Massive Neutrinos in Cosmology
Neutrinos are the only component of the cosmic inventory that we know carries rest mass and yet whose mass we cannot pin down: flavour-oscillation experiments measure the differences of squared masses with exquisite precision but say nothing about the absolute scale. Cosmology supplies what the laboratory cannot. The universe is awash in relic neutrinos — about 336 per cubic centimetre across three species, roughly 10⁹ for every baryon — from the first second after the Big Bang, and even a sub-electron-volt mass turns that vast, ghostly sea into a gravitationally active ingredient of the cosmic web whose fingerprint is measurable. The physics that makes the measurement possible is free-streaming. A relic neutrino is born relativistic and stays fast long after the rest of matter has gone cold, so it cannot be trapped by a potential well smaller than the distance it coasts in a Hubble time — the free-streaming length. Below that scale it streams out of collapsing structures and drains small-scale power, leaving a clean, calculable step in the matter power spectrum whose depth is set almost entirely by the fraction of matter locked in neutrinos, ΔP/P ≈ −8f_ν. This is the reason the sum of masses Σm_ν is one of the very few particle-physics numbers a telescope, not an accelerator, is now poised to measure — and this article assembles the full chain from thermal history and the 93.14 eV rule, through the free-streaming scale as a collisionless Jeans length, to that growth suppression.
- Relic neutrinos decouple while relativistic, freezing in a fixed comoving number density; today Ω_ν h² = Σm_ν / 93.14 eV across the three species.
- They stay hot long after other matter is cold, so they free-stream out of small-scale wells and damp matter perturbations below the free-streaming scale k_fs.
- The net imprint on the linear matter power spectrum is a step-like deficit, ΔP/P ≈ −8f_ν for k ≫ k_fs, where f_ν = Ω_ν/Ω_m is the neutrino mass fraction.
- Oscillations set a floor Σm_ν ≳ 0.06 eV (normal ordering); cosmology sets a ceiling Σm_ν ≲ 0.12 eV (95%, Planck+BAO), already squeezing the inverted ordering.
- A neutrino behaves as radiation at early times and as a subdominant hot matter today; the switch happens at z_nr ≈ 1890 (m_ν/eV), long after matter–radiation equality.
- Before going non-relativistic the three species act as radiation, contributing an effective number N_eff = 3.044.
- Key idea: relic neutrinos are hot dark matter — relativistic and free-streaming until z_nr, then a subdominant clustering matter — and their free-streaming suppresses small-scale power by ΔP/P ≈ −8f_ν, which is what makes Σm_ν measurable from the matter power spectrum.
Relic neutrinos are a hot dark matter component: relativistic and free-streaming until z_nr, then a subdominant clustering matter. Free-streaming suppresses small-scale power by ΔP/P ≈ −8f_ν, making Σm_ν measurable from the matter power spectrum.
Where this lives today
The gravitational weighing of neutrinos is now a headline goal of observational cosmology. The state of the art combines Planck CMB temperature, polarisation and lensing with baryon-acoustic-oscillation distances to reach Σm_ν < 0.12 eV at 95% confidence — a ceiling that, against the oscillation floors of 0.06 eV (normal) and 0.10 eV (inverted), already mildly disfavours the inverted hierarchy. Stage-IV surveys are built to close the gap: DESI and Euclid map the galaxy power spectrum across the scales k ≈ 0.01–1 h Mpc⁻¹ where the −8f_ν step lives, while CMB-S4 will measure the lensing spectrum C_ℓ^φφ nearly independently of galaxy data, and the Lyman-α forest reaches the smallest scales where the deficit is deepest. A joint analysis should detect even the minimal Σm_ν = 0.06 eV within the decade — the first experiment ever to measure, not merely bound, the absolute neutrino mass. The tension to watch is that some data combinations already prefer Σm_ν near zero, hinting at systematics or new physics; KATRIN's tritium endpoint and neutrinoless double-beta-decay searches supply the independent cross-check.
The physical picture
Imagine the relic neutrino sea as a gas of particles moving at nearly the speed of light while the first structures try to assemble. A potential well grows on some comoving scale and reaches out to pull matter in. Cold dark matter, with negligible velocity dispersion, obeys and falls; the neutrinos do not — they shoot across the well and out the far side before it can hold them, unless the well is larger than the distance a neutrino coasts in a Hubble time. That free-streaming length cleanly partitions the universe into two regimes: on large scales the neutrinos cluster along with everything else, while on small scales they carry their share of the density away.
As the universe expands, neutrino momenta redshift and velocities fall as (1+z). Below z_nr each species slows enough to begin clustering, but only on progressively larger scales — the small-scale power lost during the relativistic epoch is never recovered. The deficit is permanent and calculable: heavier neutrinos carry more non-clustering energy density and cut a deeper step. This is the quantity a galaxy survey actually reaches for, and the punchline of the whole subject.
Contrast the two components deliberately. Cold dark matter has essentially zero velocity dispersion and clusters on every scale down to sub-solar-mass halos; neutrinos are hot, and their thermal speed carves out a minimum clustering scale. The two leave opposite imprints on P(k) — CDM fills in small-scale power, neutrinos drain it — which is why the neutrino signature is distinctive rather than degenerate with simply lowering Ω_m.
The core equations
The whole edifice rests first on a counting statement: the relic neutrino number density is fixed by the thermal history, tied to the photon density by a pure number. Neutrinos decouple relativistically at T ≈ 1 MeV, so their comoving number freezes; the later heating of photons by e⁺e⁻ annihilation is what makes their ratio 3/11 rather than 1.
where: n_ν — number density of one species (neutrino plus antineutrino); n_γ — CMB photon number density, ≈ 411 cm⁻³; 3/11 — the fermionic ¾ statistics factor times the (4/11) entropy factor from e⁺e⁻ annihilation heating photons but not the decoupled neutrinos. Reading: three species give 3 × (3/11) × 411 ≈ 336 cm⁻³, about a billion relics per baryon. Once a species is non-relativistic its energy density is just mass times that frozen number density, and dividing by the critical density gives the single most useful formula in the subject — the 93.14 eV rule that converts a mass sum straight into a density parameter.
where: Ω_ν — present-day neutrino density parameter (once non-relativistic); h — Hubble constant in units of 100 km s⁻¹ Mpc⁻¹; Σm_ν — the summed mass of the three species; 93.14 eV — the conversion constant, which bakes in n_γ, ρ_crit, T_ν = (4/11)^{1/3} T_γ and the precise N_eff = 3.044. Reading: Σm_ν = 0.1 eV gives Ω_ν h² ≈ 0.00107, a per-mille slice of the budget. A species does not carry mass-like energy at all times. It redshifts as radiation until its mean momentum falls below its rest mass; setting ⟨p⟩ ≈ 3.15 T_ν equal to m_ν fixes the transition redshift.
where: z_nr — redshift at which a neutrino of mass m_ν turns non-relativistic; m_ν — the individual species mass. Reading: even the heaviest allowed neutrino (m_ν ≈ 0.05–0.1 eV) becomes non-relativistic at z_nr ≈ 100–200, long after matter–radiation equality at z_eq ≈ 3400 — so through equality it behaves as radiation. The scale that separates clustering from streaming is a Jeans-like wavenumber, built from the Hubble rate and the neutrino thermal speed playing the role of a sound speed. Its smallest value — the minimum free-streaming wavenumber k_nr, reached at z_nr — sets the largest scale on which suppression appears.
where: k_fs — comoving wavenumber above which clustering is suppressed; a, H(z) — scale factor and Hubble rate; c_ν ≈ ⟨v_ν⟩/c — the effective streaming (sound) speed; k_nr — the minimum of k_fs, occurring at z_nr. Reading: because ⟨v_ν⟩ ∝ 1/m at fixed z, at any epoch k_fs ∝ m, but evaluated at its own z_nr the minimum scales as m^{1/2}. The amplitude of every neutrino effect is governed by one dimensionless number — the fraction of the matter budget that is neutrinos.
where: f_ν — fraction of today's total matter in massive neutrinos; Ω_m — total matter density parameter. Reading: for Σm_ν = 0.1 eV and Ω_m h² = 0.143, Ω_ν h² = 0.00107 gives f_ν ≈ 0.0075 — under one percent of matter, yet enough to matter. The observable payoff is the step in the power spectrum. On scales well below the free-streaming length, converting a fraction f_ν of matter into a non-clustering hot component both slows growth and removes a source, and the two effects compound over the many e-folds since equality into a coefficient of eight.
where: ΔP/P — fractional change in the linear matter power spectrum relative to the f_ν = 0 case; the −8 — coefficient set by 2 × (3/5) × ln(a₀/a_eq) with ln(a₀/a_eq) ≈ 8. Reading: f_ν = 0.0075 gives ΔP/P ≈ −6%, a deficit a galaxy survey can resolve. Underlying the −8f_ν step is a small shift in the growth exponent itself: on suppressed scales the cold matter grows more slowly than the canonical δ ∝ a of an Einstein–de Sitter era.
where: δ_cdm — cold-matter density contrast on small scales; a — scale factor during matter domination. Reading: neutrinos add to the background density (setting H) but not to the clustering source, so growth is throttled by the factor (1 − f_ν) hidden in the exponent.
The core derivations
The goal is to turn the qualitative statement "neutrinos do not cluster on small scales" into the sharp, measurable coefficient ΔP/P ≈ −8f_ν, tracking cold-matter growth on scales k ≫ k_fs from matter–radiation equality to today. The essential trick is that neutrinos contribute their full share to the mean density that sets the expansion rate, but essentially nothing to the perturbation that sources gravity — so the growth equation carries a mismatched background and source. We solve that modified equation during matter domination, linearise the growth exponent in the small quantity f_ν, then integrate the deficit over the logarithmic expansion since equality.
Step 1. Write the sub-horizon linear growth equation for the cold component, δ̈_c + 2H δ̇_c = 4πG (1 − f_ν) ρ̄_m δ_c. On scales k ≫ k_fs the neutrino perturbation δ_ν is negligible, so only the fraction (1 − f_ν) of the matter density actually clumps and sources the potential, while the full ρ̄_m still governs H. Why: this single asymmetry — full weight in the background, reduced weight in the source — is the entire origin of the suppression; set f_ν = 0 and the equation reverts to standard growth.
Step 2. During matter domination H² ∝ a⁻³, and a power-law ansatz δ_c ∝ a^p turns the differential equation into the algebraic condition p(p + ½) = (3/2)(1 − f_ν), whose positive root is p = [−1 + √(1 + 24(1 − f_ν))]/4. Setting f_ν = 0 recovers p = 1, the familiar δ ∝ a. Why: reducing the source below the background value is formally identical to lowering the effective matter density in the growth equation, and the growing-mode exponent responds smoothly.
Step 3. Expand for f_ν ≪ 1: the square root gives dp/df_ν = −12/5 × 1/(2·5) evaluated at f_ν = 0, so p ≈ 1 − (3/5)f_ν. The cold matter therefore grows as δ_c ∝ a^{1−(3/5)f_ν} on suppressed scales. Why: the entire non-trivial content collapses to the number 3/5, the linear response of the growth exponent to the neutrino mass fraction.
Step 4. Integrate from equality to today. The suppressed mode falls behind by δ_c(a₀)/δ_c^{f=0}(a₀) ≈ (a₀/a_eq)^{−(3/5)f_ν}, and squaring for the power spectrum (P ∝ δ²) gives ΔP/P ≈ −2 × (3/5) × f_ν × ln(a₀/a_eq). With z_eq ≈ 3400, ln(1 + z_eq) ≈ 8.13, so 2 × (3/5) × 8.13 ≈ 9.8 — and a fuller treatment of the free-streaming turn-on trims this to the standard ΔP/P ≈ −8f_ν. Why: the coefficient is not fundamental but the accumulated product of a small per-e-fold deficit and the many e-folds of matter domination — which is why heavier structures at lower redshift feel it most. *As a check: setting f_ν = 0 returns p = 1, recovering the standard δ ∝ a; and inserting the exponent 3/5 with ln(1 + 3400) ≈ 8.13 gives 2 × (3/5) × 8.13 ≈ 9.8, reproducing the order-eight coefficient before the free-streaming turn-on correction trims it to −8f_ν.*
Go deeper · Derivation B · 2 minThe relic abundance and the ninety-three eV rule
The aim is to derive the workhorse relation Ω_ν h² = Σm_ν / 93.14 eV entirely from the frozen relic distribution, so that the constant 93.14 eV is exposed as a product of the photon density, the critical density, and the neutrino-to-photon temperature ratio. Because neutrinos decouple while relativistic, their comoving number density is locked in at decoupling and is blind to the later, unknown masses; today's mass density is then that fixed number times whatever the masses turn out to be — which is what makes the abundance a clean, one-line function of Σm_ν.
Step 1. Fix the photon density from the blackbody CMB: n_γ = (2ζ(3)/π²) T_γ³ ≈ 411 cm⁻³ at T₀ = 2.725 K. Why: the neutrino density is pegged to the photon density, so the photon count is the anchor of the whole calculation.
Step 2. Relate neutrinos to photons through the thermal history. Fermi statistics give a ¾ number factor, and e⁺e⁻ annihilation after neutrino decoupling heats photons by (11/4)^{1/3} in temperature, leaving T_ν = (4/11)^{1/3} T_γ and hence n_ν = (3/11) n_γ per species, so n_ν^tot = 3 × (3/11) × 411 ≈ 336 cm⁻³. Why: the factor 3/11, not 1, is the fingerprint of the electron–positron reheating that the neutrinos missed.
Step 3. Build today's mass density. Once each species is non-relativistic it contributes ρ_ν = Σ_i m_i n_{ν,i} = (3/11) n_γ Σm_ν, treating the three number densities as equal. Why: the frozen number density is mass-independent, so the mass enters only linearly and only here.
Step 4. Divide by the critical density ρ_crit = 1.878 × 10⁻²⁶ h² kg m⁻³, converting eV to kg. Carrying n_γ, the temperature ratio and N_eff = 3.044 through the arithmetic collapses the whole prefactor to a single denominator, Ω_ν h² = Σm_ν / (93.14 eV). Why: every physical constant — n_γ, ρ_crit, T_ν/T_γ — folds into that one number, which is why it can be memorised and applied without redoing the thermodynamics. *As a check: n_γ ≈ 411 cm⁻³ and n_ν^tot = 9/11 × 411 ≈ 336 cm⁻³ feed directly into the 93.14 eV constant, and the rule gives Ω_ν h² = 0.00107 for Σm_ν = 0.1 eV.*
Go deeper · Derivation C · 2 minThe free-streaming scale as a collisionless Jeans length
The purpose here is to construct the free-streaming wavenumber k_fs from a physical balance rather than quote it, and to explain the puzzling fact that at fixed redshift k_fs grows with mass while its minimum over cosmic time grows only as the square root of the mass. The neutrino has no pressure in the ordinary sense — it is collisionless — but its thermal velocity plays exactly the role a sound speed plays for a gas, so the Jeans argument carries over with c_s replaced by ⟨v_ν⟩. Balancing collapse against thermal escape gives the scale; evaluating it at the moment a species turns non-relativistic gives the observationally relevant minimum.
Step 1. For a self-gravitating fluid the Jeans wavenumber is k_J = √(4πGρ̄)/c_s; for the collisionless neutrinos replace the sound speed by the streaming speed, k_fs = √(3/2) aH/c_ν with c_ν = ⟨v_ν⟩/c, since during matter domination 4πGρ̄ = (3/2)H². Why: a perturbation survives only if gravity can pull matter together faster than thermal motion carries it across a wavelength — the same competition that defines the Jeans scale.
Step 2. Insert the relic thermal velocity ⟨v_ν⟩ ≈ 158 (1+z)(eV/m_ν) km s⁻¹, valid once the species is non-relativistic. At fixed redshift ⟨v_ν⟩ ∝ 1/m, so c_ν ∝ 1/m and k_fs ∝ m × aH — heavier neutrinos are slower and free-stream out to only smaller scales. Why: this is why more massive neutrinos suppress a wider range of k yet leave a shallower relative step per unit Ω_ν — the scale and amplitude respond to mass differently.
Step 3. Evaluate the minimum. k_fs decreases from equality, bottoms out at z_nr, then rises; substituting z_nr ∝ m into k_fs ∝ m aH makes the two mass dependencies partly cancel, leaving k_nr ≈ 0.018 Ω_m^{1/2}(m_ν/eV)^{1/2} h Mpc⁻¹. Why: the observable suppression begins at the largest free-streaming scale k_nr, so it is this square-root-in-mass minimum — not the fixed-redshift k_fs — that marks the left edge of the step in P(k). *As a check: evaluating the formula gives k_nr(m_ν = 0.05 eV, Ω_m = 0.31) ≈ 0.0022 h Mpc⁻¹ and k_nr(1 eV) ≈ 0.010 h Mpc⁻¹; the m^{1/2} scaling follows from z_nr ∝ m.*
Limiting cases
The neutrino contribution changes character with both scale and epoch, and every neutrino effect is a special case of the table below:
| Case | Result | Meaning |
|---|---|---|
| Relativistic era (z ≫ z_nr) | ρ_ν ∝ a⁻⁴ | Neutrinos redshift as radiation, add to N_eff = 3.044, and do not cluster at all. |
| Large scales (k ≪ k_fs) | δ_ν → δ_cdm | Free-streaming is irrelevant; neutrinos trace the matter and clustering is unaffected. |
| Small scales (k ≫ k_fs) | ΔP/P ≈ −8f_ν | Neutrinos stream out; power is suppressed by roughly eight times the mass fraction. |
| Massless limit (Σm_ν → 0) | f_ν → 0, ΔP/P → 0 | Recovers pure CDM+baryon growth; only the N_eff radiation effect survives. |
| Heavy limit (m_ν ≳ few eV) | f_ν → O(1) | Free-streaming erases galaxy-scale power — the top-down scenario ruled out by observed structure. |
Worked examples
1 · From a mass sum to a measurable power deficit (fully worked)
Take the Planck+BAO ceiling Σm_ν = 0.12 eV in a concordance model with Ω_m h² = 0.143. Find the neutrino density parameter, the mass fraction, and the small-scale power suppression, and locate the scale where it turns on.
Step 1. Apply the 93.14 eV rule: Ω_ν h² = Σm_ν / 93.14 eV = 0.12 / 93.14 = 1.29 × 10⁻³. This is the density parameter the three now-massive species carry today.
Step 2. Form the mass fraction with the total matter density: f_ν = Ω_ν h² / Ω_m h² = 1.29 × 10⁻³ / 0.143 = 0.0090. Under one percent of matter is neutrinos even at the observational ceiling.
Step 3. The small-scale deficit is ΔP/P ≈ −8f_ν = −8 × 0.0090 = −0.072, a 7.2% suppression for k ≫ k_fs. At the normal-ordering floor Σm_ν = 0.06 eV (f_ν ≈ 0.0045) it is only −3.6% — doubling the mass sum doubles the step, an essentially linear relation.
Step 4. Locate the turn-on. Splitting the sum equally gives a per-species mass near 0.04 eV, so k_nr ≈ 0.018 √0.31 × √0.04 ≈ 2 × 10⁻³ h Mpc⁻¹: the deficit begins at very large scales and deepens toward its −7% plateau across k ≈ 0.01–1 h Mpc⁻¹ — precisely the window galaxy surveys and the Lyman-α forest cover.
2 · When does a light neutrino go cold, and how fast is it?
A neutrino of mass m_ν = 0.05 eV: find its non-relativistic transition redshift and its thermal velocity today and at z_nr, and comment on why this ordering of epochs maximises the imprint.
Step 1. Transition redshift: z_nr ≈ 1890 × (0.05) = 95. The species is still relativistic through matter–radiation equality at z_eq ≈ 3400.
Step 2. Thermal velocity today: ⟨v_ν⟩ ≈ 158 × (1) × (1/0.05) = 3160 km s⁻¹, about 0.01c — small but far above any CDM dispersion, which is why the free-streaming scale is cosmologically large.
Step 3. Thermal velocity at z_nr: ⟨v_ν⟩ ≈ 158 × (96) × (20) ≈ 3.0 × 10⁵ km s⁻¹ ≈ c, confirming self-consistently that z ≈ 95 is indeed where the neutrino stops being relativistic.
Step 4. Interpretation: because z_nr < z_eq, the neutrino spends the whole radiation era and the start of matter domination streaming freely, damping small-scale modes for the longest possible time before it can cluster. Lighter neutrinos go non-relativistic later, free-stream longer, and leave a larger relative imprint per unit mass — the reason cosmological sensitivity is strongest exactly in the sub-0.1 eV range oscillations point to.
3 · How many relics, and how heavy is the sea?
Estimate the number of relic neutrinos per baryon and the fraction of critical density they carry for Σm_ν = 0.1 eV, taking Ω_b h² = 0.0224 and n_γ ≈ 411 cm⁻³.
Step 1. Relic count: n_ν^tot = 9/11 × 411 ≈ 336 cm⁻³. With n_b = Ω_b h² ρ_crit,100/m_p ≈ 2.5 × 10⁻⁷ cm⁻³, there are n_ν/n_b ≈ 1.3 × 10⁹ relic neutrinos per baryon — the sea vastly outnumbers ordinary matter in particles while carrying little mass.
Step 2. Density parameter: Ω_ν h² = 0.1 / 93.14 = 1.07 × 10⁻³, i.e. Ω_ν ≈ 0.0024 for h = 0.67 — comparable to all the stars, a fraction f_ν ≈ 0.0075 of matter. It is precisely because 10⁹ relics accompany each baryon that a per-particle mass of a fraction of an eV integrates into a gravitationally decisive effect — small mass repaid by enormous number.
Physics of the portrait
The story of massive neutrinos in cosmology is not the story of one genius but of a physical object — the cosmic neutrino background — whose consequences were teased out over half a century by a chain of theorists, each realising that a nearly weightless sea could weigh on the whole universe.
Origins · 1 mincosmic neutrino background and how this was discovered
The cosmic neutrino background is the second-oldest relic in the universe, decoupling about one second after the Big Bang — long before the CMB photons broke free at 380,000 years. It is a Fermi–Dirac sea at T_ν = (4/11)^{1/3} T_γ = 1.95 K, so cold and weakly interacting that it has never been directly detected, yet its gravitational fingerprints are everywhere. Its cosmological career began in 1966 when Gershtein and Zel'dovich noticed that a small neutrino mass would overclose the universe unless Σm_ν stayed below a few tens of eV — the first cosmological bound on a particle mass, derived years before oscillations proved the masses non-zero. Cowsik and McClelland, and independently Marx and Szalay, sharpened it in 1972; Tremaine and Gunn added a phase-space argument in 1979. When neutrinos were briefly the leading dark-matter candidate around 1980, Bond, Efstathiou and Silk showed their free-streaming would erase small-scale structure and build the cosmos top-down — a prediction White, Frenk and Davis showed in 1983 was fatally at odds with the observed galaxy distribution, retiring hot dark matter as the dominant component. The modern synthesis, in which neutrinos are a small but decisive admixture whose −8f_ν step in P(k) can be measured, was set out by Hu, Eisenstein and Tegmark in 1998. What began as a bound to avoid a problem is now the most sensitive scale we have for the absolute neutrino mass.
Going further
Texts. Dodelson, Modern Cosmology (2nd ed.), Ch. 8–9 (neutrino thermal history, transfer functions) · Ryden, Introduction to Cosmology (2nd ed.), Ch. 5, 12 · Carroll and Ostlie, An Introduction to Modern Astrophysics (2nd ed.), Ch. 30 · Mo, van den Bosch and White, Galaxy Formation and Evolution, Ch. 4 (linear growth) · Binney and Tremaine, Galactic Dynamics (2nd ed.), Ch. 5 (collisionless dynamics) · Lesgourgues and Pastor, Physics Reports 429 (2006) 307, "Massive neutrinos and cosmology" · Planck 2018 results VI, A&A 641 (2020) A6. The key numbers used throughout are: n_γ ≈ 411 cm⁻³ and n_ν^tot = 9/11 × 411 ≈ 336 cm⁻³; T_ν = (4/11)^{1/3} T_γ = 1.95 K; Ω_ν h² = Σm_ν/93.14 eV; N_eff = 3.044; z_nr ≈ 1890(m_ν/eV); ΔP/P ≈ −8f_ν; and Σm_ν < 0.12 eV (Planck 2018 + BAO, 95%). The free-streaming minimum is k_nr ≈ 0.018 Ω_m^{1/2}(m_ν/eV)^{1/2} h Mpc⁻¹; the square-root mass dependence follows from evaluating k_fs ∝ m aH at z_nr ∝ m.
Builds on: Density Perturbations and the Density Contrast · Linear Growth of Structure · Transfer Function and the Matter Power Spectrum · Jeans Instability · The ΛCDM Concordance Model. Leads to: The CMB Anisotropy Power Spectrum · Baryon Acoustic Oscillations · The Primordial Power Spectrum.
Ask the tutor: Ask me why a sub-eV neutrino mass leaves a −8f_ν step in the matter power spectrum, or to work a Σm_ν bound from Ω_ν h² by hand.
Why is cosmology so much more sensitive to neutrino mass than the laboratory?
Because it counts an astronomically larger sample. A tritium-endpoint experiment such as KATRIN watches individual beta decays and bounds the effective electron-neutrino mass at about 0.8 eV — limited by the tiny kinematic distortion a light mass makes on a decay spectrum. Cosmology instead weighs the gravitational pull of roughly 10⁹ relic neutrinos per baryon, integrated over the whole history of structure formation, where a per-mille change in the matter budget compounds into a several-percent step in P(k). The two are complementary: cosmology bounds the sum Σm_ν assuming standard thermal history, while the lab measures a flavour-weighted mass with far fewer astrophysical assumptions.
How can neutrinos be radiation and matter in the same universe?
Each species changes its behaviour with time. Early on, when the mean momentum ⟨p⟩ ≈ 3.15 T_ν far exceeds the rest mass, a neutrino moves at nearly c and its density redshifts as a⁻⁴, exactly like a photon — the regime that fixes N_eff = 3.044 and the expansion rate near equality. As the momentum redshifts, at z_nr ≈ 1890(m_ν/eV) it drops below the rest mass; thereafter the energy is dominated by mc² and the density redshifts as a⁻³, like matter. So a 0.05 eV neutrino is radiation through z ≈ 3400 equality and only becomes matter around z ≈ 95. "Radiation" and "matter" are labels for how the density scales, and a massive relic wears each in turn.
What exactly is the free-streaming length, and why does it suppress power?
It is the comoving distance a neutrino coasts, at its thermal speed, in about a Hubble time — the collisionless analogue of a fluid's Jeans length, with the sound speed replaced by the neutrino velocity. A well smaller than this length cannot hold neutrinos: they stream across and out before it can pull them in, so they never enhance the local density. On larger scales they are slow enough (relative to the well) to be captured, and cluster normally. That boundary, k_fs, is why the suppression is a step: full clustering at k ≪ k_fs, a drained plateau at k ≫ k_fs.
Where does the coefficient −8 in ΔP/P ≈ −8f_ν actually come from?
It is an accumulated, not a fundamental, number. On suppressed scales the cold matter grows a touch more slowly than the standard δ ∝ a, with exponent 1 − (3/5)f_ν, because neutrinos add to the background density that sets H but not to the source that drives collapse. That tiny per-e-fold deficit compounds over every e-fold of matter domination since equality — a span of ln(1 + z_eq) ≈ 8. Squaring the growth factor for the power spectrum brings a factor of two, so the coefficient is roughly 2 × (3/5) × 8 ≈ 8 — which is why more evolved, lower-redshift structures show the deepest suppression.
Does the neutrino mass affect the CMB, or only the galaxy power spectrum?
Both, through different handles. The primary CMB temperature spectrum is only weakly sensitive to sub-eV masses because the neutrinos are still relativistic at recombination, but they leave two marks: they shift the redshift of equality and the distance to last scattering, subtly moving the acoustic peaks; and, more powerfully, they suppress late-time growth, weakening the gravitational lensing that smooths the small-scale peaks — so the lensing spectrum C_ℓ^φφ is a direct, largely independent probe of Σm_ν. This is why the tightest bounds come from combining CMB lensing with BAO and galaxy surveys.
If neutrinos were once the leading dark-matter candidate, why were they ruled out as the whole story?
Because free-streaming builds the universe in the wrong order. If neutrinos supplied all the dark matter (f_ν → 1), their enormous free-streaming length would erase all structure below cluster scales, so galaxies could only fragment out of larger sheets afterwards — a "top-down" scenario. White, Frenk and Davis showed in 1983 that this predicts far too little small-scale structure and the wrong clustering compared with real galaxy surveys, which demand "bottom-up" growth from small halos. That killed hot dark matter as the dominant component. Massive neutrinos survive not as the main event but as a small, unavoidable admixture — enough to leave the −8f_ν step, not enough to reshape the cosmic web.