Detectability of inflation-produced gravitational waves

Michael S. Turner (Chicago/Fermilab)

REVIEW

Detectability of inflation-produced gravitational waves

Astrophysics · Submitted 12 Jul 1996 · 25 references

Pith's verdict: A 1996 synthesis that converts slow-roll tensor predictions into a detector spec: the ~10^-15 energy-sensitivity target is the durable part, while the LISA 'two orders of magnitude' verdict rests on a citation with no authors, year, or report number.

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arXiv IDastro-ph/9607066 (version 1)

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CitationMichael S. Turner (Chicago/Fermilab). "Detectability of inflation-produced gravitational waves." arXiv:astro-ph/9607066 (1996). https://pith.science/paper/U42FEAMO

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Editor's note

read the letter

The thing to know: this is not a derivation paper, it is a conversion paper. Turner takes the standard single-field slow-roll results — Eqs. (1)-(4), the consistency relation (5), the transfer fit (6), the running (8) — and turns them into a requirement on hardware: dOmega_GW h^2/dlnk of about 10^-15 to mount a real search, with LIGO short by four orders of magnitude and LISA by two. That number is why people cite it, and it holds up. I checked the maximization in Eq. (7): N of about 33 gives n_T of about -0.026, matching the text's -0.025 and Fig. 3's T/S of about 0.18 via T/S = -7 n_T. The algebra from (1) to (9) is clean, and I found no error in the chain. Credit where due: the paper is unusually honest about its own limits. Ref. [18] quantifies the power-law extrapolation at about 33% for chaotic V = a phi^m; Fig. 4 shows the accessible T/S band widening by orders of magnitude as dn_T/dlnk varies over plus or minus 10^-3; and the closing paragraph explicitly lists bubble nucleation, pre-big-bang, and Grishchuk scenarios that could beat the estimate. The footnote to [9] restricts the treatment to single-field smooth potentials. So the extrapolation soft spot the stress-test flags is acknowledged in the text, not hidden, and should not be scored as a new finding. Where I would push: the LISA number. 'Pre-phase A Design Study for LISA' [22] carries the load-bearing 10^-13 with no authors, year, or report number, and that single citation drives the 'misses by two orders of magnitude' line in the summary. It is a real gap, though a fixable one, since the order-of-magnitude conclusion survives even if LISA's number moves by a factor of a few. The COBE normalization [14] listed as in press is a milder version of the same issue; the value is externally reproducible, so it is not load-bearing the way [22] is. Also worth noting: the paper itself says the white-dwarf binary foreground could dominate above about 10^-4 Hz [21], which makes the LISA comparison foreground-limited rather than purely instrumental. The paper mentions this but does not fold it into the verdict. One correction to the reader's report: the classifier tags this as a review with no results. That is wrong. Eq. (7) and the 10^-15 target are a quantitative result, and they are the paper's whole point. Who it is for: anyone setting sensitivity specs for CMB polarization or space interferometers, and anyone who needs the standard citation for why the tensor background is hard to reach. It deserves a serious referee; it is a PRL Letter doing what a Letter should.

Referee report

Significance

The paper's value is that it reduces a large and diffuse question to a single, falsifiable figure of merit: dΩ_GW h²/d ln k ~ 10⁻¹⁵, with a clear statement of what each observational route can and cannot reach. I checked the maximization that underlies Eq. (7) and the quoted nT ≈ −0.025 (equivalently T/S ≈ 0.18 via Eq. 5) and it is self-consistent, as is the strain-to-energy conversion in Eq. (9). The paper is also candid in ways that are unusual and useful: it cites the opposing view (ref. [8], Lyth) rather than ignoring it, it states the accuracy of its own power-law extrapolation (ref. [18], ~33% for chaotic models), it shows the ±10⁻³ running band explicitly in Fig. 4, and its closing paragraph names the non-standard scenarios that could invalidate the bottom line. The prediction is falsifiable in the strict sense: a positive detection at the quoted sensitivity, or a CBR measurement of T/S, would test it. These are real strengths and they should be weighed in the assessment.

Major objections (1)

[Direct Detection, ref. [22] and the LISA verdict] The statement that 'the design study for LISA indicates an energy sensitivity of around dΩ_GW h²/d ln k ~ 10⁻¹³, which misses by two orders of magnitude [22]' rests entirely on a citation given only as 'Pre-phase A Design Study for LISA,' with no authors, year, or report number. This single number carries one of the paper's two headline conclusions, and in its present form it cannot be checked by a reader. Please supply the full citation (authors, year, report number, and the specific sensitivity curve or table from which 10⁻¹³ is taken), and state whether that figure is broadband or refers to a particular frequency near 10⁻⁴ Hz. Relatedly, the paper itself notes 'a worrisome background of coalescing white-dwarf binaries, which could dominate inflation at frequencies greater than around 10⁻⁴ Hz [21],' but this caveat is not folded into the LISA verdict. If the effective sensitivity at 10⁻⁴ Hz is foreground-limited rather than instrument-limited, the 'two orders of magnitude' statement should be qualified accordingly, or the foreground amplitude should be quoted so the reader can judge. This is a support and framing issue rather than an internal inconsistency, and it is fixable within the Letter's scope.

Minor objections (8)

[Eq. (7), definition of N] The constant in N ≡ ln(k/H0) ≈ 33 + ln(f/10⁻⁴ Hz) + ln(0.6/h) appears inconsistent with H0 = 100 h km s⁻¹ Mpc⁻¹. Taking H0 = 3.24 × 10⁻¹⁸ h s⁻¹, I find f/H0 ≈ 3.1 × 10¹³/h at f = 10⁻⁴ Hz, so N ≈ 31.6 for h = 0.6 rather than 33. The difference is small in its effect on exp(nT N) and does not change any conclusion, but the constant should be corrected or its convention stated.
[Eq. (8), sign of the running] Please verify the sign in dnT/d ln k = −nT[(n − 1) − nT]. Evaluating the slow-roll expressions directly for V = ½m²φ², where n − 1 = −4ε and nT = −2ε, gives dnT/d ln k = +4ε², whereas the printed relation gives −4ε². Since the text describes the typical value as ≈ −10⁻³ and Fig. 4 explores both signs, the sign does not affect the conclusions, but the relation as written should be checked against the direct derivative.
[Eq. (6), transfer-function fit] The bracketed factor in Eq. (6) is described as 'a numerical fit to the transfer function for gravitational waves' without any statement of its accuracy over the relevant range of k/kEQ. A sentence giving the fit's fractional error, or the range over which it was calibrated, would let the reader judge whether it contributes to the overall uncertainty budget alongside the running.
[COBE normalization, ref. [14]] The footnote to ref. [14] states that the dependence of the COBE normalization on n and T/S 'can be ignored' for the accuracy needed here. It would strengthen the paper to quantify this: at T/S = 0.18 the scalar quadrupole is reduced by roughly 18% in amplitude relative to the pure-scalar normalization, which is a modest effect compared with the orders-of-magnitude conclusions but is larger than the ~33% power-law extrapolation error quoted in ref. [18].
[Fig. 4 and the quoted shortfalls] The sentences stating that LIGO 'misses the mark by four orders of magnitude' and that LISA 'misses by two orders of magnitude' are stated as fixed numbers, while Fig. 4 shows the accessible T/S band widening substantially as dnT/d ln k is varied over ±10⁻³. It would be more informative to quote the shortfall as a range tied to Fig. 4 and to name the assumed running at the point where the number is asserted, so that the headline claim carries its own uncertainty.
[Direct Detection, LIGO sensitivity figures] The quoted strain sensitivities (10⁻²¹ initial, 10⁻²⁴ at f = 10² Hz) should be labeled as broadband or narrowband/tuned, and the frequency at which the four-orders-of-magnitude comparison is made should be stated explicitly, since Eq. (9) shows the energy-density sensitivity scales as f² at fixed strain.
[CBR polarization discussion] The statement that polarization could reach T/S ~ 0.02 at '0.01% precision on large-angular scales' would benefit from specifying whether this is a per-multipole noise level or an aggregate over a range of l, and what sky coverage and foreground treatment are assumed, since these determine whether the quoted limit is realistic.
[Abstract] The abstract says only that 'We discuss the potential' of the two routes. Since the Letter's contribution is a specific quantitative requirement (~10⁻¹⁵) and a specific comparison with planned instruments, the abstract would serve readers better by stating that result rather than describing the paper's activity.

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The paper

The paper claims

Inflation should have filled the universe with a faint, nearly scale-free background of gravitational waves, and this paper puts a number on how hard that background is to see: an energy sensitivity near $10^{-15}$ per logarithmic frequency

The reading

Inflation makes three predictions that can be checked: a spatially flat universe, a nearly scale-invariant spectrum of density perturbations, and a nearly scale-invariant spectrum of gravitational waves. The first two are already being tested by ordinary cosmology; the third is the cleanest signature of inflation and by far the hardest to observe. Turner asks how large the gravity-wave background should be and whether any instrument then planned — CMB anisotropy and polarization satellites, the ground-based laser interferometers LIGO and VIRGO/GEO, or the space-based LISA — could actually reach it. His answer is quantitative: the background is expected at an energy density per logarithmic frequency interval near $10^{-15}$, about four orders of magnitude below the most optimistic ground-based strain sensitivity and two orders below the LISA design study. The paper's contribution is to convert the slogan 'gravity waves test inflation' into a specific number that any detector proposal can be measured against.

What carries the argument

The load-bearing device is the consistency relation $n_T = -T/S/7$, which ties the overall amplitude of the tensor spectrum to its tilt. That relation lets the author rewrite the gravity-wave energy density entirely in terms of $n_T$ alone, giving Eq. (7): a prefactor near $5.1times10^{-15}(g_*/3.36)$ multiplied by $n_T/(n_T-1/7)$ and by an exponential of $n_T N + N^2(dn_T/dln k)/2$, where $N = ln(k/H_0)$ is about 33 at $f = 10^{-4}$ Hz. Because the spectrum is anchored at the Hubble scale and carried some fifteen orders of magnitude in frequency, the exponential matters: raising $n_T$ raises the prefactor but tilts the spectrum down at high frequency, and the energy density at a fixed frequency is maximized for $n_T$ near $-0.025$, a value realized in chaotic inflation models. A second piece of machinery is sampling variance, the $1/2l+1$ floor on how well any multipole variance can be measured, which sets the CMB ceiling independently of experimental care.

Extended reading notes

Core claim

The paper is an accounting, not a detection, and its central claim is a matched pair of limits. On the CMB side, the tensor contribution to the quadrupole is limited by sampling variance: because only $2l+1$ multipole amplitudes exist at each $l$, the variance of the multipoles can be estimated only to a relative precision of $1/2l+1$, so the tensor-to-scalar ratio $T/S$ must exceed roughly 0.1 before the gravity-wave signature in anisotropy can be separated. Polarization is in principle more sensitive, reaching $T/S$ as small as 0.02, but only with roughly 0.01% precision on large angular scales, and it is strongly dependent on the ionization history. On the direct-detection side, the author normalizes the tensor spectrum to the COBE quadrupole at the Hubble scale and extrapolates it to interferometer frequencies, finding that an energy sensitivity $dOmega_{GW}h^2/dln k$ of about $10^{-15}$ is needed for a serious search. LIGO's most optimistic energy sensitivity misses that mark by four orders of magnitude, and the LISA design study, at about $10^{-13}$, misses by two. The author states the conclusion plainly and then tempers it, noting that the calculation assumes single-field slow-roll inflation with a smooth potential, and that bubble nucleation, pre-big-bang models, or other early-Universe sources could put far more gravity-wave energy in reach.

Load-bearing premise

The tensor spectrum is anchored to the COBE quadrupole at the Hubble scale and then carried about fifteen orders of magnitude in frequency by a power law with only first-order running, assuming single-field slow-roll inflation with a smooth potential; the paper itself notes in reference [18] that the approximation is good to only about 33% for chaotic models and that the accessible band of $T/S$ widens by orders of magnitude as the running is varied over $pm10^{-3}$.

Editorial extensions

If this is right

If the estimate holds, a CMB anisotropy experiment alone needs $T/S$ greater than about 0.1 to see the tensor signature, and the sampling-variance floor cannot be beaten by building a better anisotropy instrument.
Polarization could in principle reach $T/S$ near 0.02, but only by measuring large-angular-scale polarization to roughly 0.01% and only if the ionization history cooperates.
A detector reaching $dOmega_{GW}h^2/dln k$ near $10^{-15}$ at $10^{-4}$ Hz would probe $T/S$ down to about 0.01; a sensitivity of $10^{-16}$ would open the window wide.
Combining a CMB determination of $T/S$ with an interferometric determination of the average tilt would allow the running $dn_T/dln k$ to be inferred, giving a further handle on the shape of the inflationary potential.
Reconstructing the inflaton potential at all requires the gravity-wave spectrum; without it, only particular combinations of $V$ and its first derivative are accessible from density perturbations.
Additional relativistic species beyond photons and three neutrinos raise the gravity-wave energy density, perhaps by a factor of three, and so improve the prospects for direct detection.

Reading between the lines

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

The two-orders-of-magnitude LISA gap is as much a statement about the assumed potential as about LISA: because the amplitude at $10^{-4}$ Hz depends exponentially on $n_T N$ with $N$ about 33, a modest change in the running moves the target by orders of magnitude, so the verdict is conditional on smooth single-field slow roll.
The sharpest falsifiable content is the consistency relation itself. Independent CMB determinations of $T/S$ and $n_T$ that violate $n_T = -T/S/7$ would rule out the single-field slow-roll class, which is a narrower and more informative outcome than ruling out inflation.
The same calculation makes the stochastic background a probe of the relativistic degrees of freedom: a detection above the minimal prediction would count species present at early times, complementing the big-bang nucleosynthesis bound on $g_*$.
Because the tensor quadrupole fixes the height of the potential through $T = 0.61(V_*/m_{Pl}^4)$, an upper bound on $T/S$ from any experiment is directly an upper bound on the energy scale of inflation, so a null search constrains the inflaton potential rather than merely the detector.

Original abstract

Detection of the gravitational waves excited during inflation as quantum mechanical fluctuations is a key test of inflation and crucial to learning about the specifics of the inflationary model. We discuss the potential of Cosmic Background Radiation (CBR) anisotropy and polarization and of laser interferometers such as LIGO, VIRGO/GEO and LISA to detect these gravity waves.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,301 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.07 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.17 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

Detection of the gravitational waves excited during inflation as quantum mechanical fluctuations is a key test of inflation and crucial to learning about the specifics of the inflationary model. We discuss the potential of Cosmic Background Radiation (CBR) anisotropy and polarization and of laser interferometers such as LIGO, VIRGO/GEO and LISA to detect these gravity waves.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Detection of the gravitational waves excited during inflation as quantum mechanical fluctuations is a key test of inflation and crucial to learning about the specifics of the inflationary model.	background 1.00	0.04	0.12	asserted 1.00
s1	We discuss the potential of Cosmic Background Radiation (CBR) anisotropy and polarization and of laser interferometers such as LIGO, VIRGO/GEO and LISA to detect these gravity waves.	contribution 0.86	0.27	0.13	hedged possibility 0.51

What kind of paper the classifier reports this is: review 0.74 · evidence from argument 0.94.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

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

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

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.** FIG. 1. Angular power spectra (

Showing 2 of the paper's 4 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:astro-ph/9607066 by the authors.

References

Works cited (25)

[18]

Given the inﬂationary potential the gravity-wave spec - trum can be computed without assuming a power law; we have done this for chaotic inﬂation models, V (φ) = aφm, to judge the accuracy of our approximation. It is typically better than 33%. In addition, since there is no standard model of inﬂation, the use of nT and dnT /d ln k to extrap- olate the gra...

View on Pith
[9]

All formulae are given to lowest order in the deviation from scale invariance and are strictly applicable to single - ﬁeld models with smooth potentials; see A.R. Liddle and M.S. Turner, Phys. Rev. D 50, 758 (1994)

View on Pith 1994
[22]

Pre-phase A Design Study for LISA

View on Pith
[14]

K. Gorski et al., Astrophys. J. , in press (1996). The nor- malization depends upon both n and T /S; for the accu- racy needed here, this complication can be ignored

View on Pith 1996
[21]

K.S. Thorne, in 300 Years of Gravitation , eds. S.W. Hawking and W. Israel (Cambridge Univ. Press, Cambridge, 1987), p.330

View on Pith 1987
[8]

One opinion has it that inﬂation-produced gravity waves are undetectably small in any reasonable model of inﬂa- tion (D. Lyth, hep-ph/9606387)

View on Pith Pith review arXiv
[1]

See e.g., E.W. Kolb and M.S. Turner, The Early Universe (Addison-Wesley, Redwood City, CA, 1990), Ch. 8. Not all agree that Ω TOT = 1 is a robust prediction of inﬂa- tion; see e.g., M. Bucher A.S. Goldhaber, and N. Turok, Phys. Rev. D 52, 3314 (1995); J.R. Gott, Nature 295, 304 (1992)

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[2]

A. H. Guth and S.-Y. Pi, Phys. Rev. Lett. 49, 1110 (1982); S. W. Hawking, Phys. Lett. B 115, 295 (1982); A. A. Starobinskii, ibid 117, 175 (1982); J. M. Bardeen, P. J. Steinhardt, and M. S. Turner, Phys. Rev. D 28, 697 (1983)

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V.A. Rubakov, M. Sazhin, and A. Veryaskin, Phys. Lett. B 115, 189 (1982); R. Fabbri and M. Pollock, ibid 125, 445 (1983); A.A. Starobinskii, Sov. Astron. Lett. 9, 302 (1983)

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[4]

See e.g., C.J. Copi, D.N. Schramm and M.S. Turner, Sci- ence 267, 192 (1995)

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[5]

See e.g., G. Blumenthal et al., Nature 311, 517 (1984)

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[6]

S. Dodelson, E. Gates and M.S. Turner, astro- ph/9603081

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E.J. Copeland, E.W. Kolb, A.R. Liddle, and J.E. Lid- sey, Phys. Rev. Lett. 71, 219 (1993); Phys. Rev. D 48, 2529 (1993); M.S. Turner, ibid, 3502 (1993); ibid 48, 5539 (1993)

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R. Davis et al., Phys. Rev. Lett. 69, 1856 (1992); F. Lucchin, S. Mattarese, and S. Mollerach, Astrophys. J. 401, L49 (1992); D. Salopek, Phys. Rev. Lett. 69, 3602 (1992); A. Liddle and D. Lyth, Phys. Lett. B 291, 391 (1992); J.E. Lidsey and P. Coles, Mon. Not. R. astron. Soc. 258, 57p (1992); T. Souradeep and V. Sahni, Mod. Phys. Lett. A 7, 3541 (1992)

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[11]

Isotropy in the mean implies that ⟨alm⟩ = 0; the gaus- sian nature of the inﬂationary metric ﬂuctuations im- plies that the multipole amplitudes have gaussian distri- butions, fully speciﬁed by their predicted variances

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[12]

L. Knox and M.S. Turner, Phys. Rev. Lett. 73, 3347 (1994)

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M.S. Turner, J.E. Lidsey and M. White, Phys. Rev. D 48, 4613 (1993)

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S. Dodelson, G. Gyuk, and M.S. Turner, Phys. Rev. Lett 72, 3578 (1994); J.R. Bond and G. Efstathiou, Phys. Lett. B 265, 245 (1991)

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[16]

This is simple to understand: the drop in energy den- sity from the Hubble scale to the plateau is given by the redshift of matter – radiation equality, which is inversely proportional to the energy density in radiation

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[17]

A. Kosowsky and M.S. Turner, Phys. Rev. D 52, R1739 (1995)

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[19]

M.S. Turner and M. White, Phys. Rev. D 53, 6822 (1996)

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[20]

See e.g., A. Abramovici et al., Science 256, 325 (1992) and in Particle and Nuclear Astrophysics and Cosmology in the Next Millennium , eds. E.W. Kolb and R.D. Pec- cei (World Scientiﬁc, Singapore, 1995), p. 398;N. Chris- tensen, Phys. Rev. D 46, 5250 (1992); B. Allen, qr- gc/9604033

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Forward citations

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How the literature uses it

46 arXiv papers cite this in 55 sentences

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

Papers citing it · 20 sentences without a confident label

Uses its method or tool 1 paper

To convert the primordial spectrum, Eq. ([ref]), to the spectrum which would be observed today we employ the numerically obtained transfer function [ref]

The gravitational wave background from super-inflation in Loop Quantum Cosmology · The present-day spectrum · 0.97

Cites it as background 31 papers

On the other hand, superhorizon tensor perturbations form the so-called primordial gravitational waves (GWs), which result in the stochastic GW background in the present universe with various frequencies, ranging from cosmological scales to terrestrial scales [ref] .

High Frequency Spectrum of Primordial Gravitational Waves · Introduction · 0.92
Along with this background, that will be measured [ref] by the end of the mission of LVK [ref] or at the latest by future ground-based (CE, ET) [ref] and space-based (LISA, BBO, MAGIS, DECIGO) [ref] experiments, there could lie a primordial signal generated in the early cosmological history [ref] .

Precision cosmology with primordial GW backgrounds in presence of astrophysical foregrounds · Introduction · 0.82
The possible sources for this stochastic gravitational wave signal are: first order phase transitions in the early Universe after inflation [ref] , inflation itself [ref] , and topological defects (domain walls [ref] and cosmic strings [ref] ).

Testing Unification and Dark Matter with Gravitational Waves · Introduction · 0.96

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.

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Given the inﬂationary potential the gravity-wave spec - trum can be computed without assuming a power law; we have done this for chaotic inﬂation models, V (φ) = aφm, to judge the accuracy of our approximation. It is typically better than 33%. In addition, since there is no standard model of inﬂation, the use of nT and dnT /d ln k to extrap- olate the gra...

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All formulae are given to lowest order in the deviation from scale invariance and are strictly applicable to single - ﬁeld models with smooth potentials; see A.R. Liddle and M.S. Turner, Phys. Rev. D 50, 758 (1994)

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Pre-phase A Design Study for LISA

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K. Gorski et al., Astrophys. J. , in press (1996). The nor- malization depends upon both n and T /S; for the accu- racy needed here, this complication can be ignored

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K.S. Thorne, in 300 Years of Gravitation , eds. S.W. Hawking and W. Israel (Cambridge Univ. Press, Cambridge, 1987), p.330

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One opinion has it that inﬂation-produced gravity waves are undetectably small in any reasonable model of inﬂa- tion (D. Lyth, hep-ph/9606387)

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[7] [1]

See e.g., E.W. Kolb and M.S. Turner, The Early Universe (Addison-Wesley, Redwood City, CA, 1990), Ch. 8. Not all agree that Ω TOT = 1 is a robust prediction of inﬂa- tion; see e.g., M. Bucher A.S. Goldhaber, and N. Turok, Phys. Rev. D 52, 3314 (1995); J.R. Gott, Nature 295, 304 (1992)

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A. H. Guth and S.-Y. Pi, Phys. Rev. Lett. 49, 1110 (1982); S. W. Hawking, Phys. Lett. B 115, 295 (1982); A. A. Starobinskii, ibid 117, 175 (1982); J. M. Bardeen, P. J. Steinhardt, and M. S. Turner, Phys. Rev. D 28, 697 (1983)

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V.A. Rubakov, M. Sazhin, and A. Veryaskin, Phys. Lett. B 115, 189 (1982); R. Fabbri and M. Pollock, ibid 125, 445 (1983); A.A. Starobinskii, Sov. Astron. Lett. 9, 302 (1983)

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See e.g., C.J. Copi, D.N. Schramm and M.S. Turner, Sci- ence 267, 192 (1995)

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[11] [5]

See e.g., G. Blumenthal et al., Nature 311, 517 (1984)

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S. Dodelson, E. Gates and M.S. Turner, astro- ph/9603081

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E.J. Copeland, E.W. Kolb, A.R. Liddle, and J.E. Lid- sey, Phys. Rev. Lett. 71, 219 (1993); Phys. Rev. D 48, 2529 (1993); M.S. Turner, ibid, 3502 (1993); ibid 48, 5539 (1993)

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R. Davis et al., Phys. Rev. Lett. 69, 1856 (1992); F. Lucchin, S. Mattarese, and S. Mollerach, Astrophys. J. 401, L49 (1992); D. Salopek, Phys. Rev. Lett. 69, 3602 (1992); A. Liddle and D. Lyth, Phys. Lett. B 291, 391 (1992); J.E. Lidsey and P. Coles, Mon. Not. R. astron. Soc. 258, 57p (1992); T. Souradeep and V. Sahni, Mod. Phys. Lett. A 7, 3541 (1992)

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Isotropy in the mean implies that ⟨alm⟩ = 0; the gaus- sian nature of the inﬂationary metric ﬂuctuations im- plies that the multipole amplitudes have gaussian distri- butions, fully speciﬁed by their predicted variances

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[16] [12]

L. Knox and M.S. Turner, Phys. Rev. Lett. 73, 3347 (1994)

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M.S. Turner, J.E. Lidsey and M. White, Phys. Rev. D 48, 4613 (1993)

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S. Dodelson, G. Gyuk, and M.S. Turner, Phys. Rev. Lett 72, 3578 (1994); J.R. Bond and G. Efstathiou, Phys. Lett. B 265, 245 (1991)

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This is simple to understand: the drop in energy den- sity from the Hubble scale to the plateau is given by the redshift of matter – radiation equality, which is inversely proportional to the energy density in radiation

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A. Kosowsky and M.S. Turner, Phys. Rev. D 52, R1739 (1995)

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M.S. Turner and M. White, Phys. Rev. D 53, 6822 (1996)

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See e.g., A. Abramovici et al., Science 256, 325 (1992) and in Particle and Nuclear Astrophysics and Cosmology in the Next Millennium , eds. E.W. Kolb and R.D. Pec- cei (World Scientiﬁc, Singapore, 1995), p. 398;N. Chris- tensen, Phys. Rev. D 46, 5250 (1992); B. Allen, qr- gc/9604033

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A. Kosowsky, M.S. Turner, and R. Watkins, Phys. Rev. Lett. 69, 2026 (1992)

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R. Brustein, M. Gasperini, M. Giovannini, and G. Veneziano, Phys. Lett. B 361, 45 (1995)

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S.P. Grishchuk, Zh. Eksp. Teor. Fiz. 67, 825 (1974) [ Sov. Phys. JETP 40, 409 (1975)]. 4

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