Adiabatic and Isocurvature Initial Modes

An adiabatic perturbation shifts every component along the same background history. An isocurvature perturbation instead changes a relative abundance or relative velocity while leaving the leading total curvature absent. For the standard radiation-era mixture of photons, baryons, cold dark matter, and free-streaming neutrinos, regular scalar initial data form one adiabatic mode and four independent isocurvature modes when the primordial particle distributions carry no independently excited higher angular multipoles.

Helpful background. Einstein Equations and relativistic matter introduce the species and their constraints. Formulations of general relativity explains residual gauge freedom, the cosmology overview locates the background history, and PDEs and Green functions introduces Fourier expansions. Bardeen Potentials and Curvature Perturbations derives the curvature variables used below.

Relative entropy perturbations

Use conformal time $\eta$, with a prime denoting $d/d\eta$ and $\mathcal H=a'/a$, at first order about spatially flat FLRW. For each component $i$ with $\rho_i'\ne0$, define its curvature perturbation on a uniform-$i$-density slice,

$$\zeta_i=-\psi-\mathcal H\frac{\delta\rho_i}{\rho_i'},$$

and define an ordered relative entropy perturbation by

$$\boxed{ S_{ij}=3(\zeta_i-\zeta_j) }.$$

Both quantities are gauge invariant at first order. Indeed, $\delta\rho_i\to\delta\rho_i-\rho_i'T$ and $\psi\to\psi+\mathcal HT$, so the time shift cancels. If each background component is separately conserved and has constant $w_i=p_i/\rho_i$, then

$$S_{ij} =\frac{\delta_i}{1+w_i} -\frac{\delta_j}{1+w_j}, \qquad \delta_i\equiv\frac{\delta\rho_i}{\rho_i}.$$

Thus, for photons $\gamma$, baryons $b$, cold dark matter $c$, and massless neutrinos $\nu$,

$$S_{c\gamma}=\delta_c-\frac34\delta_\gamma, \qquad S_{b\gamma}=\delta_b-\frac34\delta_\gamma, \qquad S_{\nu\gamma}=\frac34(\delta_\nu-\delta_\gamma).$$

Adiabaticity means that all components share a common local time shift,

$$\frac{\delta\rho_i}{\rho_i'} =\frac{\delta\rho_j}{\rho_j'} \quad\hbox{for every pair }i,j,$$

so all $S_{ij}$ vanish. This definition is stronger and clearer than saying only that the total pressure is momentarily adiabatic. Malik and Wands develop the component curvatures, intrinsic and relative nonadiabatic pressure, and energy-transfer qualifications in Malik and Wands 2009, §8.2.3, printed pp. 49–51, eqs. (8.38)–(8.50), and §9.1, pp. 54–57, especially eqs. (9.1)–(9.11) (Open PDF).

The word “isocurvature” describes the initial condition; it does not promise zero curvature forever. For separately conserved barotropic fluids in the regular long-wavelength limit, each $\zeta_i$ is constant, but

$$\zeta=\sum_i\frac{\rho_i'}{\rho'}\zeta_i =\sum_i\frac{\rho_i+p_i}{\rho+p}\zeta_i$$

has time-dependent weights. For example, in a toy photon–CDM mixture with $\zeta_\gamma=0$ and $\zeta_c=S_{c\gamma}/3$,

$$\zeta=\frac{\rho_c}{4\rho_\gamma+3\rho_c}S_{c\gamma}.$$

It is suppressed deep in radiation domination and tends to $S_{c\gamma}/3$ in matter domination. No component entropy was created: the changing background mixture converted a relative-density perturbation into total curvature. This example neglects gradients and interactions, so it is not a finite-wavelength transfer function.

The five regular modes

Take initial data after neutrino decoupling and electron–positron annihilation, deep in radiation domination, with photons and baryons tightly coupled, massless collisionless neutrinos, pressureless CDM, and locally thermal, causally prepared distributions whose independent primordial neutrino data stop at the monopole and dipole. Expand the coupled Einstein–Boltzmann system as a regular series in $k\eta$ and in the small matter-to-radiation ratio. After removing residual gauge solutions and excluding singular or decaying primordial branches, the standard basis is:

Mode	Leading physical distinction
Adiabatic (AD)	All $S_{ij}=0$; the regular growing curvature mode is nonzero.
CDM isocurvature (CDI)	$S_{c\gamma}\ne0$ while the leading radiation-era curvature vanishes.
Baryon isocurvature (BI)	$S_{b\gamma}\ne0$ while the leading radiation-era curvature vanishes.
Neutrino-density isocurvature (NID)	$R_\gamma\delta_\gamma+R_\nu\delta_\nu=0$ initially but $S_{\nu\gamma}\ne0$.
Neutrino-velocity isocurvature (NIV)	Photon–baryon and neutrino velocities differ while their leading total momentum density cancels.

Here $R_\gamma=\rho_\gamma/(\rho_\gamma+\rho_\nu)$ and $R_\nu=1-R_\gamma$. Density entropies do not characterize NIV: its invariant seed is the relative velocity $V_\nu-V_\gamma$. With $V_i=\delta q_i/(\rho_i+p_i)$ and $\theta_i=-k^2V_i$, the leading radiation momentum cancellation is

$$R_\gamma V_\gamma+R_\nu V_\nu=0, \qquad V_\nu-V_\gamma\ne0.$$

The common gauge time shift cancels in the velocity difference.

Bucher, Moodley, and Turok derive the regular series and the compensation conditions in Bucher, Moodley, and Turok 2000, §II, printed pp. 6–14, especially eqs. (22)–(26), pp. 10–13 (Open PDF). Their classification also shows why regularity is not equivalent to demanding that every Newtonian-gauge potential remain finite: the NIV potential has a singular early-time form, while a regular synchronous description and the dimensionless four-dimensional Ricci scalar $a^2\eta^2\,{}^{(4)}R$ remain nonsingular. Physical curvature and stress behavior, residual gauge freedom, and the domain of the series must be checked together.

The five-mode statement is not a theorem for an arbitrary collisionless distribution. Independent regular neutrino multipoles of order $\ell\ge3$ can seed additional solutions. The standard basis excludes those primordial angular structures; the free-streaming hierarchy generates higher multipoles dynamically from its monopole and dipole. The hierarchy assumptions and their causal interpretation are discussed in Blas, Lesgourgues, and Tram 2011, Appendix B.1, printed pp. 30–31 (Open PDF).

BI and CDI are independent basis vectors, but a compensated matter isocurvature perturbation is a particular linear combination satisfying

$$\rho_bS_{b\gamma}+\rho_cS_{c\gamma}=0.$$

On a uniform-photon slice this reduces to $\delta\rho_b+\delta\rho_c=0$. The invariant statement is needed because a density cancellation on an unspecified time slicing is not itself physical. The compensated combination can be weakly gravitating at leading order, yet baryon pressure, recombination physics, and higher-order effects prevent it from being universally invisible.

Amplitudes, correlations, and transfer

Let $A_m(\mathbf k)$ denote the amplitude of mode $m\in\{\mathrm{AD,CDI,BI,NID,NIV}\}$. Statistical homogeneity and isotropy allow the primordial covariance

$$\left\langle A_m(\mathbf k)A_n^*(\mathbf k')\right\rangle =(2\pi)^3\delta^{(3)}(\mathbf k-\mathbf k')P_{mn}(k).$$

The $5\times5$ matrix $P_{mn}$ is Hermitian and positive semidefinite; for real parity-even scalar fields it is conventionally taken real and symmetric. An “uncorrelated” mixture has vanishing off-diagonal entries, whereas a correlated mixture does not. Mode amplitudes depend on the chosen basis and normalization, but $S_{ij}$ and the evolved observables do not.

The five-mode matrix description is given in Bucher, Moodley, and Turok 2000, §I, printed pp. 1–5, and §III, pp. 14–16, eq. (33) (Open PDF). Transfer functions subsequently map each initial basis vector into CMB and matter observables. Current empirical limits depend on a declared subset of modes, correlations, spectra, priors, and datasets, so they belong to observational inference rather than to this classification page.

This basis is conditional on the stated species and interactions. Extra dark radiation, interacting neutrinos, additional conserved charges, defects, or late active sources change the regularity problem and can add modes. Decaying solutions and pure gauge modes are not new primordial isocurvature species.

Exercises

Suppose at an initial time

$$\delta_\gamma=4A, \qquad \delta_\nu=4A, \qquad \delta_b=3A+B, \qquad \delta_c=3A+C.$$

Find the three density entropy perturbations above. State the adiabatic condition and the condition for a compensated baryon–CDM perturbation.

Solution

Substitution gives

$$S_{b\gamma}=B, \qquad S_{c\gamma}=C, \qquad S_{\nu\gamma}=0.$$

The displayed density data satisfy the adiabatic relations when $B=C=0$. The full perturbation is adiabatic only when its velocities and hierarchy data also match the regular AD mode. A nontrivial compensated baryon–CDM mode instead has

$$\rho_bB+\rho_cC=0,$$

or $C=-(\rho_b/\rho_c)B$, while $A$ may be set to zero to isolate the isocurvature part. The compensation cancels the total nonrelativistic density at that instant; it does not identify BI and CDI as the same basis mode.

References

• Blas, D., J. Lesgourgues, and T. Tram, “The Cosmic Linear Anisotropy Solving System (CLASS) II: Approximation Schemes,” Journal of Cosmology and Astroparticle Physics 2011(07), 034 (2011), DOI, Open PDF.
• Bucher, M., K. Moodley, and N. Turok, “The General Primordial Cosmic Perturbation,” Physical Review D 62, 083508 (2000), DOI, Open PDF.
• Malik, K. A., and D. Wands, “Cosmological Perturbations,” Physics Reports 475, 1–51 (2009), DOI, Open PDF.