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Cosmological structure formation is the growth of primordial density perturbations into sheets, filaments, halos, galaxies, and clusters under gravity.

Nature 311, pp. 517–525. The paper links weak free streaming of cold collisionless particles to the fluctuation spectrum that seeds galaxies and clusters.

A matter component with relativistic motion during an important part of structure formation. Its free streaming suppresses small-scale perturbations. Making a light thermal neutrino Majorana rather than Dirac does not by itself turn this component into cold dark matter.

cold dark matter is nonrelativistic and effectively collisionless on the scales of interest. Its small velocity dispersion limits free streaming, allowing small-scale gravitational wells to survive. baryons coupled to photons undergo acoustic motion and damping before recombination; dark matter does not share that photon pressure, and provides potential wells into which baryons can later fall.

For a subhorizon pressureless perturbation, once a Newtonian treatment is applicable,During radiation domination the matter contribution to the background gravity is small; neglecting it gives $\delta \simeq C_1+C_2\log t$. This slow subhorizon growth is the Mészáros effect. During matter domination,in an Einstein-de Sitter universe. Once $\delta$ approaches unity, linear theory fails and collapse, virialization and merging become important.

Modes entering the horizon during radiation domination therefore grow less before equality than longer modes entering later. This scale dependence is encoded in a transfer function, with a schematic linear matter power spectrum $P(k,a)=A{k}^{n_s}{T}^2(k)D^2(a)$. cold dark matter supports hierarchical assembly; a sufficiently large warm- or hot-dark-matter free-streaming length suppresses small-scale structure. Superhorizon density contrasts change under a gauge transformation in cosmological perturbation theory, so the subhorizon growth laws should not be applied indiscriminately outside the horizon. The original cold-dark-matter formation model explicitly connects weak free streaming to the spectrum that seeds galaxies and clusters.^[ref]

Small primordial fluctuations seed later structure. For adiabatic cosmological perturbations on scales outside the horizon, the uniform-density curvature perturbation is conserved when the non-adiabatic pressure perturbation and the decaying mode can be neglected. A density contrast outside the horizon is gauge-dependent, so its apparent power-law growth should not be confused with directly measured local collapse.

For subhorizon, pressureless matter, the linearized continuity equation, Euler equation and Poisson equation combine intoHubble expansion damps peculiar motion; self-gravity amplifies overdensities. During radiation domination, radiation pressure drives acoustic oscillations and cold dark matter growth after horizon entry is only logarithmic. During matter domination the growing and decaying solutions are $\delta \propto a$ and $\delta \propto a^{-3/2}$. Dark-energy domination later suppresses growth relative to the matter-era law.

For a pressure-supported fluid the Fourier-mode equation also contains $+c_s^2{k}^2\delta /a^2$, giving a Jeans length separating acoustic support from gravitational growth. Before recombination, baryons are tightly coupled to photons and oscillate rather than freely following dark matter. Silk damping from photon diffusion damps small-scale photon-baryon fluctuations; after decoupling, baryons can fall into dark-matter potential wells. free streaming smooths species-dependent scales and distinguishes cold from warm or hot dark matter.

Schematically, the linear matter power spectrum can be written $P_{\delta }(k,a)=A{k}^{n_s}{T}^2(k)D^2(a)$ under the usual primordial-curvature normalization. The cosmological transfer function records horizon entry, matter-radiation equality, acoustic physics and damping; the linear growth factor records later amplification. density contrasts approaching unity require nonlinear collapse and halo formation beyond this approximation. cosmic microwave background anisotropies and galaxy clustering test the same initial perturbations through different transfer histories; galaxy bias, redshift-space distortions and observational selection enter the galaxy comparison.

Galaxy formation combines gravitational assembly of dark-matter haloes with dissipative baryonic physics. Primordial density fluctuations supply a statistical seed distribution. In linear theory, each Fourier mode evolves through a growth factor and a scale-dependent transfer function; the power spectrum describes the variance by scale. Radiation pressure, free streaming and the time of horizon entry influence the small-scale transfer, so one should not replace the spectrum by a single universal growing power of time.

When fluctuations become nonlinear, overdense regions collapse, virialize and build larger structures through accretion and mergers. Cold dark matter permits a broad hierarchy of bound haloes. A spherical-collapse threshold and a mass-variance prescription, such as the Press-Schechter formalism, organize an approximate halo abundance; departures from spherical collapse and correlated environments limit its literal interpretation.

Gas falling into a halo is compressed and may be shock-heated toward a virial scaleUnlike collisionless dark matter, it can radiate away thermal energy. A rough cooling time iswhere the denominator specifies a chosen optically thin astrophysical cooling function convention. Cooling is effective for condensation when this time is short compared with the relevant dynamical or supply time. Atomic transitions, molecular processes, ionization state, density and metal abundance matter. A stable virial shock is not obligatory for all accretion: cold dense inflows can supply gas in regimes where shock-heated gas would cool rapidly.

The distinction between a gravitationally assembled massive halo and a radiatively condensing luminous component is fundamental. In the classic two-stage picture, non-dissipative clustering builds potential wells while cooling allows gas to concentrate much further than the halo component.^[ref]

Angular momentum inherited from tidal torques and later accretion helps gas settle into a rotationally supported disc after cooling; disc size depends on angular-momentum retention and transport. Locally self-gravitating gas fragments and forms stars. Stellar radiation, winds and supernovae then heat gas, drive turbulence, enrich it with heavy elements and can expel it from shallow potential wells. Metal enrichment can also accelerate later cooling, so feedback is more than a single positive or negative energy term.

Galaxy growth is usefully expressed as a gas-budget competition:Here $\mathcal{R}$ is the promptly returned stellar mass fraction; unlike the simplification in Question1, a general evolutionary model need not set it to zero. The equation separates newly supplied, converted, expelled and recycled gas rather than assuming that halo mass uniquely fixes stellar mass.

Accreting central black holes can provide radiative and mechanical feedback. Heating a halo, ejecting gas or maintaining a hot atmosphere can reduce fresh fuel; removal of a satellite's gas supply and ram pressure in a surrounding medium provide additional environmental routes. Quiescent galaxy assembly can still add stellar mass through dry mergers, so cessation of star formation is not identical to cessation of all mass assembly.

Mergers and disc instabilities can rearrange morphology and trigger bursts, while smooth accretion can rebuild discs. Cosmological expansion changes the supply and merger environment; accelerated late-time expansion also slows linear growth. A predictive model must connect these mechanisms to the stellar mass function, gas fractions, star-formation histories, sizes, chemical abundances and clustering. Those observables test different pieces of the assembly-and-feedback picture; none alone implies a one-to-one instantaneous relation between galaxy luminosity and total halo mass.

The known active neutrinos are too light and too fast to supply the dominant cold dark matter. Their relativistic decoupling leaves substantial free streaming, which erases rather than seeds sufficiently small-scale structure; their allowed mass density is also inadequate for the dominant cold component. Changing a light active neutrino from Dirac to Majorana does not remove those problems. What can change the picture is an enlarged neutral-fermion sector containing sterile states with different masses, production and decay rates. A sufficiently long-lived sterile neutrino in a Majorana state can be a dark-matter candidate; Majorana character alone supplies neither stability nor the correct abundance.

For the spinor calculation, put $a=p_z/(E+m)$ and define $A=N(1-a)/2$, $B=N(1+a)/2$. Applying the given chiral projectors givesConsequentlywhere the last equality uses $E^2-p_z^2=m^2$. Same-chirality Lorentz scalar products instead vanish: for example ${\overline{U}}_{1L}{U}_{1L}=0$. The nonzero overlaps demonstrate the Lorentz scalar structures coupling left and right chirality. For $U$ to $U$ this is a particle-number-preserving Dirac coupling; for $U$ to the charge-conjugate $V$ sector it can describe a lepton-number-violating Majorana coupling.

There is a normalization subtlety: both displayed overlaps equal the on-shell mass parameter used to define the spinors. A chiral scalar spinor overlap verifies a coupling structure without fixing an independent Lagrangian coefficient $M_L$. Repeating the on-shell construction with mass parameter $M_L$ would instead give the corresponding overlap $M_L$; this is a different input, rather than a newly derived parameter. The independent coefficients of a Dirac mass term and a Majorana mass term are defined byThe Majorana terms connect a field with its charge conjugate and violate the corresponding continuous lepton phase symmetry by two units. With only unbroken Standard Model gauge symmetry, a left-handed Majorana mass also requires electroweak symmetry breaking or an appropriate higher-dimensional operator; it is not an arbitrary gauge-invariant bare mass for a lepton doublet.

Use an all-left-handed Weyl basis ${\mathbf{n}}_L=({\nu }_L,{\nu }_R^c{)}^T$. For real parameters the one-family Dirac-Majorana mass matrix isWriting $({\nu }_{1L},{\nu }_{2L}{)}^T=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)({\nu }_L,{\nu }_R^c{)}^T$ gives an orthogonal diagonalization, with $\tan 2\theta =2m/(M_R-M_L)$ up to the rotation-sign convention. Massive-particle chirality and helicity are not identical; the familiar left/right helicity language becomes an approximation in the relativistic regime.

The characteristic polynomial isIts roots areFor Majorana fermions the nonnegative physical masses, as explained by physical Majorana mass and signed eigenvalue, are $|{\lambda }_{\pm }|$ after a possible field rephasing; a negative real eigenvalue is not a negative physical rest mass. When $M_R\gg |m|,|M_L|$,The light state is predominantly ${\nu }_L-(m/M_R){\nu }_R^c$, while the heavy state is predominantly ${\nu }_R^c+(m/M_R){\nu }_L$. Each mass eigenstate combines its Weyl component and charge conjugate to form a Majorana field.

For $M_R=10^{12}\mathrm{GeV}$, the heavy mass is approximately $10^{12}\mathrm{GeV}$. In the type-I limit $M_L\simeq 0$,The seesaw light-mass estimate needs a Dirac scale: no unique numerical light mass follows because the Dirac mass and $M_L$ were not supplied. As an explicitly conditional illustration, choosing $m=100\mathrm{GeV}$ and $M_L=0$ gives $10\mathrm{eV}$; this is an assumed Dirac scale, not additional exam data or a prediction from $M_R$ alone. A heavy seesaw state is also not automatically stable enough to be dark matter.

The conventional minimal Pati-Salam model gauge group isThe group printed in this paper is different: it contains two $SU(4)$ factors and only one $SU(2)$. This is a discrepancy in the original PDF, not just in the transcription. The standard-model discussion below uses the conventional intended Pati-Salam model model. A literal theory with the printed product would require additional fermion representations and symmetry-breaking data that are not specified.

The Pati-Salam model unifies the three quark color charges with lepton number as a fourth color charge. A family forms $(4,2,1)$ left-handed and $(4,1,2)$ right-handed multiplets, including a right-handed neutrino. The extended color charge group contains quark-lepton gauge bosons, while the right-handed weak factor supplies right-handed charged currents after symmetry breaking. The Pati-Salam charge embedding isThe original paper explicitly develops the smaller local subgroup $SU(2{)}_L\times SU(2{)}_R\times SU(4{)}_C$, as well as a larger underlying global structure; neither distinction licenses silently replacing the printed group.^[ref] The product group is a quark-lepton unification scheme rather than a simple gauge group with one automatically common coupling. Breaking the right-handed and $B-L$ symmetries can generate a large $M_R$ and light/heavy Majorana neutrinos through the seesaw. The gauge group alone does not uniquely fix their masses or guarantee a dark-matter lifetime.

The most important laboratory test of Majorana character is neutrinoless double beta decay,with no outgoing neutrinos. Under light-neutrino exchange, the amplitude involvesThe coherent squared mixing elements allow phase cancellations. A null result constrains this combination within the assumed mechanism, rather than proving that all neutrinos are Dirac.

The observable is a peak in summed electron energy at the decay Q value, unlike the continuous two-neutrino spectrum. Enriched source material, large exposure, good energy resolution, underground operation and radiopure construction suppress backgrounds. germanium arrays use the source as a semiconductor detector; xenon time-projection chambers combine ionization, light and event-position information; bolometers measure deposited energy through a temperature rise. The GERDA collaboration's Phase-I study is a concrete example of an enriched-germanium source-detector search with background suppression.^[ref] energy calibration, resolution tails, radon and material backgrounds, cosmic-ray vetoes, signal-like event topology and sideband fits all need independent checks before a Q-value excess is convincing. Collider same-sign dilepton processes can test heavier Majorana states, but require a specified production and decay mechanism. Lepton-number violation, absolute-mass measurements and oscillation data provide complementary information; oscillations alone do not settle Majorana character.

A neutral fermion without the ordinary active-neutrino weak gauge couplings. Mixing can still let it interact or decay. Its possible dark-matter role depends on lifetime, production and free streaming, rather than on Majorana character alone.