

View on Pith Pith review arXiv 1907
[93]

N. Bhaumik and R. K. Jain, Stochastic induced gravitational waves and lowest mass limit of primordial black holes with the eﬀects of reheating , 2009.10424

View on Pith Pith review arXiv 2009
[94]

H. Ragavendra, P. Saha, L. Sriramkumar and J. Silk, PBHs and secondary GWs from ultra slow roll and punctuated inﬂation , 2008.12202

View on Pith Pith review arXiv 2008
[95]

J. Lin, Q. Gao, Y. Gong, Y. Lu, C. Zhang and F. Zhang, Primordial black holes and secondary gravitational waves from k and G inﬂation, Phys. Rev. D 101 (2020) 103515 [ 2001.05909]

View on Pith Pith review arXiv 2020
[96]

Z. Yi, Q. Gao, Y. Gong and Z.-h. Zhu, Primordial black holes and secondary gravitational waves from inﬂationary model with a non-canonical kinetic term , 2011.10606

View on Pith Pith review arXiv 2011
[97]

F. Zhang, Y. Gong, J. Lin, Y. Lu and Z. Yi, Primordial Non-Gaussianity from k/G inﬂation , 2012.06960

View on Pith Pith review arXiv 2012
[98]

J. L. Cook and L. Sorbo, Particle production during inﬂation and gravitational waves detectable by ground-based interferometers, Phys. Rev. D 85 (2012) 023534 [ 1109.0022]

View on Pith Pith review arXiv 2012
[99]

N. Barnaby, E. Pajer and M. Peloso, Gauge Field Production in Axion Inﬂation: Consequences for Monodromy, non-Gaussianity in the CMB, and Gravitational Waves at Interferometers, Phys. Rev. D 85 (2012) 023525 [ 1110.3327]

View on Pith Pith review arXiv 2012
[100]

R. Namba, M. Peloso, M. Shiraishi, L. Sorbo and C. Unal, Scale-dependent gravitational waves from a rolling axion , JCAP 01 (2016) 041 [ 1509.07521]

View on Pith Pith review arXiv 2016
[101]

J. Garcia-Bellido, M. Peloso and C. Unal, Gravitational waves at interferometer scales and primordial black holes in axion inﬂation , JCAP 1612 (2016) 031 [ 1610.03763]

View on Pith Pith review arXiv 2016
[102]

O. ¨Ozsoy, Gravitational Waves from a Rolling Axion Monodromy , 2005.10280

View on Pith Pith review arXiv 2005
[103]

O. ¨Ozsoy and Z. Lalak, Primordial black holes as dark matter and gravitational waves from axion inﬂation, 2008.07549

View on Pith Pith review arXiv 2008
[104]

T. Vachaspati and A. Vilenkin, Gravitational Radiation from Cosmic Strings , Phys. Rev. D 31 (1985) 3052

View on Pith 1985
[105]

M. Sakellariadou, Gravitational waves emitted from inﬁnite strings , Phys. Rev. D 42 (1990) 354

View on Pith 1990
[106]

M. Hindmarsh, Gravitational radiation from kinky inﬁnite strings , Phys. Lett. B 251 (1990) 28. – 32 –

View on Pith 1990
[107]

C. Ringeval and T. Suyama, Stochastic gravitational waves from cosmic string loops in scaling, JCAP 12 (2017) 027 [ 1709.03845]

View on Pith Pith review arXiv 2017
[108]

Y. Matsui, K. Horiguchi, D. Nitta and S. Kuroyanagi, Gravitational wave spectrum from kinks on inﬁnite cosmic superstrings with Y-junctions , JCAP 11 (2020) 039 [ 2001.01241]

View on Pith Pith review arXiv 2020
[109]

A. Kosowsky, M. S. Turner and R. Watkins, Gravitational waves from ﬁrst order cosmological phase transitions, Phys. Rev. Lett. 69 (1992) 2026

View on Pith 1992
[110]

M. Kamionkowski, A. Kosowsky and M. S. Turner, Gravitational radiation from ﬁrst order phase transitions, Phys. Rev. D 49 (1994) 2837 [ astro-ph/9310044]

View on Pith Pith review arXiv 1994
[111]

T. Damour and A. Vilenkin, Gravitational wave bursts from cosmic strings , Phys. Rev. Lett. 85 (2000) 3761 [ gr-qc/0004075]

View on Pith Pith review arXiv 2000
[112]

S. J. Huber and T. Konstandin, Gravitational Wave Production by Collisions: More Bubbles , JCAP 09 (2008) 022 [ 0806.1828]

View on Pith Pith review arXiv 2008
[113]

T. Robson, N. J. Cornish and C. Liu, The construction and use of LISA sensitivity curves , Class. Quant. Grav. 36 (2019) 105011 [ 1803.01944]

View on Pith Pith review arXiv 2019
[114]

C. Caprini, R. Durrer, T. Konstandin and G. Servant, General Properties of the Gravitational Wave Spectrum from Phase Transitions , Phys. Rev. D 79 (2009) 083519 [ 0901.1661]

View on Pith Pith review arXiv 2009
[115]

S. Kuroyanagi, T. Chiba and T. Takahashi, Probing the Universe through the Stochastic Gravitational Wave Background, JCAP 1811 (2018) 038 [ 1807.00786]

View on Pith Pith review arXiv 2018
[116]

C. Caprini et al., Detecting gravitational waves from cosmological phase transitions with LISA: an update , JCAP 03 (2020) 024 [ 1910.13125]

View on Pith Pith review arXiv 2020
[117]

J. Garcia-Bellido and D. G. Figueroa, A stochastic background of gravitational waves from hybrid preheating, Phys. Rev. Lett. 98 (2007) 061302 [ astro-ph/0701014]

View on Pith Pith review arXiv 2007
[118]

J. F. Dufaux, A. Bergman, G. N. Felder, L. Kofman and J.-P. Uzan, Theory and Numerics of Gravitational Waves from Preheating after Inﬂation , Phys. Rev. D 76 (2007) 123517 [0707.0875]

View on Pith Pith review arXiv 2007
[119]

J. Garcia-Bellido, D. G. Figueroa and A. Sastre, A Gravitational Wave Background from Reheating after Hybrid Inﬂation , Phys. Rev. D 77 (2008) 043517 [ 0707.0839]

View on Pith Pith review arXiv 2008
[120]

J.-F. Dufaux, G. Felder, L. Kofman and O. Navros, Gravity Waves from Tachyonic Preheating after Hybrid Inﬂation , JCAP 03 (2009) 001 [ 0812.2917]

View on Pith Pith review arXiv 2009
[121]

J.-F. Dufaux, D. G. Figueroa and J. Garcia-Bellido, Gravitational Waves from Abelian Gauge Fields and Cosmic Strings at Preheating , Phys. Rev. D 82 (2010) 083518 [ 1006.0217]

View on Pith Pith review arXiv 2010
[122]

D. G. Figueroa, J. Garc´ ıa-Bellido and F. Torrent´ ı,Gravitational wave production from the decay of the standard model Higgs ﬁeld after inﬂation , Phys. Rev. D 93 (2016) 103521 [1602.03085]

View on Pith Pith review arXiv 2016
[123]

R.-G. Cai, S. Pi, S.-J. Wang and X.-Y. Yang, Resonant multiple peaks in the induced gravitational waves, JCAP 05 (2019) 013 [ 1901.10152]

View on Pith Pith review arXiv 2019
[124]

R.-G. Cai, Z.-K. Guo, J. Liu, L. Liu and X.-Y. Yang, Primordial black holes and gravitational waves from parametric ampliﬁcation of curvature perturbations , JCAP 06 (2020) 013 [1912.10437]

View on Pith Pith review arXiv 2020
[125]

S.-Y. Zhou, E. J. Copeland, R. Easther, H. Finkel, Z.-G. Mou and P. M. Saﬃn, Gravitational Waves from Oscillon Preheating , JHEP 10 (2013) 026 [ 1304.6094]

View on Pith Pith review arXiv 2013
[126]

S. Antusch, F. Cefala and S. Orani, Gravitational waves from oscillons after inﬂation , Phys. Rev. Lett. 118 (2017) 011303 [ 1607.01314]

View on Pith Pith review arXiv 2017
[127]

S. Antusch, F. Cefala and S. Orani, What can we learn from the stochastic gravitational wave background produced by oscillons?, JCAP 03 (2018) 032 [ 1712.03231]. – 33 –

View on Pith Pith review arXiv 2018
[128]

J. Liu, Z.-K. Guo, R.-G. Cai and G. Shiu, Gravitational Waves from Oscillons with Cuspy Potentials, Phys. Rev. Lett. 120 (2018) 031301 [ 1707.09841]

View on Pith Pith review arXiv 2018
[129]

M. A. Amin, J. Braden, E. J. Copeland, J. T. Giblin, C. Solorio, Z. J. Weiner et al., Gravitational waves from asymmetric oscillon dynamics? , Phys. Rev. D 98 (2018) 024040 [1803.08047]

View on Pith Pith review arXiv 2018
[130]

H. An, K.-F. Lyu, L.-T. Wang and S. Zhou, A unique gravitational wave signal from phase transition during inﬂation , 2009.12381

View on Pith Pith review arXiv 2009
[131]

A. Ashoorioon, A. Rostami and J. T. Firouzjaee, Charting the Landscape in Our Neighborhood from the PBHs Mass Distribution and GWs , 2012.02817

View on Pith Pith review arXiv 2012
[132]

S. Garcia-Saenz, S. Renaux-Petel and J. Ronayne, Primordial ﬂuctuations and non-Gaussianities in sidetracked inﬂation , JCAP 07 (2018) 057 [ 1804.11279]

View on Pith Pith review arXiv 2018
[133]

S. Garcia-Saenz, L. Pinol and S. Renaux-Petel, Revisiting non-Gaussianity in multiﬁeld inﬂation with curved ﬁeld space , JHEP 01 (2020) 073 [ 1907.10403]

View on Pith Pith review arXiv 2020
[134]

J. Fumagalli, S. Garcia-Saenz, L. Pinol, S. Renaux-Petel and J. Ronayne, Hyper-Non-Gaussianities in Inﬂation with Strongly Nongeodesic Motion , Phys. Rev. Lett. 123 (2019) 201302 [ 1902.03221]

View on Pith Pith review arXiv 2019
[135]

T. Bjorkmo and M. D. Marsh, Hyperinﬂation generalised: from its attractor mechanism to its tension with the ‘swampland conditions’ , JHEP 04 (2019) 172 [ 1901.08603]

View on Pith Pith review arXiv 2019
[136]

S. Young and C. T. Byrnes, Primordial black holes in non-Gaussian regimes , JCAP 08 (2013) 052 [1307.4995]

View on Pith Pith review arXiv 2013
[137]

J. Garcia-Bellido, M. Peloso and C. Unal, Gravitational Wave signatures of inﬂationary models from Primordial Black Hole Dark Matter , JCAP 09 (2017) 013 [ 1707.02441]

View on Pith Pith review arXiv 2017
[138]

G. Franciolini, A. Kehagias, S. Matarrese and A. Riotto, Primordial Black Holes from Inﬂation and non-Gaussianity , JCAP 03 (2018) 016 [ 1801.09415]

View on Pith Pith review arXiv 2018
[139]

V. Atal and C. Germani, The role of non-gaussianities in Primordial Black Hole formation , Phys. Dark Univ. 24 (2019) 100275 [ 1811.07857]

View on Pith Pith review arXiv 2019
[140]

V. De Luca, G. Franciolini, A. Kehagias, M. Peloso, A. Riotto and C. ¨Unal, The Ineludible non-Gaussianity of the Primordial Black Hole Abundance , JCAP 1907 (2019) 048 [1904.00970]

View on Pith Pith review arXiv 2019
[141]

C.-M. Yoo, J.-O. Gong and S. Yokoyama, Abundance of primordial black holes with local non-Gaussianity in peak theory , JCAP 09 (2019) 033 [ 1906.06790]

View on Pith Pith review arXiv 2019
[142]

J. M. Ezquiaga, J. Garc´ ıa-Bellido and V. Vennin, The exponential tail of inﬂationary ﬂuctuations: consequences for primordial black holes , JCAP 03 (2020) 029 [ 1912.05399]

View on Pith Pith review arXiv 2020
[143]

R.-g. Cai, S. Pi and M. Sasaki, Gravitational Waves Induced by non-Gaussian Scalar Perturbations, Phys. Rev. Lett. 122 (2019) 201101 [ 1810.11000]

View on Pith Pith review arXiv 2019
[144]

C. Unal, Imprints of Primordial Non-Gaussianity on Gravitational Wave Spectrum , Phys. Rev. D 99 (2019) 041301 [ 1811.09151]

View on Pith Pith review arXiv 2019
[146]

V. F. Mukhanov, H. Feldman and R. H. Brandenberger, Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part 3. Extensions , Phys. Rept. 215 (1992) 203

View on Pith 1992
[147]

C. Gordon, D. Wands, B. A. Bassett and R. Maartens, Adiabatic and entropy perturbations from inﬂation, Phys. Rev. D63 (2001) 023506 [ astro-ph/0009131]. – 34 –

View on Pith Pith review arXiv 2001
[148]

F. Di Marco, F. Finelli and R. Brandenberger, Adiabatic and isocurvature perturbations for multiﬁeld generalized Einstein models , Phys. Rev. D67 (2003) 063512 [ astro-ph/0211276]

View on Pith Pith review arXiv 2003
[149]

S. Cremonini, Z. Lalak and K. Turzynski, Strongly Coupled Perturbations in Two-Field Inﬂationary Models, JCAP 1103 (2011) 016 [ 1010.3021]. – 35 –

View on Pith Pith review arXiv 2011

Forward citations

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How later papers used it

73 later papers cite it

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Although we consider a full investigation outside the scope of this work, we have generalized some of the models in [ref] to three fields and have found that the features' behavior persists, and we see decaying oscillations in both turn rate and torsion during the trajectory's realignment.

Primordial Stochastic Gravitational Wave Backgrounds from a Sharp Feature in Three-field Inflation II: The Inflationary Era (2024) · Phenomenology · 0.99

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69 arXiv papers cite this in 112 sentences

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In [ref] it has been shown that multi-field models with strong turns in field space evade the single-field bound on the growth of $P_ζ$, and we demonstrate here that the same holds when the growth of the curvature perturbation is also due to the strong curvature of the hyperbolic field space.

Revisiting small-scale fluctuations in $\alpha$-attractor models of inflation · Robustness of single-field predictions · 0.89
Although we consider a full investigation outside the scope of this work, we have generalized some of the models in [ref] to three fields and have found that the features' behavior persists, and we see decaying oscillations in both turn rate and torsion during the trajectory's realignment.

Primordial Stochastic Gravitational Wave Backgrounds from a Sharp Feature in Three-field Inflation II: The Inflationary Era · Phenomenology · 0.99

Uses its method or tool 1 paper

We use the 2 field version of BINGO [ref] [This version, which was also used to produce the results of Refs. [ref] , is not yet publicly available.] to solve the model.

Comparing multi-field primordial feature models with the Planck data · Numerical solution · 1.00

Compares with it 1 paper

References [ref] studied several such two-field models that produce observable signals in LISA – these features typically are not one top hat as we've studied here, but a series of sharp turns with decaying amplitude, although the observed power spectra and radiation-era SGWBs (inflationary-era SGWBs are not studied in their work) are qualitatively very similar to the two-field limit of ours.

Primordial Stochastic Gravitational Wave Backgrounds from a Sharp Feature in Three-field Inflation II: The Inflationary Era · Phenomenology · 0.99

Cites it as background 27 papers

The so-called induced GWs have received a lot of attention recently [ref] .

Probing non-Gaussianities with the high frequency tail of induced gravitational waves · Introduction · 0.98
This signal can be explained by the stochastic GW with $Ω_GW∼10^-9$ at $f∼10^-8$, which can be sourced by cosmic string [ref] , phase transitions [ref] , large density fluctuations in primordial black hole (PBH) formation models [ref] and other origin [ref] .

Gravitational waves from type II axion-like curvaton model and its implication for NANOGrav result · Introduction · 0.96
In particular, there are numerous theoretical studies on the formation of PBHs from inflationary models, such as Refs. [ref] .

Exploring Critical Overdensity Thresholds in Inflationary Models of Primordial Black Holes Formation · Introduction · 0.94

What it cites · 128 sentences without a confident label

Contests 1 paper

However, as already mentioned before, the peak in the power spectrum produced in our models is different from a log-normal one, and the formulae derived in Ref. [ref] do not reproduce our results.], and to understand how they arise in our models, as well as to make contact with the types of feature produced in inflationary models reviewed in the Introduction and in Section [ref], we now divide our results in four classes and discuss them in turn. * Bump feature.

Gravitational Waves Induced by Scalar Perturbations with a Lognormal Peak · The approach of template estimators · contest 0.76

Builds on 1 paper

As a third example, we consider the following modification by changing the shape of potential in the $Θ$ direction, again introduced in Refs. [ref] : [equation] The potential in the $Θ$ direction now acquires the form of chaotic inflation, therefore inflation ends smoothly without the need for an extra hybrid field to assist it.

Models of the Primordial Standard Clock · Model III · 0.90

Uses the method or tool of 11 papers

In this paper, we make use of the full numerical pipeline, developed in Ref. [ref] , to compute the PPS, the SGWB density and the abundance of PBHs.

Generating PBHs and small-scale GWs in two-field models of inflation · A two-field model and useful formulae · 0.98
The expression for the energy density of the gravitational waves is [ref] : [equation] In the equation above, $Ω_r,0≃8.6×10^-5$ is the density of radiation today and the functions in the integral are explicitly given by [equation] where $T(k η)$ is the standard transfer function in the radiation dominated era [equation] A complete analytical expression of the functions $ℐ_c,s$ is given in Eqs. (D.1) and (D.2) of Ref. [ref] (see also Ref. [ref] ).

Semianalytic Calculation of Gravitational Wave Spectrum Nonlinearly Induced from Primordial Curvature Perturbations · Generated SGWB and detectability with future space-based interferometers · 0.89
The expression for the energy density of the gravitational waves is [ref] : [equation] In the equation above, $Ω_r,0≃8.6×10^-5$ is the density of radiation today and the functions in the integral are explicitly given by [equation] where $T(k η)$ is the standard transfer function in the radiation dominated era [equation] A complete analytical expression of the functions $ℐ_c,s$ is given in Eqs. (D.1) and (D.2) of Ref. [ref] (see also Ref. [ref] ).

A Cosmological Signature of the SM Higgs Instability: Gravitational Waves · Generated SGWB and detectability with future space-based interferometers · 0.89

Uses data or values from 1 paper

The NANOGrav result can be interpreted as a SGWB signal fitted by a power-law with amplitude at $Ω_GW(5.5 nHz) ∈[3 ×10^-10, 2 ×10^-9]$ and a spectral index $α∈[-1.5, 0.5]$ at 2$σ$ [ref] .

The NANOGrav 12.5-year Data Set: Search For An Isotropic Stochastic Gravitational-Wave Background · The approach of template estimators · 0.96

Compares with 5 papers

With more advanced techniques such as those presented in Refs. [ref] , some of the bins in Fig. [ref] could indeed be merged without affecting the reconstruction at all.

Improved reconstruction of a stochastic gravitational wave background with LISA · The approach of signal reconstruction · 0.84
With more advanced techniques such as those presented in Refs. [ref] , some of the bins in Fig. [ref] could indeed be merged without affecting the reconstruction at all.

Reconstructing the spectral shape of a stochastic gravitational wave background with LISA · The approach of signal reconstruction · 0.84
A potential very similar Eq. [ref] was also studied in Ref. [ref] where two of us presented results similar to the ones shown in this Subsection.

Generating PBHs and small-scale GWs in two-field models of inflation · Model III · 0.96

Cites as support 6 papers

Even more importantly, non-Gaussianities can significantly change the spectral shape of the SGWB [ref] possibly enhancing or washing out the oscillatory signal, which motivates further investigations in this direction.

PBHs and secondary GWs from ultra slow roll and punctuated inflation · Conclusions · 0.98
Even more importantly, non-Gaussianities can significantly change the spectral shape of the SGWB [ref] possibly enhancing or washing out the oscillatory signal, which motivates further investigations in this direction.

Gravitational Waves Induced by non-Gaussian Scalar Perturbations · Conclusions · 0.98
Even more importantly, non-Gaussianities can significantly change the spectral shape of the SGWB [ref] possibly enhancing or washing out the oscillatory signal, which motivates further investigations in this direction.

Imprints of Primordial Non-Gaussianity on Gravitational Wave Spectrum · Conclusions · 0.98

Cites as background 51 papers

Some physical mechanisms that are known to generate single or multi peaked structure of the SGWB are single-field inflationary models with a nearly inflection point [ref] or a non-canonical kinetic term [ref] , certain variants of axion inflation with explosive production of gauge fields [ref] , cosmic strings [ref] and strong first order phase transitions during the thermal era of the Universe [ref] .

Scale-dependent gravitational waves from a rolling axion · Comparison with other sources of SGWB · 0.87
Some physical mechanisms that are known to generate single or multi peaked structure of the SGWB are single-field inflationary models with a nearly inflection point [ref] or a non-canonical kinetic term [ref] , certain variants of axion inflation with explosive production of gauge fields [ref] , cosmic strings [ref] and strong first order phase transitions during the thermal era of the Universe [ref] .

Gravitational Wave Production by Collisions: More Bubbles · Comparison with other sources of SGWB · 0.87
Some physical mechanisms that are known to generate single or multi peaked structure of the SGWB are single-field inflationary models with a nearly inflection point [ref] or a non-canonical kinetic term [ref] , certain variants of axion inflation with explosive production of gauge fields [ref] , cosmic strings [ref] and strong first order phase transitions during the thermal era of the Universe [ref] .

Particle production during inflation and gravitational waves detectable by ground-based interferometers · Comparison with other sources of SGWB · 0.87

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [47]

M. Braglia, D. K. Hazra, F. Finelli, G. F. Smoot, L. Sriramkumar and A. A. Starobinsky, Generating PBHs and small-scale GWs in two-ﬁeld models of inﬂation , JCAP 08 (2020) 001 [2005.02895]

View on Pith Pith review arXiv 2020

[2] [145]

J. Fumagalli, S. Renaux-Petel and L. T. Witkowski, Oscillations in the stochastic gravitational wave background from sharp features and particle production during inﬂation , 2012.02761

View on Pith Pith review arXiv 2012

[3] [1]

A. A. Starobinsky, Spectrum of adiabatic perturbations in the universe when there are singularities in the inﬂation potential , JETP Lett. 55 (1992) 489

View on Pith 1992

[4] [2]

X. Chen, R. Easther and E. A. Lim, Generation and Characterization of Large Non-Gaussianities in Single Field Inﬂation , JCAP 04 (2008) 010 [ 0801.3295]

View on Pith Pith review arXiv 2008

[5] [3]

J. A. Adams, B. Cresswell and R. Easther, Inﬂationary perturbations from a potential with a step, Phys. Rev. D 64 (2001) 123514 [ astro-ph/0102236]

View on Pith Pith review arXiv 2001

[6] [4]

X. Chen, R. Easther and E. A. Lim, Large Non-Gaussianities in Single Field Inﬂation , JCAP 06 (2007) 023 [ astro-ph/0611645]

View on Pith Pith review arXiv 2007

[7] [5]

D. K. Hazra, M. Aich, R. K. Jain, L. Sriramkumar and T. Souradeep, Primordial features due to a step in the inﬂaton potential , JCAP 10 (2010) 008 [ 1005.2175]

View on Pith Pith review arXiv 2010

[8] [6]

P. Adshead, C. Dvorkin, W. Hu and E. A. Lim, Non-Gaussianity from Step Features in the Inﬂationary Potential, Phys. Rev. D 85 (2012) 023531 [ 1110.3050]

View on Pith Pith review arXiv 2012

[9] [7]

D. K. Hazra, A. Shaﬁeloo, G. F. Smoot and A. A. Starobinsky, Wiggly Whipped Inﬂation, JCAP 08 (2014) 048 [ 1405.2012]

View on Pith Pith review arXiv 2014

[10] [8]

R. Bean, X. Chen, G. Hailu, S.-H. Tye and J. Xu, Duality Cascade in Brane Inﬂation , JCAP 03 (2008) 026 [ 0802.0491]

View on Pith Pith review arXiv 2008

[11] [9]

V. Miranda, W. Hu and P. Adshead, Warp Features in DBI Inﬂation , Phys. Rev. D 86 (2012) 063529 [1207.2186]

View on Pith Pith review arXiv 2012

[12] [10]

A. Achucarro, J.-O. Gong, S. Hardeman, G. A. Palma and S. P. Patil, Features of heavy physics in the CMB power spectrum , JCAP 01 (2011) 030 [ 1010.3693]

View on Pith Pith review arXiv 2011

[13] [11]

X. Chen, Primordial Features as Evidence for Inﬂation , JCAP 01 (2012) 038 [ 1104.1323]

View on Pith Pith review arXiv 2012

[14] [12]

X. Gao, D. Langlois and S. Mizuno, Inﬂuence of heavy modes on perturbations in multiple ﬁeld inﬂation, JCAP 10 (2012) 040 [ 1205.5275]

View on Pith Pith review arXiv 2012

[15] [13]

X. Chen and M. H. Namjoo, Standard Clock in Primordial Density Perturbations and Cosmic Microwave Background, Phys. Lett. B 739 (2014) 285 [ 1404.1536]

View on Pith Pith review arXiv 2014

[16] [14]

X. Chen, M. H. Namjoo and Y. Wang, Models of the Primordial Standard Clock , JCAP 1502 (2015) 027 [ 1411.2349]

View on Pith Pith review arXiv 2015

[17] [15]

R. Flauger, L. McAllister, E. Pajer, A. Westphal and G. Xu, Oscillations in the CMB from Axion Monodromy Inﬂation, JCAP 06 (2010) 009 [ 0907.2916]

View on Pith Pith review arXiv 2010

[18] [16]

R. Flauger and E. Pajer, Resonant Non-Gaussianity, JCAP 01 (2011) 017 [ 1002.0833]

View on Pith Pith review arXiv 2011

[19] [17]

X. Chen, Folded Resonant Non-Gaussianity in General Single Field Inﬂation , JCAP 12 (2010) 003 [ 1008.2485]

View on Pith Pith review arXiv 2010

[20] [18]

M. Aich, D. K. Hazra, L. Sriramkumar and T. Souradeep, Oscillations in the inﬂaton potential: Complete numerical treatment and comparison with the recent and forthcoming CMB datasets, Phys. Rev. D 87 (2013) 083526 [ 1106.2798]

View on Pith Pith review arXiv 2013

[21] [19]

X. Chen, Fingerprints of Primordial Universe Paradigms as Features in Density Perturbations, Phys. Lett. B 706 (2011) 111 [ 1106.1635]

View on Pith Pith review arXiv 2011

[22] [20]

X. Chen, Primordial Non-Gaussianities from Inﬂation Models , Adv. Astron. 2010 (2010) 638979 [1002.1416]

View on Pith Pith review arXiv 2010

[23] [21]

J. Chluba, J. Hamann and S. P. Patil, Features and New Physical Scales in Primordial Observables: Theory and Observation , Int. J. Mod. Phys. D 24 (2015) 1530023 [ 1505.01834]

View on Pith Pith review arXiv 2015

[24] [22]

A. Slosar et al., Scratches from the Past: Inﬂationary Archaeology through Features in the Power Spectrum of Primordial Fluctuations , 1903.09883. – 28 –

View on Pith Pith review arXiv 1903

[25] [23]

Planck collaboration, Y. Akrami et al., Planck 2018 results. X. Constraints on inﬂation , 1807.06211

View on Pith Pith review arXiv 2018

[26] [24]

D. K. Hazra, D. Paoletti, M. Ballardini, F. Finelli, A. Shaﬁeloo, G. F. Smoot et al., Probing features in inﬂaton potential and reionization history with future CMB space observations , JCAP 02 (2018) 017 [ 1710.01205]

View on Pith Pith review arXiv 2018

[27] [25]

Z. Huang, L. Verde and F. Vernizzi, Constraining inﬂation with future galaxy redshift surveys , JCAP 04 (2012) 005 [ 1201.5955]

View on Pith Pith review arXiv 2012

[28] [26]

D. K. Hazra, Changes in the halo formation rates due to features in the primordial spectrum , JCAP 03 (2013) 003 [ 1210.7170]

View on Pith Pith review arXiv 2013

[29] [27]

X. Chen, C. Dvorkin, Z. Huang, M. H. Namjoo and L. Verde, The Future of Primordial Features with Large-Scale Structure Surveys, JCAP 11 (2016) 014 [ 1605.09365]

View on Pith Pith review arXiv 2016

[30] [28]

M. Ballardini, F. Finelli, C. Fedeli and L. Moscardini, Probing primordial features with future galaxy surveys, JCAP 10 (2016) 041 [ 1606.03747]

View on Pith Pith review arXiv 2016

[31] [29]

G. A. Palma, D. Sapone and S. Sypsas, Constraints on inﬂation with LSS surveys: features in the primordial power spectrum, JCAP 06 (2018) 004 [ 1710.02570]

View on Pith Pith review arXiv 2018

[32] [30]

B. L’Huillier, A. Shaﬁeloo, D. K. Hazra, G. F. Smoot and A. A. Starobinsky, Probing features in the primordial perturbation spectrum with large-scale structure data , Mon. Not. Roy. Astron. Soc. 477 (2018) 2503 [ 1710.10987]

View on Pith Pith review arXiv 2018

[33] [31]

M. Ballardini, F. Finelli, R. Maartens and L. Moscardini, Probing primordial features with next-generation photometric and radio surveys , JCAP 04 (2018) 044 [ 1712.07425]

View on Pith Pith review arXiv 2018

[34] [32]

F. Beutler, M. Biagetti, D. Green, A. Slosar and B. Wallisch, Primordial Features from Linear to Nonlinear Scales, Phys. Rev. Res. 1 (2019) 033209 [ 1906.08758]

View on Pith Pith review arXiv 2019

[35] [33]

M. Ballardini, R. Murgia, M. Baldi, F. Finelli and M. Viel, Non-linear damping of superimposed primordial oscillations on the matter power spectrum in galaxy surveys , JCAP 04 (2020) 030 [ 1912.12499]

View on Pith Pith review arXiv 2020

[36] [34]

I. Debono, D. K. Hazra, A. Shaﬁeloo, G. F. Smoot and A. A. Starobinsky, Constraints on features in the inﬂationary potential from future Euclid data , Mon. Not. Roy. Astron. Soc. 496 (2020) 3448 [ 2003.05262]

View on Pith Pith review arXiv 2020

[37] [35]

X. Chen, P. Meerburg and M. M¨ unchmeyer,The Future of Primordial Features with 21 cm Tomography, JCAP 09 (2016) 023 [ 1605.09364]

View on Pith Pith review arXiv 2016

[38] [36]

Y. Xu, J. Hamann and X. Chen, Precise measurements of inﬂationary features with 21 cm observations, Phys. Rev. D 94 (2016) 123518 [ 1607.00817]

View on Pith Pith review arXiv 2016

[39] [37]

S. Matarrese, O. Pantano and D. Saez, General relativistic dynamics of irrotational dust: Cosmological implications, Phys. Rev. Lett. 72 (1994) 320 [ astro-ph/9310036]

View on Pith Pith review arXiv 1994

[40] [38]

S. Matarrese, S. Mollerach and M. Bruni, Second order perturbations of the Einstein-de Sitter universe, Phys. Rev. D58 (1998) 043504 [ astro-ph/9707278]

View on Pith Pith review arXiv 1998

[41] [39]

V. Acquaviva, N. Bartolo, S. Matarrese and A. Riotto, Second order cosmological perturbations from inﬂation, Nucl. Phys. B667 (2003) 119 [ astro-ph/0209156]

View on Pith Pith review arXiv 2003

[42] [40]

C. Carbone and S. Matarrese, A Uniﬁed treatment of cosmological perturbations from super-horizon to small scales , Phys. Rev. D71 (2005) 043508 [ astro-ph/0407611]

View on Pith Pith review arXiv 2005

[43] [41]

K. N. Ananda, C. Clarkson and D. Wands, The Cosmological gravitational wave background from primordial density perturbations, Phys. Rev. D 75 (2007) 123518 [ gr-qc/0612013]

View on Pith Pith review arXiv 2007

[44] [42]

D. Baumann, P. J. Steinhardt, K. Takahashi and K. Ichiki, Gravitational Wave Spectrum Induced by Primordial Scalar Perturbations, Phys. Rev. D76 (2007) 084019 [hep-th/0703290]. – 29 –

View on Pith Pith review arXiv 2007

[45] [43]

H. Assadullahi and D. Wands, Gravitational waves from an early matter era , Phys. Rev. D 79 (2009) 083511 [ 0901.0989]

View on Pith Pith review arXiv 2009

[46] [44]

D. Polarski and A. A. Starobinsky, Spectra of perturbations produced by double inﬂation with an intermediate matter dominated stage , Nucl. Phys. B385 (1992) 623

View on Pith 1992

[47] [45]

D. Polarski and A. A. Starobinsky, Isocurvature perturbations in multiple inﬂationary models , Phys. Rev. D50 (1994) 6123 [ astro-ph/9404061]

View on Pith Pith review arXiv 1994

[48] [46]

D. Polarski and A. A. Starobinsky, Structure of primordial gravitational waves spectrum in a double inﬂationary model , Phys. Lett. B356 (1995) 196 [ astro-ph/9505125]

View on Pith Pith review arXiv 1995

[49] [48]

S. Renaux-Petel and K. Turzy´ nski,Geometrical Destabilization of Inﬂation , Phys. Rev. Lett. 117 (2016) 141301 [ 1510.01281]

View on Pith Pith review arXiv 2016

[50] [49]

N. Bartolo, V. De Luca, G. Franciolini, A. Lewis, M. Peloso and A. Riotto, Primordial Black Hole Dark Matter: LISA Serendipity , Phys. Rev. Lett. 122 (2019) 211301 [ 1810.12218]

View on Pith Pith review arXiv 2019

[51] [50]

B. Carr, K. Kohri, Y. Sendouda and J. Yokoyama, Constraints on Primordial Black Holes , 2002.12778

View on Pith Pith review arXiv 2002

[52] [51]

D. K. Hazra, L. Sriramkumar and J. Martin, BINGO: A code for the eﬃcient computation of the scalar bi-spectrum, JCAP 1305 (2013) 026 [ 1201.0926]

View on Pith Pith review arXiv 2013

[53] [52]

S. Tsujikawa, D. Parkinson and B. A. Bassett, Correlation - consistency cartography of the double inﬂation landscape, Phys. Rev. D 67 (2003) 083516 [ astro-ph/0210322]

View on Pith Pith review arXiv 2003

[54] [53]

Z. Lalak, D. Langlois, S. Pokorski and K. Turzynski, Curvature and isocurvature perturbations in two-ﬁeld inﬂation , JCAP 0707 (2007) 014 [ 0704.0212]

View on Pith Pith review arXiv 2007

[55] [54]

X. Chen, M. H. Namjoo and Y. Wang, On the equation-of-motion versus in-in approach in cosmological perturbation theory, JCAP 01 (2016) 022 [ 1505.03955]

View on Pith Pith review arXiv 2016

[56] [55]

G. A. Palma, S. Sypsas and C. Zenteno, Seeding primordial black holes in multi-ﬁeld inﬂation , 2004.06106

View on Pith Pith review arXiv 2004

[57] [56]

J. Fumagalli, S. Renaux-Petel, J. W. Ronayne and L. T. Witkowski, Turning in the landscape: a new mechanism for generating Primordial Black Holes , 2004.08369

View on Pith Pith review arXiv 2004

[58] [57]

J. Garcia-Bellido and E. Ruiz Morales, Primordial black holes from single ﬁeld models of inﬂation, Phys. Dark Univ. 18 (2017) 47 [ 1702.03901]

View on Pith Pith review arXiv 2017

[59] [58]

G. Ballesteros and M. Taoso, Primordial black hole dark matter from single ﬁeld inﬂation , Phys. Rev. D 97 (2018) 023501 [ 1709.05565]

View on Pith Pith review arXiv 2018

[60] [59]

I. Dalianis, A. Kehagias and G. Tringas, Primordial black holes from α-attractors, JCAP 01 (2019) 037 [ 1805.09483]

View on Pith Pith review arXiv 2019

[61] [60]

I. Dalianis and K. Kritos, Exploring the Spectral Shape of Gravitational Waves Induced by Primordial Scalar Perturbations and Connection with the Primordial Black Hole Scenarios , 2007.07915

View on Pith Pith review arXiv 2007

[62] [61]

G. Ballesteros, J. Rey, M. Taoso and A. Urbano, Primordial black holes as dark matter and gravitational waves from single-ﬁeld polynomial inﬂation , JCAP 07 (2020) 025 [ 2001.08220]

View on Pith Pith review arXiv 2020

[63] [62]

M. Braglia, D. K. Hazra, L. Sriramkumar and F. Finelli, Generating primordial features at large scales in two ﬁeld models of inﬂation , 2004.00672

View on Pith Pith review arXiv 2004

[64] [63]

S. Pi, Y.-l. Zhang, Q.-G. Huang and M. Sasaki, Scalaron fromR2-gravity as a heavy ﬁeld , JCAP 1805 (2018) 042 [ 1712.09896]. – 30 –

View on Pith Pith review arXiv 2018

[65] [64]

Y. Aldabergenov, A. Addazi and S. V. Ketov, Primordial black holes from modiﬁed supergravity, Eur. Phys. J. C 80 (2020) 917 [ 2006.16641]

View on Pith Pith review arXiv 2020

[66] [65]

D. V. Nanopoulos, V. C. Spanos and I. D. Stamou, Primordial Black Holes from No-Scale Supergravity, Phys. Rev. D 102 (2020) 083536 [ 2008.01457]

View on Pith Pith review arXiv 2020

[67] [66]

Y. Aldabergenov, A. Addazi and S. V. Ketov, Testing Primordial Black Holes as Dark Matter in Supergravity from Gravitational Waves , 2008.10476

View on Pith Pith review arXiv 2008

[68] [67]

A. Gundhi, S. V. Ketov and C. F. Steinwachs, Primordial black hole dark matter in dilaton-extended two-ﬁeld Starobinsky inﬂation , 2011.05999

View on Pith Pith review arXiv 2011

[69] [68]

A. Gundhi and C. F. Steinwachs, Scalaron-Higgs inﬂation reloaded: Higgs-dependent scalaron mass and primordial black hole dark matter , 2011.09485

View on Pith Pith review arXiv 2011

[70] [69]

M. A. Garcia, M. A. Amin and D. Green, Curvature Perturbations From Stochastic Particle Production During Inﬂation, JCAP 06 (2020) 039 [ 2001.09158]

View on Pith Pith review arXiv 2020

[71] [70]

C. T. Byrnes, P. S. Cole and S. P. Patil, Steepest growth of the power spectrum and primordial black holes, JCAP 06 (2019) 028 [ 1811.11158]

View on Pith Pith review arXiv 2019

[72] [71]

P. Carrilho, K. A. Malik and D. J. Mulryne, Dissecting the growth of the power spectrum for primordial black holes , Phys. Rev. D 100 (2019) 103529 [ 1907.05237]

View on Pith Pith review arXiv 2019

[73] [72]

A. D. Gow, C. T. Byrnes, P. S. Cole and S. Young, The power spectrum on small scales: Robust constraints and comparing PBH methodologies , 2008.03289

View on Pith Pith review arXiv 2008

[74] [73]

C. Unal, E. D. Kovetz and S. P. Patil, Multi-messenger Probes of Inﬂationary Fluctuations and Primordial Black Holes , 2008.11184

View on Pith Pith review arXiv 2008

[75] [74]

J. R. Espinosa, D. Racco and A. Riotto, A Cosmological Signature of the SM Higgs Instability: Gravitational Waves , JCAP 1809 (2018) 012 [ 1804.07732]

View on Pith Pith review arXiv 2018

[76] [75]

K. Kohri and T. Terada, Semianalytic calculation of gravitational wave spectrum nonlinearly induced from primordial curvature perturbations, Phys. Rev. D 97 (2018) 123532 [1804.08577]

View on Pith Pith review arXiv 2018

[77] [76]

G. Dom` enech,Induced gravitational waves in a general cosmological background , Int. J. Mod. Phys. D 29 (2020) 2050028 [ 1912.05583]

View on Pith Pith review arXiv 2020

[78] [77]

G. Dom` enech, S. Pi and M. Sasaki, Induced gravitational waves as a probe of thermal history of the universe , JCAP 08 (2020) 017 [ 2005.12314]

View on Pith Pith review arXiv 2020

[79] [78]

LISA collaboration, P. Amaro-Seoane et al., Laser Interferometer Space Antenna, 1702.00786

View on Pith Pith review arXiv

[80] [79]

K. Yagi and N. Seto, Detector conﬁguration of DECIGO/BBO and identiﬁcation of cosmological neutron-star binaries, Phys. Rev. D83 (2011) 044011 [ 1101.3940]

View on Pith Pith review arXiv 2011

[81] [80]

C. Caprini, D. G. Figueroa, R. Flauger, G. Nardini, M. Peloso, M. Pieroni et al., Reconstructing the spectral shape of a stochastic gravitational wave background with LISA , 1906.09244

View on Pith Pith review arXiv 1906

[82] [81]

R. Flauger, N. Karnesis, G. Nardini, M. Pieroni, A. Ricciardone and J. Torrado, Improved reconstruction of a stochastic gravitational wave background with LISA , 2009.11845

View on Pith Pith review arXiv 2009

[83] [82]

T. L. Smith and R. Caldwell, LISA for Cosmologists: Calculating the Signal-to-Noise Ratio for Stochastic and Deterministic Sources , Phys. Rev. D 100 (2019) 104055 [ 1908.00546]

View on Pith Pith review arXiv 2019

[84] [83]

E. Thrane and J. D. Romano, Sensitivity curves for searches for gravitational-wave backgrounds, Phys. Rev. D 88 (2013) 124032 [ 1310.5300]

View on Pith Pith review arXiv 2013

[85] [84]

D. Alonso, C. R. Contaldi, G. Cusin, P. G. Ferreira and A. I. Renzini, Noise angular power spectrum of gravitational wave background experiments , Phys. Rev. D 101 (2020) 124048 [2005.03001]. – 31 –

View on Pith Pith review arXiv 2020

[86] [85]

S. Pi and M. Sasaki, Gravitational Waves Induced by Scalar Perturbations with a Lognormal Peak, JCAP 09 (2020) 037 [ 2005.12306]

View on Pith Pith review arXiv 2020

[87] [86]

NANOGrav collaboration, Z. Arzoumanian et al., The NANOGrav 12.5-year Data Set: Search For An Isotropic Stochastic Gravitational-Wave Background , 2009.04496

View on Pith Pith review arXiv 2009

[88] [87]

V. Vaskonen and H. Veerm¨ ae,Did NANOGrav see a signal from primordial black hole formation?, 2009.07832

View on Pith Pith review arXiv 2009

[89] [88]

K. Kohri and T. Terada, Solar-Mass Primordial Black Holes Explain NANOGrav Hint of Gravitational Waves, 2009.11853

View on Pith Pith review arXiv 2009

[90] [89]

V. De Luca, G. Franciolini and A. Riotto, NANOGrav Hints to Primordial Black Holes as Dark Matter, 2009.08268

View on Pith Pith review arXiv 2009

[91] [90]

C. Caprini et al., Science with the space-based interferometer eLISA. II: Gravitational waves from cosmological phase transitions, JCAP 04 (2016) 001 [ 1512.06239]

View on Pith Pith review arXiv 2016

[92] [91]

C. Caprini and D. G. Figueroa, Cosmological Backgrounds of Gravitational Waves , Class. Quant. Grav. 35 (2018) 163001 [ 1801.04268]

View on Pith Pith review arXiv 2018

[93] [92]

N. Bhaumik and R. K. Jain, Primordial black holes dark matter from inﬂection point models of inﬂation and the eﬀects of reheating , 1907.04125

View on Pith Pith review arXiv 1907

[94] [93]

N. Bhaumik and R. K. Jain, Stochastic induced gravitational waves and lowest mass limit of primordial black holes with the eﬀects of reheating , 2009.10424

View on Pith Pith review arXiv 2009

[95] [94]

H. Ragavendra, P. Saha, L. Sriramkumar and J. Silk, PBHs and secondary GWs from ultra slow roll and punctuated inﬂation , 2008.12202

View on Pith Pith review arXiv 2008

[96] [95]

J. Lin, Q. Gao, Y. Gong, Y. Lu, C. Zhang and F. Zhang, Primordial black holes and secondary gravitational waves from k and G inﬂation, Phys. Rev. D 101 (2020) 103515 [ 2001.05909]

View on Pith Pith review arXiv 2020

[97] [96]

Z. Yi, Q. Gao, Y. Gong and Z.-h. Zhu, Primordial black holes and secondary gravitational waves from inﬂationary model with a non-canonical kinetic term , 2011.10606

View on Pith Pith review arXiv 2011

[98] [97]

F. Zhang, Y. Gong, J. Lin, Y. Lu and Z. Yi, Primordial Non-Gaussianity from k/G inﬂation , 2012.06960

View on Pith Pith review arXiv 2012

[99] [98]

J. L. Cook and L. Sorbo, Particle production during inﬂation and gravitational waves detectable by ground-based interferometers, Phys. Rev. D 85 (2012) 023534 [ 1109.0022]

View on Pith Pith review arXiv 2012

[100] [99]

N. Barnaby, E. Pajer and M. Peloso, Gauge Field Production in Axion Inﬂation: Consequences for Monodromy, non-Gaussianity in the CMB, and Gravitational Waves at Interferometers, Phys. Rev. D 85 (2012) 023525 [ 1110.3327]

View on Pith Pith review arXiv 2012

[101] [100]

R. Namba, M. Peloso, M. Shiraishi, L. Sorbo and C. Unal, Scale-dependent gravitational waves from a rolling axion , JCAP 01 (2016) 041 [ 1509.07521]

View on Pith Pith review arXiv 2016

[102] [101]

J. Garcia-Bellido, M. Peloso and C. Unal, Gravitational waves at interferometer scales and primordial black holes in axion inﬂation , JCAP 1612 (2016) 031 [ 1610.03763]

View on Pith Pith review arXiv 2016

[103] [102]

O. ¨Ozsoy, Gravitational Waves from a Rolling Axion Monodromy , 2005.10280

View on Pith Pith review arXiv 2005

[104] [103]

O. ¨Ozsoy and Z. Lalak, Primordial black holes as dark matter and gravitational waves from axion inﬂation, 2008.07549

View on Pith Pith review arXiv 2008

[105] [104]

T. Vachaspati and A. Vilenkin, Gravitational Radiation from Cosmic Strings , Phys. Rev. D 31 (1985) 3052

View on Pith 1985

[106] [105]

M. Sakellariadou, Gravitational waves emitted from inﬁnite strings , Phys. Rev. D 42 (1990) 354

View on Pith 1990

[107] [106]

M. Hindmarsh, Gravitational radiation from kinky inﬁnite strings , Phys. Lett. B 251 (1990) 28. – 32 –

View on Pith 1990

[108] [107]

C. Ringeval and T. Suyama, Stochastic gravitational waves from cosmic string loops in scaling, JCAP 12 (2017) 027 [ 1709.03845]

View on Pith Pith review arXiv 2017

[109] [108]

Y. Matsui, K. Horiguchi, D. Nitta and S. Kuroyanagi, Gravitational wave spectrum from kinks on inﬁnite cosmic superstrings with Y-junctions , JCAP 11 (2020) 039 [ 2001.01241]

View on Pith Pith review arXiv 2020

[110] [109]

A. Kosowsky, M. S. Turner and R. Watkins, Gravitational waves from ﬁrst order cosmological phase transitions, Phys. Rev. Lett. 69 (1992) 2026

View on Pith 1992

[111] [110]

M. Kamionkowski, A. Kosowsky and M. S. Turner, Gravitational radiation from ﬁrst order phase transitions, Phys. Rev. D 49 (1994) 2837 [ astro-ph/9310044]

View on Pith Pith review arXiv 1994

[112] [111]

T. Damour and A. Vilenkin, Gravitational wave bursts from cosmic strings , Phys. Rev. Lett. 85 (2000) 3761 [ gr-qc/0004075]

View on Pith Pith review arXiv 2000

[113] [112]

S. J. Huber and T. Konstandin, Gravitational Wave Production by Collisions: More Bubbles , JCAP 09 (2008) 022 [ 0806.1828]

View on Pith Pith review arXiv 2008

[114] [113]

T. Robson, N. J. Cornish and C. Liu, The construction and use of LISA sensitivity curves , Class. Quant. Grav. 36 (2019) 105011 [ 1803.01944]

View on Pith Pith review arXiv 2019

[115] [114]

C. Caprini, R. Durrer, T. Konstandin and G. Servant, General Properties of the Gravitational Wave Spectrum from Phase Transitions , Phys. Rev. D 79 (2009) 083519 [ 0901.1661]

View on Pith Pith review arXiv 2009

[116] [115]

S. Kuroyanagi, T. Chiba and T. Takahashi, Probing the Universe through the Stochastic Gravitational Wave Background, JCAP 1811 (2018) 038 [ 1807.00786]

View on Pith Pith review arXiv 2018

[117] [116]

C. Caprini et al., Detecting gravitational waves from cosmological phase transitions with LISA: an update , JCAP 03 (2020) 024 [ 1910.13125]

View on Pith Pith review arXiv 2020

[118] [117]

J. Garcia-Bellido and D. G. Figueroa, A stochastic background of gravitational waves from hybrid preheating, Phys. Rev. Lett. 98 (2007) 061302 [ astro-ph/0701014]

View on Pith Pith review arXiv 2007

[119] [118]

J. F. Dufaux, A. Bergman, G. N. Felder, L. Kofman and J.-P. Uzan, Theory and Numerics of Gravitational Waves from Preheating after Inﬂation , Phys. Rev. D 76 (2007) 123517 [0707.0875]

View on Pith Pith review arXiv 2007

[120] [119]

J. Garcia-Bellido, D. G. Figueroa and A. Sastre, A Gravitational Wave Background from Reheating after Hybrid Inﬂation , Phys. Rev. D 77 (2008) 043517 [ 0707.0839]

View on Pith Pith review arXiv 2008

[121] [120]

J.-F. Dufaux, G. Felder, L. Kofman and O. Navros, Gravity Waves from Tachyonic Preheating after Hybrid Inﬂation , JCAP 03 (2009) 001 [ 0812.2917]

View on Pith Pith review arXiv 2009

[122] [121]

J.-F. Dufaux, D. G. Figueroa and J. Garcia-Bellido, Gravitational Waves from Abelian Gauge Fields and Cosmic Strings at Preheating , Phys. Rev. D 82 (2010) 083518 [ 1006.0217]

View on Pith Pith review arXiv 2010

[123] [122]

D. G. Figueroa, J. Garc´ ıa-Bellido and F. Torrent´ ı,Gravitational wave production from the decay of the standard model Higgs ﬁeld after inﬂation , Phys. Rev. D 93 (2016) 103521 [1602.03085]

View on Pith Pith review arXiv 2016

[124] [123]

R.-G. Cai, S. Pi, S.-J. Wang and X.-Y. Yang, Resonant multiple peaks in the induced gravitational waves, JCAP 05 (2019) 013 [ 1901.10152]

View on Pith Pith review arXiv 2019

[125] [124]

R.-G. Cai, Z.-K. Guo, J. Liu, L. Liu and X.-Y. Yang, Primordial black holes and gravitational waves from parametric ampliﬁcation of curvature perturbations , JCAP 06 (2020) 013 [1912.10437]

View on Pith Pith review arXiv 2020

[126] [125]

S.-Y. Zhou, E. J. Copeland, R. Easther, H. Finkel, Z.-G. Mou and P. M. Saﬃn, Gravitational Waves from Oscillon Preheating , JHEP 10 (2013) 026 [ 1304.6094]

View on Pith Pith review arXiv 2013

[127] [126]

S. Antusch, F. Cefala and S. Orani, Gravitational waves from oscillons after inﬂation , Phys. Rev. Lett. 118 (2017) 011303 [ 1607.01314]

View on Pith Pith review arXiv 2017

[128] [127]

S. Antusch, F. Cefala and S. Orani, What can we learn from the stochastic gravitational wave background produced by oscillons?, JCAP 03 (2018) 032 [ 1712.03231]. – 33 –

View on Pith Pith review arXiv 2018

[129] [128]

J. Liu, Z.-K. Guo, R.-G. Cai and G. Shiu, Gravitational Waves from Oscillons with Cuspy Potentials, Phys. Rev. Lett. 120 (2018) 031301 [ 1707.09841]

View on Pith Pith review arXiv 2018

[130] [129]

M. A. Amin, J. Braden, E. J. Copeland, J. T. Giblin, C. Solorio, Z. J. Weiner et al., Gravitational waves from asymmetric oscillon dynamics? , Phys. Rev. D 98 (2018) 024040 [1803.08047]

View on Pith Pith review arXiv 2018

[131] [130]

H. An, K.-F. Lyu, L.-T. Wang and S. Zhou, A unique gravitational wave signal from phase transition during inﬂation , 2009.12381

View on Pith Pith review arXiv 2009

[132] [131]

A. Ashoorioon, A. Rostami and J. T. Firouzjaee, Charting the Landscape in Our Neighborhood from the PBHs Mass Distribution and GWs , 2012.02817

View on Pith Pith review arXiv 2012

[133] [132]

S. Garcia-Saenz, S. Renaux-Petel and J. Ronayne, Primordial ﬂuctuations and non-Gaussianities in sidetracked inﬂation , JCAP 07 (2018) 057 [ 1804.11279]

View on Pith Pith review arXiv 2018

[134] [133]

S. Garcia-Saenz, L. Pinol and S. Renaux-Petel, Revisiting non-Gaussianity in multiﬁeld inﬂation with curved ﬁeld space , JHEP 01 (2020) 073 [ 1907.10403]

View on Pith Pith review arXiv 2020

[135] [134]

J. Fumagalli, S. Garcia-Saenz, L. Pinol, S. Renaux-Petel and J. Ronayne, Hyper-Non-Gaussianities in Inﬂation with Strongly Nongeodesic Motion , Phys. Rev. Lett. 123 (2019) 201302 [ 1902.03221]

View on Pith Pith review arXiv 2019

[136] [135]

T. Bjorkmo and M. D. Marsh, Hyperinﬂation generalised: from its attractor mechanism to its tension with the ‘swampland conditions’ , JHEP 04 (2019) 172 [ 1901.08603]

View on Pith Pith review arXiv 2019

[137] [136]

S. Young and C. T. Byrnes, Primordial black holes in non-Gaussian regimes , JCAP 08 (2013) 052 [1307.4995]

View on Pith Pith review arXiv 2013

[138] [137]

J. Garcia-Bellido, M. Peloso and C. Unal, Gravitational Wave signatures of inﬂationary models from Primordial Black Hole Dark Matter , JCAP 09 (2017) 013 [ 1707.02441]

View on Pith Pith review arXiv 2017

[139] [138]

G. Franciolini, A. Kehagias, S. Matarrese and A. Riotto, Primordial Black Holes from Inﬂation and non-Gaussianity , JCAP 03 (2018) 016 [ 1801.09415]

View on Pith Pith review arXiv 2018

[140] [139]

V. Atal and C. Germani, The role of non-gaussianities in Primordial Black Hole formation , Phys. Dark Univ. 24 (2019) 100275 [ 1811.07857]

View on Pith Pith review arXiv 2019

[141] [140]

V. De Luca, G. Franciolini, A. Kehagias, M. Peloso, A. Riotto and C. ¨Unal, The Ineludible non-Gaussianity of the Primordial Black Hole Abundance , JCAP 1907 (2019) 048 [1904.00970]

View on Pith Pith review arXiv 2019

[142] [141]

C.-M. Yoo, J.-O. Gong and S. Yokoyama, Abundance of primordial black holes with local non-Gaussianity in peak theory , JCAP 09 (2019) 033 [ 1906.06790]

View on Pith Pith review arXiv 2019

[143] [142]

J. M. Ezquiaga, J. Garc´ ıa-Bellido and V. Vennin, The exponential tail of inﬂationary ﬂuctuations: consequences for primordial black holes , JCAP 03 (2020) 029 [ 1912.05399]

View on Pith Pith review arXiv 2020

[144] [143]

R.-g. Cai, S. Pi and M. Sasaki, Gravitational Waves Induced by non-Gaussian Scalar Perturbations, Phys. Rev. Lett. 122 (2019) 201101 [ 1810.11000]

View on Pith Pith review arXiv 2019

[145] [144]

C. Unal, Imprints of Primordial Non-Gaussianity on Gravitational Wave Spectrum , Phys. Rev. D 99 (2019) 041301 [ 1811.09151]

View on Pith Pith review arXiv 2019

[146] [146]

V. F. Mukhanov, H. Feldman and R. H. Brandenberger, Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part 3. Extensions , Phys. Rept. 215 (1992) 203

View on Pith 1992

[147] [147]

C. Gordon, D. Wands, B. A. Bassett and R. Maartens, Adiabatic and entropy perturbations from inﬂation, Phys. Rev. D63 (2001) 023506 [ astro-ph/0009131]. – 34 –

View on Pith Pith review arXiv 2001

[148] [148]

F. Di Marco, F. Finelli and R. Brandenberger, Adiabatic and isocurvature perturbations for multiﬁeld generalized Einstein models , Phys. Rev. D67 (2003) 063512 [ astro-ph/0211276]

View on Pith Pith review arXiv 2003

[149] [149]

S. Cremonini, Z. Lalak and K. Turzynski, Strongly Coupled Perturbations in Two-Field Inﬂationary Models, JCAP 1103 (2011) 016 [ 1010.3021]. – 35 –

View on Pith Pith review arXiv 2011