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For either tensor helicity, neutrino free streaming produces the memory equationThe neutrino tensor free-streaming kernel encodes the angular propagation of the perturbation. Energy transferred to neutrino directional anisotropy damps the oscillation amplitude of a gravitational wave after horizon entry. This effect was calculated by Weinberg.

Assume standard radiation-dominated expansion, negligible chemical potential, one neutrino helicity and one antineutrino helicity, and a stable species $X$ with no later entropy injection. Use natural units and unreduced Planck mass $m_{\mathrm{pl}}\simeq 1.22\times 10^{19}\mathrm{GeV}$. The relativistic energy density is ${\rho }_R={\pi }^2{g}_{\ast }{T}^4/30$. The Friedmann equation then givesFor thermal decoupling in cosmology, equate the interaction rate $G_A^2{T}_D^5$ to $H(T_D)$:This order-of-magnitude estimate assumes the supplied rate law remains applicable there.

A decoupled relativistic species cools as $T_X\propto a^{-1}$, while annihilations heat the coupled bath. Treat the supplied high-temperature count $100$ as the remaining bath's entropy count, approximately equal to its energy count. Cosmological entropy conservation down to ordinary neutrino decoupling givesSubsequent electron-positron annihilation in cosmology heats photons but neither neutrino population. Therefore the two-stage entropy dilution of an early-decoupled neutrino givesThe ordinary Cosmic neutrino background has $T_{\nu }/T_{\gamma }=(4/11{)}^{1/3}$. If the rounded count $100$ instead includes the new two helicities, the remaining bath count is $98.25$; this changes the answer only slightly, to about $0.341$. Keeping the extra species' entropy separate is essential in either convention.

For the two relativistic fermion states, $g_{\ast s,X}=(7/8)2=7/4$. Photon and three standard neutrino species have present entropy degrees of freedom, referred to $T_{\gamma }$,Writing $\theta =T_X/T_{\gamma }$, the fractional entropy of a relativistically decoupled species isThus approximately 1.7% of the thermal particle entropy is in the new neutrinos and antineutrinos. If $100$ counts all states before the new decoupling, the fraction is $1.75\%$. These are the same rounded estimate. The question concerns the radiation/relic thermal entropy budget; it does not supply a model for entropy in later black holes or other gravitational structures.

If $X$ has a small mass but decoupled relativistically, its comoving particle number is still conserved. The relativistic-neutrino number-density ratio givesOnce nonrelativistic, ${\rho }_X\simeq m_X{n}_X$. The density bound on a relativistically decoupled massive relic is thereforeUsing the standard present photon abundance and critical-density normalization, this is approximately ${\Omega }_X{h}^2\simeq m_X/(9\times 10^2\mathrm{eV})$, where $h=H_0/(100\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1})$ is the present Hubble normalization, not a magnetic field. A stable relic must not exceed the permitted matter density; for example, if ${\Omega }_M{h}^2=0.15$, even assigning all matter to it gives only an order-$10^2\mathrm{eV}$ mass ceiling.

A stronger restriction can follow from neutrino free streaming and the observed formation of large-scale structure of the universe: a light thermal population acts as hot rather than cold dark matter, erasing structure below its streaming scale. If at most a fraction $f_{\max }$ of the matter may be in this population, replace the overclosure ceiling by $m_X\lesssim 9\times 10^2{f}_{\max }{\Omega }_M{h}^2\mathrm{eV}$. The allowed fraction is observational input and is not supplied, so the paper does not determine a unique tighter mass limit. A species still relativistic today is radiation rather than matter; its contribution to the effective number of neutrino species is $\Delta N_{\mathrm{eff}}=(T_X/T_{\nu }{)}^4\simeq 0.051$, and must instead satisfy the radiation-density constraints. These conclusions assume stable thermally populated states; decay or nonstandard entropy histories change the bounds.

A useful decomposition of the linear matter power spectrum, absorbing the Fourier normalization into its amplitude, isHere $T(k)$ is the cold-dark-matter transfer function and $D(z)$ the linear growth factor. On scales entering the horizon after matter-radiation equality, $T\simeq 1$, so $P\propto k^{n_s}$ with $n_s$ near one. Modes entering during radiation domination undergo suppressed growth relative to later entrants, producing a turnover near the equality scale and, well above it, $T\propto \ln (k/k_{\mathrm{eq}})/k^2$. Thus the linear spectrum asymptotes approximately to $k^{n_s-4}{\ln }^2(k/k_{\mathrm{eq}})$. Baryon acoustic oscillations superpose a weak oscillatory pattern. At sufficiently short scales nonlinear gravitational instability invalidates this linear extrapolation.

The plotted curve illustrates the shape rather than a precision cosmological fit; the bands are overlapping approximate scale sensitivities, not sharp experiment boundaries. Cosmic microwave background anisotropies constrain primordial amplitudes and transfer physics on very large through acoustic scales; converting them to a late-time matter spectrum requires a cosmological model. Galaxy redshift surveys constrain approximately $k\sim 0.01$–$0.2h{\mathrm{Mpc}}^{-1}$, with galaxy bias and redshift-space distortions modeled. Peculiar velocities constrain large-scale mass fluctuations; cluster abundances constrain the fluctuation normalization near cluster scales. Weak gravitational lensing measures projected matter on intermediate and smaller scales. The Lyman-alpha forest extends constraints to roughly $k\sim 0.3$–a few $h{\mathrm{Mpc}}^{-1}$ at high redshift, requiring gas, temperature and ionizing-background modeling. These are complementary measurements of the underlying spectrum rather than interchangeable direct Fourier measurements.

The matter fraction controls the equality turnover. From $a_{\mathrm{eq}}={\Omega }_{r,0}/{\Omega }_{m,0}$ and the Friedmann equation at equality,The measured background radiation fixes ${\Omega }_{r,0}{h}^2$, so the physical turnover scale measures approximately ${\Omega }_{m,0}{h}^2$. In units $h{\mathrm{Mpc}}^{-1}$ its location mainly constrains ${\Omega }_{m,0}h$; independent knowledge of $h$ is needed to extract ${\Omega }_{m,0}$. Standard radiation content gives the useful estimate $k_{\mathrm{eq}}\simeq 0.073{\Omega }_{m,0}{h}^2{\mathrm{Mpc}}^{-1}$, approximately $0.015h{\mathrm{Mpc}}^{-1}$ for ${\Omega }_{m,0}=0.3$, $h=0.7$.

The cosmological baryon density sets the inertia of the pre-decoupling photon–baryon fluid and changes its sound horizon and the oscillation pattern. The oscillation contrast and broadband suppression depend on the baryon fraction ${\Omega }_{b,0}/{\Omega }_{m,0}$; the acoustic positions depend on the sound horizon and the distance conversion. The cosmic microwave background peak-height pattern independently constrains ${\Omega }_{b,0}{h}^2$. Combining these with the equality scale and a distance or Hubble constant determination permits estimates of both requested density fractions. Spectral tilt, neutrino free streaming, galaxy bias and nonlinear evolution must be fitted rather than assigning every shape change to baryons. The acoustic signature was already observed in the 2005 galaxy correlation measurement; the first-year microwave-background parameter analysis demonstrates the complementary parameter constraints.

Assume the new neutrino is stable and retains the abundance established by relativistic thermal decoupling in cosmology. While it remains relativistic, its contribution to the effective number of neutrino species isThus its effect on the radiation density and expansion during Big Bang nucleosynthesis is reduced relative to a fully heated extra neutrino species.

After it becomes nonrelativistic, its energy density is approximately $m_X{n}_X$. A two-state relativistic fermion relic has $n_X/n_{\gamma }=(3/4)(T_X/T_{\gamma }{)}^3\simeq 0.0275$. The density bound on a relativistically decoupled massive relic then givesHere ${\Omega }_{X,\max }$ is the allowed present matter-density fraction and $⟨E_{\gamma }⟩={\pi }^4{T}_{\gamma ,0}/(30\zeta (3))$. Choosing the total available matter fraction yields the basic overclosure bound. Its mass cannot be raised freely at a fixed relic number density.

There is also a perturbation constraint. Neutrino free streaming suppresses clustering below the distance traveled before the species becomes nonrelativistic. The ratio $T_X/m_X$ controls this distance, and the relic matter fraction controls the strength of the suppression. A significant hot or warm relic component must therefore satisfy the observed structure-formation constraints in addition to the abundance bound. Its allowed dark-matter contribution is limited jointly by these two effects. Numerical observational bounds depend on the chosen matter budget and perturbation data; the temperature and abundance dilution derived here are the factors needed to apply them.

A large neutrino degeneracy parameter raises the relativistic energy density and therefore the expansion rate. Big Bang nucleosynthesis tests this through primordial light-element abundances; an electron-neutrino chemical potential also shifts neutron--proton chemical equilibrium directly. The Cosmic microwave background power spectrum tests the changed expansion rate, matter-radiation equality, sound horizon, damping scale, and neutrino anisotropic stress. Finally, large-scale structure of the universe and the matter power spectrum test the altered equality scale and neutrino free streaming. These observables distinguish a degenerate-neutrino cosmology from the standard thermal relic scenario in complementary epochs.

Neutrino free streaming gives ${\delta }_{\nu }\simeq 0$ on the stated scales, so ${\delta }_m=(1-f_{\nu }){\delta }_{cb}$. With derivatives with respect to $N=\log a$, the growth equation becomesFor ${\delta }_{cb}\propto a^p$,Henceto first order.

A tensor cosmological perturbation is the transverse, trace-free part of a metric perturbation about a Friedmann-Lemaître-Robertson-Walker metric. In conformal coordinates, its spatial metric has the form $g_{ij}=a^2({\delta }_{ij}+h_{ij})$, with ${\partial }_i{h}_{ij}=0$ and $h_{ii}=0$. Its two independent polarizations describe a gravitational wave. Neutrino tensor anisotropic stress can damp its evolution through neutrino free streaming.