REVIEW
Damping of Tensor Modes in Cosmology
Astrophysics · Submitted 16 Jun 2003 · 11 references
Pith's verdict: A short, correct paper that puts a parameter-free number on neutrino free-streaming damping of tensor modes — 35.6% in squared amplitude for radiation-era horizon entry — and the number has held up.
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Referee report
The paper
The paper claims
Free-streaming neutrinos damp cosmological gravitational waves: for modes entering the horizon during radiation domination, the squared tensor amplitude is 35.6% smaller than a perfect-fluid treatment predicts.
The reading
What carries the argument
The load-bearing object is the kernel $K(s) = -3$; it is the angular average of the neutrino phase-space response, and it converts the gravitational wave's past shear history into a present-day stress. Two further pieces carry the argument. First, the superhorizon conservation law proved in the Appendix: by treating the zero-wavenumber field equations as invariant under a restricted set of coordinate transformations, Weinberg shows that a tensor mode with $h_{ij}$ constant outside the horizon is a physical mode, not merely a gauge artifact, and survives even when the anisotropic stress is not negligible. That is what licenses the initial condition $h'_{ij}(0) = 0$ at the time neutrinos begin free streaming, which is what makes the damping a clean multiplicative factor rather than a functional of unknown initial data. Second, the change of variables $y = a/a_{EQ}$ and $u = 2Q(-1$; this repackages the equation into a form where the answer depends on cosmology only through where last scattering sits.
Extended reading notes
Core claim
The paper establishes that neutrino free streaming is a real, calculable source of damping for cosmological tensor modes, and that it can be captured in closed form. Starting from a phase-space Boltzmann equation for neutrinos in a perturbed metric, Weinberg obtains an analytic expression for the traceless transverse anisotropic stress, $T^{ij} = 16$; it is a memory integral over the past history of $h'_{ij}$ weighted by a kernel $K(s) = -3$; substituting this into the tensor wave equation turns it into an integro-differential equation. For modes that re-enter the horizon during radiation domination the equation simplifies, and a numerical solution shows the amplitude settling onto $A = 0.8026$ times the undamped oscillating solution, i.e. a 35.6% reduction of the squared amplitude, independent of any cosmological parameters. For modes entering later, the suppression is smaller: the ratio of damped to undamped $|g'(y_L)|^2$ runs from 0.893 at $1$; the paper also notes that at large $Q$ the ratio of two oscillating functions produces narrow spikes at the zeros of the undamped derivative, which are artifacts of comparing nearly out-of-phase oscillations rather than physically interesting features.
Load-bearing premise
The result rests on setting the time derivative of the gravitational-wave amplitude to zero at the moment neutrinos begin free streaming, which is justified only by the Appendix argument that the constant superhorizon tensor mode is the physical one; footnote 8 records that Misner made the opposite choice and found a decaying solution, and that a later paper left the cosmological initial condition open.
Editorial extensions
If this is right
- Every tensor contribution to the microwave-background temperature and B-mode polarization spectra is reduced by 35.6% for multipoles dominated by modes that entered the horizon during radiation domination, so forecasts and parameter fits that omit the effect overstate the tensor signal on those scales.
- At the largest angular scales the reduction is between roughly 9% and 11% for $0.10 \le \Omega_M h^2 \le 0.15$, so the correction cannot be absorbed into a single overall rescaling of the tensor spectrum.
- Because the short-wavelength suppression is independent of cosmological parameters, it can be applied as a fixed factor in existing Boltzmann codes without re-running the full Einstein-Boltzmann hierarchy.
- The Appendix establishes that tensor as well as scalar perturbations obey a superhorizon conservation law even with non-negligible anisotropic stress, extending the standard constancy argument beyond perfect fluids.
Reading between the lines
- If the 35.6% figure holds, then an inflationary tensor-to-scalar ratio $r$ inferred from B-mode data by a pipeline that ignores free streaming would be biased low relative to the true primordial value; the correction is large enough to matter for the energy-scale targets of next-generation experiments.
- The damping depends on the fraction of energy density in whatever is free-streaming, so the same kernel machinery should apply to dark radiation, sterile neutrinos, or gravitinos; a measured scale-dependence of tensor damping would in principle count relativistic species at decoupling, though the degeneracy with other parameters would need to be checked.
- The constancy of $h_{ij}$ outside the horizon is what makes the answer robust, so any pre-decoupling epoch with non-negligible anisotropic stress, or any non-adiabatic tensor component, would break the clean multiplicative factor and turn the suppression into an initial-condition-dependent one.
- Photons also carry anisotropic stress before recombination, and the paper sets that contribution aside as smaller; a joint neutrino-plus-photon treatment would push the total damping modestly beyond 35.6% at the scales where both are active.
Original abstract
An analytic formula is given for the traceless transverse part of the anisotropic stress tensor due to free streaming neutrinos, and used to derive an integro-differential equation for the propagation of cosmological gravitational waves. The solution shows that anisotropic stress reduces the squared amplitude by 35.6 % for wavelengths that enter the horizon during the radiation-dominated phase, independent of any cosmological parameters. This decreases the tensor temperature and polarization correlation functions for these wavelengths by the same amount. The effect is less for wavelengths that enter the horizon at later times. At the longest wavelengths the decrease in the tensor correlation functions due to neutrino free streaming ranges from 10.7% for $\Omega_Mh^2=0.1$ to 9.0% for $\Omega_Mh^2=0.15$. An Appendix gives a general proof that tensor as well as scalar modes satisfy a conservation law for perturbations outside the horizon, even when the anisotropic stress tensor is not negligible.
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.
The abstract, sentence by sentence
An analytic formula is given for the traceless transverse part of the anisotropic stress tensor due to free streaming neutrinos, and used to derive an integro-differential equation for the propagation of cosmological gravitational waves. The solution shows that anisotropic stress reduces the squared amplitude by 35.6 % for wavelengths that enter the horizon during the radiation-dominated phase, independent of any cosmological parameters. This decreases the tensor temperature and polarization correlation functions for these wavelengths by the same amount. The effect is less for wavelengths that enter the horizon at later times. At the longest wavelengths the decrease in the tensor correlation functions due to neutrino free streaming ranges from 10.7% for $\Omega_Mh^2=0.1$ to 9.0% for $\Omega_Mh^2=0.15$. An Appendix gives a general proof that tensor as well as scalar modes satisfy a conservation law for perturbations outside the horizon, even when the anisotropic stress tensor is not negligible.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | An analytic formula is given for the traceless transverse part of the anisotropic stress tensor due to free streaming neutrinos, and used to derive an integro-differential equation for the propagation of cosmological gravitational waves. | method 0.54 | 0.93 | 0.07 | asserted 1.00 |
| s1 | The solution shows that anisotropic stress reduces the squared amplitude by 35.6 % for wavelengths that enter the horizon during the radiation-dominated phase, independent of any cosmological parameters. | result 1.00 | 0.93 | 0.75 | asserted 1.00 |
| s2 | This decreases the tensor temperature and polarization correlation functions for these wavelengths by the same amount. | result 0.81 | 0.89 | 0.16 | asserted 1.00 |
| s3 | The effect is less for wavelengths that enter the horizon at later times. | result 0.99 | 0.86 | 0.05 | asserted 1.00 |
| s4 | At the longest wavelengths the decrease in the tensor correlation functions due to neutrino free streaming ranges from 10.7% for $\Omega_Mh^2=0.1$ to 9.0% for $\Omega_Mh^2=0.15$. | result 1.00 | 0.92 | 0.10 | asserted 1.00 |
| s5 | An Appendix gives a general proof that tensor as well as scalar modes satisfy a conservation law for perturbations outside the horizon, even when the anisotropic stress tensor is not negligible. | contribution 0.74 | 0.89 | 0.06 | asserted 1.00 |
What was examined, and what was not
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References
How later papers used it
95 later papers cite it
2 later papers by other authors are listed below: 1 used its method or tool and 1 used its data or values.
Uses its method or tool 1 of 4 papers with this label
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In order to derive the adiabatic mode for the vector field, we follow Ref. [ref] .
Cosmological Gravitational Waves from Ultralight Vector Dark Matter (2026) · Adiabatic mode for the vector field from Weinberg's construction · 0.99
Uses its data or values 1 of 3 papers with this label
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Used this paper's fraction of the total energy density in neutrinos.
Applications of Cosmological Perturbation Theory in the Late Universe (2021) · 0.95 · paraphrased: the licence does not allow quoting
How the literature uses it
84 arXiv papers cite this in 155 sentences
Papers citing it · 120 sentences without a confident label
Contests it 1 paper
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The asymptotic results of Ref. [ref] are not applicable in this limit and a full numerical solution of Eq. ([ref]) is required for accurate results.
Relativistic cosmology and large-scale structure · Tensor perturbations · contest 0.71
Builds on it 2 papers
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Note that in the massless limit, $u_q(τ) = k τ$, from which follows that in a radiation dominated universe Eq. [ref] reduces to the previously known expression [ref] [equation]
Precision Early Universe Cosmology from Stochastic Gravitational Waves · Derivation of the damping · 0.86
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[ref] within the restricted Horndeski framework.
Damping of cosmological tensor modes in Horndeski theories after GW170817 · Introduction · 1.00
Uses its method or tool 3 papers
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[ref] , when treating relativistic neutrino gas by classical kinetic theory, the linearized collisionless Boltzmann equation gives [equation] where $ρ_ν$ indicates the neutrino energy density and $𝒦$ is the kernel [equation]
Damping of cosmological tensor modes in Horndeski theories after GW170817 · Tensor modes and first-order free-streaming neutrinos anisotropic inertia · 0.87
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We do this by using Weinberg construction of the adiabatic mode [ref] (see Sec. [ref]).
Cosmological Gravitational Waves from Ultralight Vector Dark Matter · Introduction · 1.00
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We implement this part as for an additional neutrino species, where $Π^ϕ$ arises due to the effect of free-streaming [ref] .
Converting non-relativistic dark matter to radiation · Perturbations · 0.84
Uses its data or values 2 papers
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The fraction of the total energy density in neutrinos is [ref]
Applications of Cosmological Perturbation Theory in the Late Universe · Approximating tensor perturbations with anisotropic stress · 0.95
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The fraction of the total energy density in neutrinos is [ref]
Gravitational waves in a flat radiation-matter universe including anisotropic stress · Approximating tensor perturbations with anisotropic stress · 0.97
Compares with it 4 papers
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To compare this result with [ref] , it should be noted that for left-handed relativistic neutrinos, we have $g_f = 1$.
Non-Markovian open quantum system approach to the early universe: I. Damping of gravitational waves by matter · Damping of GWs by decoupled neutrinos · 0.96
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Interestingly, as in the case of the GR result of ref [ref] , the rH framework preserves a linear dependence of $ℓ_k$ on only $𝒬_k$.
Damping of cosmological tensor modes in Horndeski theories after GW170817 · Multipole order effects · 1.00
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This is similar to what happens in the Boltzmann equation [ref] for the leading even-parity density perturbation $δn_ν+ ν(t, x, p)$: [equation] The relation between the field theory approach and the one based on kinetic theory has been recently established in [ref] , where the authors have considered the effects of triangle anomalies without any metric perturbation.
Kinetic theory and evolution of cosmological fluctuations with neutrino number asymmetry · Introduction · 0.85
Cites it as support 3 papers
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In fact, as made clear by a theorem of Steven Weinberg regarding the existence of an adiabatic solution to the perturbed Einstein equations at superhorizon scales [ref] , the initial homogeneous state of the axion is already adiabatic, and it will proceed to remain adiabatic into the regime of its DM-like behavior.
Axion Perturbations: A General Analytical Treatment · Introduction · 0.97
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This decision is based on the findings of previous investigations ( [ref] ), which have shown that the impact of these factors is negligible.
Probing the speed of scalar-induced gravitational waves with pulsar timing arrays · Introduction · 1.00
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Here, we neglect the effects of first-order gravitational waves, vector perturbations, and anisotropic stress, as previous studies ( [ref] ) have demonstrated that their contributions are small.
Confronting sound speed resonance with pulsar timing arrays · SIGWs and PBHs · 0.98
Cites it as background 12 papers
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The latter has been extensively studied in the context of neutrino's free streaming [ref] , while less attention has been devoted to shear viscosity [ref] .
Freeze-out and spectral running of primordial gravitational waves in viscous cosmology · The transfer function and the relative energy power spectrum with shear viscous damping · 0.84
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A number of works in the literature have analyzed the possibility of emergence of such phenomenon in the case of gravitational waves in GR, considering either a flat Minkowski or an expanding FLRW background [ref] .
Landau damping for gravitational waves in parity-violating theories · Introduction · 0.95
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One interesting effect is provided by free-streaming neutrinos, as proposed by Steven Weinberg in Ref. [ref] .
Birefringence in fermion-attenuated gravitational wave power spectrum · Introduction · 0.81
Discussion
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