Adiabatic Density Perturbations

• Adiabatic density perturbations are large-scale energy fluctuations where all components vary in fixed proportions, maintaining a common cosmic curvature.
• Gauge-invariant analysis shows that under vanishing non-adiabatic pressure, these perturbations remain conserved on superhorizon scales.
• Their role is crucial in modeling CMB anisotropies, large-scale structure formation, and constraining early universe and dark matter production scenarios.

Adiabatic density perturbations are large-scale fluctuations in the energy density of the universe in which all components (radiation, matter, dark matter, etc.) fluctuate in such a way that their relative ratios are conserved, corresponding to a common curvature perturbation across all species. These perturbations, fundamentally rooted in the adiabatic (curvature) mode generated by inflationary or analogous mechanisms, constitute the dominant source for the observed temperature anisotropies and large-scale structure in the universe. Adiabatic density perturbations stand in contrast to isocurvature (entropy) perturbations, where the total energy density remains fixed but the relative composition of the different species fluctuates.

1. Gauge-Invariant Definitions and Physical Content

In general-relativistic cosmological perturbation theory, adiabaticity has a precise, gauge-invariant definition. For any fluid component $a$ (e.g., photons, dark matter, baryons), the curvature perturbation on hypersurfaces of uniform species density is

$$\zeta_a \equiv -\psi - H\, \frac{\delta\rho_a}{\dot\rho_a}$$

where $\psi$ is the spatial curvature perturbation and $H$ the Hubble parameter. The adiabatic mode is characterized by all $\zeta_a$ being equal on superhorizon scales:

$\zeta_a = \zeta_b = \zeta \quad \forall\, a, b$

Correspondingly, the relative (isocurvature) entropy perturbations

$S_{ab} \equiv 3(\zeta_a - \zeta_b)$

vanish for purely adiabatic perturbations (Racco et al., 2022, Miedema, 2014).

Gauge-invariant evolution equations make it manifest that, in adiabatic evolution, the entropy perturbation $S_{ab}$ is either zero or a decaying homogeneous solution.

2. Evolution and Conservation of Adiabatic Modes

On superhorizon scales, the evolution of the adiabatic curvature perturbation $\zeta$ (or the equivalent comoving curvature perturbation ${\cal R}$) is governed by energy conservation and the intrinsic and relative non-adiabatic pressure perturbations. In a general matter sector,

$$\dot\zeta = -\frac{H}{\rho+P} \delta P_{\rm nad} + \mathcal{O}\left( \frac{k^2}{a^2 H^2} \right)$$

where

$$\delta P_{\rm nad} = \delta P - c_w^2\, \delta\rho\,,\quad c_w^2 = \frac{\dot P}{\dot\rho}$$

and the term vanishes for a strictly barotropic fluid or a single canonical scalar field, leading to conservation of $\zeta$ and ${\cal R}$ on superhorizon scales if $c_s^2 \neq c_w^2$ (Romano et al., 2015). This conservation law is gravity theory independent as it follows from momentum conservation and the vanishing of non-adiabatic pressure perturbations.

A nontrivial result is that the equivalence between the comoving, uniform-density, and proper-time slicings—and thus conservation of $\zeta$ and ${\cal R}$—holds only when $\delta P_{\rm nad} \approx 0$ and $c_s^2 \neq c_w^2$. In so-called ultra slow-roll inflation, where $c_s^2 = c_w^2 = 1$ and $\delta P_{\rm nad} = 0$, both ${\cal R}_c$ and $\zeta$ are not conserved due to the next-to-leading order gradient expansion contributions, highlighting the subtleties underlying adiabaticity and conservation laws (Romano et al., 2015).

3. Classification, Construction, and Physical Realization

Adiabatic density perturbations have an interpretation in terms of large coordinate (diffeomorphism) transformations at $\vec q=0$, i.e., residual gauge modes at infinite wavelength. Pajer & Jazayeri (Pajer et al., 2017) systematically classify all possible adiabatic modes in spatially-flat FRW backgrounds:

• Weinberg's First (Constant) Scalar Mode: Associated with spatial dilations and a constant curvature perturbation. It corresponds to the growing, non-decaying superhorizon mode:

$$\zeta_{\vec q}(t) = A_{\vec q} + B_{\vec q}\int^t\frac{dt'}{\epsilon(t') a^3(t')}$$

with $A_{\vec q}$ constant.

• Weinberg's Second (Decaying) Scalar Mode: Corresponds to pure time shifts; it decays with the expansion ($a^{-1}$) and disappears in comoving gauge.
• Special Conformal and Higher Gradient Modes: Related to spatial conformal transformations, contributing only at gradient order.

For a single perfect-fluid or minimally coupled scalar-field background, the adiabatic mode is unique and generates mode-independent predictions for superhorizon-scale correlators, leading to cosmological soft theorems such as Maldacena's consistency relations.

4. Mechanisms for Adiabatic Density Perturbation Generation

The standard origin of adiabatic perturbations is quantum vacuum fluctuations during inflation, which, upon horizon exit and subsequent decoherence, freeze as classical statistical perturbations and remain “adiabatic”:

• In single-field slow-roll inflation, curvature perturbations are produced with strictly adiabatic initial conditions; the spectrum is almost scale-invariant and matches CMB data.
• In the two-field ekpyrotic scenario, entropy (non-adiabatic) perturbations are generated but later converted to adiabatic modes through a bending of the background trajectory. The conversion mechanism is accompanied by quantum squeezing and decoherence, resulting in semi-classical, nearly scale-invariant curvature perturbations (Battarra et al., 2013).

A formal approach with the extended $\delta N$ formalism demonstrates that in multi-field inflation (e.g. mixed inflaton-curvaton or double inflation scenarios), the linear superhorizon curvature perturbation is always of the adiabatic type, but isocurvature contributions can be present, potentially leading to significant non-Gaussianity if isocurvature fields contribute non-linearly (0809.4646).

5. Adiabaticity in Cosmological Scenarios: Axions, Freeze-in Dark Matter, and Quantum Corrections

Certain cosmological mechanisms robustly guarantee adiabaticity:

• Freeze-in Dark Matter: In models where dark matter is produced via the freeze-in mechanism, the density perturbations of the dark matter sector remain strictly adiabatic. The key is that the only “clock” during the freeze-in epoch is the Standard Model bath temperature, so produced dark matter inherits the primordial adiabatic curvature mode, and any initial entropy perturbation decays rapidly. This leads to a complete absence of isocurvature signatures, making the freeze-in scenario consistent with current CMB constraints (Racco et al., 2022).
• Axion and ALP Scenarios: For axion dark matter, the adiabatic mode dominates when quantum fluctuations are suppressed (e.g., in pre-inflationary Peccei–Quinn symmetry-breaking scenarios with $f_a/H_{\rm inf} \gtrsim 1.25 \times 10^4$), and adiabatic plasma temperature fluctuations feed directly into the axion density on large scales. This can result in axion minicluster formation even in the purely adiabatic regime, removing the traditional connection between miniclusters and non-adiabatic initial conditions (Ayad et al., 7 Mar 2025, Allali et al., 8 Oct 2025).
• Loop Quantum Cosmology: In effective hydrodynamical LQC with inverse-triad corrections, the conservation of the adiabatic curvature perturbation is preserved, but quantum gravity corrections can couple the adiabatic and entropy modes, allowing the adiabatic curvature to source entropy perturbations in a way absent in standard classical cosmology (Li et al., 2011).

6. Mathematical Structure and Thermodynamic Consistency

The adiabatic condition is characterized by the vanishing of the entropy perturbation:

$\delta n - \frac{\delta\epsilon}{1 + w} = 0$

where $\delta n$ is the particle number density contrast, $\delta\epsilon$ the energy density contrast, and $w = p/\epsilon$ (Miedema, 2014). In a barotropic fluid ($p = p(\epsilon)$), particle number density fluctuations do not contribute to pressure, ensuring adiabatic perturbation evolution. This condition generalizes to multi-fluid systems and is preserved under gauge transformations when the proper gauge-invariant variables are used.

For generic perturbations, the total power spectrum can be sourced by both adiabatic and isocurvature modes. However, for a strictly adiabatic spectrum, all observable large-scale features derive from a single primordial curvature perturbation variable, whose evolution and non-Gaussianity can be systematically computed with the $\delta N$ expansion (0809.4646).

7. Observational and Theoretical Implications

Adiabatic density perturbations leave a characteristic imprint: large-scale temperature and polarization anisotropies in the CMB display consistency relations and acoustic peak structures that match the predictions from primordial adiabatic curvature perturbations. The Planck satellite constraints on isocurvature modes restrict their fractional contribution to $|S|/|\zeta| \lesssim 10^{-2}$, essentially requiring that primordial density perturbations be dominantly adiabatic (Racco et al., 2022).

The model-independence and universality of the adiabatic mode underly its robustness as a cosmological initial condition—for inflation, contraction, or more exotic models—subject to technical caveats such as the breakdown of superhorizon conservation in cases like ultra slow-roll inflation (Romano et al., 2015).

Adiabatic perturbations represent the physical, gauge-invariant content of the initial conditions leading to cosmic structure and continue to be central both in precision cosmology and the discrimination of early universe scenarios.