Super-Comoving Variables in Cosmological Flows

• Super-comoving variables are time-dependent transformations that rescale density, velocity, and pressure to absorb background expansion effects, enabling direct comparisons with non-expanding systems.
• They simplify the hydrodynamic and gravitational equations by localizing the expansion factor primarily in the Poisson equation, reducing computational complexity in cosmological simulations.
• This framework facilitates analytic mappings between turbulent, unstable, contracting, and expanding flows, enhancing our understanding of instabilities and turbulence in dynamic geometries.

Super-comoving variables, also known as rescaled variables, are a set of time-dependent transformations applied to fluid or kinetic equations, allowing the systematic removal of explicit background expansion or contraction effects. This formalism was initially developed for cosmological structure formation simulations, where its key utility lies in reformulating the hydrodynamic and gravitational equations such that the expansion factor appears in only a minimal subset of terms, usually the Poisson equation. Recent developments extend these techniques beyond Cartesian cosmology to non-Euclidean or non-isotropic backgrounds, notably to local models of spherically contracting or expanding flows. The super-comoving mapping enables analytic solutions and direct correspondence with known results for standard Cartesian flows, fundamentally simplifying the analysis of turbulence, instabilities, and overall dynamics in backgrounds with time-dependent geometry (Lynch et al., 16 Jan 2026, Gnedin et al., 2011).

1. Formal Definition and Transformations

Super-comoving variables are introduced by applying time-dependent scalings to density, velocity, and pressure, together with a rescaled time coordinate. For a generic background with time-dependent scale factors $L_1(t), L_2(t), L_3(t)$ and metric $g_{ij} = \operatorname{diag}(L_1^2, L_2^2, L_3^2)$:

• The physical-to-super-comoving mapping is

$$\rho(\mathbf{r}, t) = \rho_0(t)\, \tilde{\rho}(\mathbf{r}, t),\quad v(\mathbf{r}, t) = v_0(t)\, \tilde{v}(\mathbf{r}, t),\quad p(\mathbf{r}, t) = p_0(t)\, \tilde{p}(\mathbf{r}, t),$$

with $\rho_0(t)$ and related scaling functions chosen to absorb background divergences.

• The super-comoving time is defined by

$\tau(t) = \int^t v_0(t')\, dt'.$

For isotropic flows ($L_1 = L_2 = L_3 = L(t)$) and the conformal choice $v_0 = \|g\|^{-1/3}$, this simplifies many prefactors.

• In cosmological applications, with the expansion factor $a(t)$, the mapping (for example, position and time) is

$$\mathbf{r} = a(t)\, \tilde{\mathbf{x}}, \qquad d\tilde{t} = \frac{dt}{a^2(t)}.$$

The transformed equations acquire canonical forms, typically identical to their non-expanding equivalents, with time and amplitude source terms isolated (Gnedin et al., 2011, Lynch et al., 16 Jan 2026).

2. Reformulation of Hydrodynamic and Gravitational Equations

In the super-comoving formalism, the equations of motion for mass, momentum, and (when relevant) energy simplify structurally:

• The continuity and Euler (momentum) equations take the same form as in non-expanding space:

$$\frac{\partial \tilde{\rho}}{\partial \tau} + \nabla \cdot (\tilde{\rho} \tilde{\mathbf{v}}) = 0$$

$$\frac{\partial \tilde{\mathbf{v}}}{\partial \tau} + (\tilde{\mathbf{v}} \cdot \nabla) \tilde{\mathbf{v}} = -\nabla \tilde{p} / \tilde{\rho}$$

• All explicit expansion or contraction terms disappear, except for metric derivatives in anisotropic backgrounds.
• The Poisson equation retains a dependence on the scale factor:

$$\nabla_{\tilde{x}}^2 \tilde{\varphi} = 4\pi G\, a(t)\left[ \tilde{\rho} - \overline{\tilde{\rho}} \right]$$

This property underpins the method’s efficacy for both cosmological N-body codes and for the analysis of spherical contraction/expansion—the only coupling to background expansion is relegated to the gravity solver or, in the absence of gravity, entirely eliminated (Gnedin et al., 2011, Lynch et al., 16 Jan 2026).

3. Mapping Analytical Solutions and Physical Interpretation

Super-comoving variables permit direct mapping of analytical solutions in non-stationary backgrounds to solutions of canonical Cartesian hydrodynamics:

• For any known solution $$(\tilde{\rho}(\mathbf{X}, \tau), \tilde{\mathbf{v}}(\mathbf{X}, \tau), \tilde{p}(\mathbf{X}, \tau))$$ to the standard Euler equations, a solution to the spherically contracting/expanding system is immediately given by:

$$x_i = L_i(t) X_i,\quad \rho(x, t) = \rho_0(t) \tilde{\rho}(X, \tau),\quad v(x, t) = v_\text{bg}(x, t) + v_0(t) \tilde{v}(X, \tau)$$

with $v_\text{bg}$ capturing Hubble-type stretching terms associated with $\dot{L}_i/L_i$.

• Example: For a vortex in $2$D with $L_1 = L_2$, and with a suitable relation $L_3 \propto L_1^{-2(\gamma-2)/(\gamma-1)}$, the rescaled vortex solution maps directly to the evolving spherical case with transformed amplitudes and coordinates (Lynch et al., 16 Jan 2026).

This mapping establishes a rigorous connection between models of turbulent or unstable flows in contracting/expanding media and their standard Euclidean analogs, providing substantial analytic tractability.

4. Linear Instabilities and Mode Freezing

Linear perturbations in the rescaled frame evolve as $e^{\sigma \tau}$, but their actual growth rate in physical time is modulated by the background scaling:

$$A(t) \sim \exp\left[ \sigma \tau(t) \right], \qquad \frac{d}{dt} \ln A(t) = \sigma v_0(t)$$

For isotropic exponential expansion, $L(t)=e^{Ht}$, this analysis shows that instabilities with $\sigma < 2H$ are suppressed: their amplitude saturates before they can realize a full e-fold in the original $t$ coordinate. For collapsing flows, a similar time constraint applies, with the feasible growth period limited by the maximum attainable $\tau$ before dominant nonlinear or geometrical effects intervene (Lynch et al., 16 Jan 2026). This phenomenon is fundamental to understanding mode selection and the survivability of instabilities in dynamic backgrounds.

5. Applications and Implementation in Cosmological Simulations

Super-comoving variables are widely used in numerical cosmology for both collisionless (dark matter) and baryonic fluid solvers:

• The kinetic, hydrodynamic, and particle equations become formally invariant under expansion, facilitating direct use of standard solvers.
• Cosmic expansion and "DC mode" treatment is isolated to the Poisson equation:

$$\nabla_x^2 \tilde{\varphi} = 4\pi G\, a_\text{box}(t) \overline{\tilde{\rho}}\, \delta(\tilde{\mathbf{x}}, \tilde{t})$$

with $a_\text{box}(t)$ incorporating the large-scale overdensity of the simulation domain.

• All other modules—hydrodynamics, radiative transfer, star formation—are unaffected by the expansion history, except where explicit background rates or redshifts are required (Gnedin et al., 2011).

This partitioning significantly reduces computational and conceptual overhead, yielding minimal-coupling algorithms for evolving complex cosmic structures.

6. Turbulence, Kolmogorov Spectrum, and the Incompressible Limit

In isotropically contracting or expanding media, the super-comoving mapping directly relates the small-scale (low Mach number, $\sigma \ll 1$) limit to standard incompressible Euler turbulence:

• For $\gamma = 5/3$ and $L_1 = L_2 = L_3 = L(t)$, with the conformal rescaling, the system possesses no source terms. The small-scale turbulence is described by:

$\tilde{P}_k(\tau) \sim \epsilon^{2/3} k^{-5/3}$

• The physical spectrum is related by

$P_k(t) = L^{-5/3} \tilde{P}_k(\tau(t))$

with the Fourier amplitudes rescaled due to the background contraction or expansion.

A key implication is that isotropically evolving systems "inherit" a rescaled Kolmogorov spectrum, with amplitudes and temporal evolution modulated by the background metric's volume contraction or expansion (Lynch et al., 16 Jan 2026). This offers an analytic handle on turbulent cascade properties under global changes in scale.

7. Conceptual Significance and Broader Applications

The super-comoving framework establishes a general method for reducing time-dependent metric effects in evolution equations to explicit source terms or single-location coefficients (e.g., gravity solvers). Its efficacy in absorbing background divergences and universalizing the structure of hydrodynamic and Vlasov-Poisson equations simplifies both analytic and numerical work. These transformations underpin contemporary simulation codes in cosmology (Gnedin et al., 2011), and the extension to spherically symmetric and anisotropic flows (Lynch et al., 16 Jan 2026) further broadens the formalism's versatility, enabling direct cross-comparison between solutions in different geometries and dynamical backgrounds. This formalism provides a unified platform for investigating instabilities, turbulence, and nonlinear structure formation in time-evolving backgrounds.