Gradient, Tachyonic, and Jeans Instabilities

An instability is meaningful only after its wavelength and growth time are compared with the background lifetime and the effective-theory cutoff. The same negative value of $\omega^2$ can describe three physically different situations: a gradient instability whose rate grows with wavenumber, a tachyonic instability controlled by a finite mass scale, or the ordinary Jeans growth that lets gravitating matter form structure. The dispersion relation, not the word “negative,” decides which case applies.

Required background. Variational and Symplectic Geometry supplies the reduced physical variables, while Ordinary Differential Equations supplies exponential-mode and timescale analysis.

Helpful background. Einstein Equations provide the gravitational source term behind Jeans growth, Gravitational Actions and Boundary Terms fix the variational problem, and Formulations of General Relativity clarify which gauge-fixed perturbations and constraints enter a dispersion calculation.

Dispersion from the common reduced action

Use the same physical quadratic action as on the kinetic-sign page,

$$S_{\mathrm{red}}^{(2)}=\frac12\int dt\,d^3x\,a^3 \left[ \dot q^{\mathsf T}K\dot q -\frac{1}{a^2}\partial_iq^{\mathsf T}G\partial_iq -q^{\mathsf T}M^2q +2\dot q^{\mathsf T}Bq \right].$$

It is obtained from a declared action only after the background, gauge-fixed formulation, constraint reduction, and solution branch have been chosen. Its coefficients are trusted only inside a stated EFT regime. On scales short compared with their variation time, set $q\propto e^{-i\omega t+i\mathbf k\cdot\mathbf x}$. The local dispersion relation is

$$\det\!\left[-\omega^2K-2i\omega B +k_{\mathrm{phys}}^2G+M^2\right]=0, \qquad k_{\mathrm{phys}}=\frac{k}{a}.$$

For one canonically normalized eigenmode with negligible $B$, this reduces to

$$\omega^2=c_s^2k_{\mathrm{phys}}^2+m_{\mathrm{eff}}^2.$$

The kinetic condition $K\gt0$ and the gradient condition $c_s^2\gt0$ are logically distinct. In a coupled system, squared high-frequency speeds are generalized eigenvalues of $Gv=c_s^2Kv$. A positive $K$ does not prevent a negative eigenvalue of this pencil.

When the matrices vary with time, an instantaneous diagonalization also contributes terms proportional to the rotation rate of the eigenvectors. Those terms belong with $B$ and can mix modes even when $K$ and $G$ are momentarily diagonal. The frozen-coefficient dispersion relation is reliable only when $|\dot K|/|K|$, $|\dot G|/|G|$, and the eigenbasis rotation are small compared with the frequency being tested. Near an eigenvalue crossing, one must evolve the coupled system rather than attach a separate sound speed to each instantaneous label.

Gradient instability: ultraviolet-fast growth

If $c_s^2=-\lvert c_s\rvert^2\lt0$, then at sufficiently large wavenumber

$$\omega=\pm i|c_s|k_{\mathrm{phys}}, \qquad \tau_{\mathrm{grad}}(k)=\frac{1}{|c_s|k_{\mathrm{phys}}}.$$

The rate increases toward the ultraviolet. Therefore a negative high-frequency sound speed normally invalidates the background: within the interval where the quadratic EFT is reliable, shorter modes grow faster. The precise test is whether there exists a controlled window

$$L_{\mathrm{bg}}^{-1}\ll k_{\mathrm{phys}}\ll\Lambda_{\mathrm{EFT}}$$

in which the WKB approximation holds and $c_s^2\lt0$. If the sign reversal occurs only at or above $\Lambda_{\mathrm{EFT}}$, the truncated action cannot decide what the completion does. If it occurs well below the cutoff, appealing to unknown ultraviolet physics does not rescue the branch.

The gradient criteria extracted from reduced scalar and tensor actions in scalar–tensor cosmology are presented in Kobayashi 2019, § 3.2.1. The background and gauge choice matter because integrating out a constraint can introduce momentum-dependent coefficients; the sign must be checked in the actual physical regime rather than inferred from a covariant operator in isolation.

A small positive $c_s^2$ is not a gradient instability, but it raises a different question. Canonical interactions may become enhanced by inverse powers of the sound speed, lowering the strong-coupling scale. Conversely, a superluminal characteristic speed is not the same algebraic condition as $c_s^2\lt0$: it concerns causal cones and possible ultraviolet completion, whereas a negative value produces exponential high-frequency growth. Conflating these diagnoses obscures which calculation has actually failed.

Tachyonic instability: a finite clock

Let $c_s^2\gt0$ but $m_{\mathrm{eff}}^2=-\mu^2\lt0$. Only modes satisfying $c_s^2k_{\mathrm{phys}}^2\lt\mu^2$ grow, with rate

$$\Gamma(k)=\sqrt{\mu^2-c_s^2k_{\mathrm{phys}}^2}, \qquad \tau_{\mathrm{tach}}(k)=\Gamma(k)^{-1}.$$

The maximum rate is finite, $\Gamma(0)=\mu$. A negative mass squared may signal that the chosen background should roll or bifurcate to a new branch; it is not a negative-norm state. Whether it is unacceptable depends on the intended duration $T_{\mathrm{bg}}$. A conservative test is $\Gamma T_{\mathrm{bg}}\ll1$ for modes actually present in the data class. If growth is slower than the age or residence time of the background, imposing a blanket “no tachyon” prior can discard viable models.

This timescale qualification is central to Gsponer and Noller 2022, §§ III–IV, which contrasts ghost and gradient exclusions with tachyonic priors in dark-energy models. Their conclusion is not that all negative mass eigenvalues are harmless: a rate much faster than the background evolution, or a mode excited with appreciable amplitude, still destroys the assumed solution.

For several fields, the relevant mass matrix is not $M^2$ by itself. After making the positive kinetic matrix canonical, inspect $K^{-1/2}M^2K^{-1/2}$ together with the mixing induced by $B$. A negative diagonal entry in the original basis need not be a negative physical eigenvalue. On an expanding background the mode equation also contains friction, schematically $\ddot q+3H\dot q-\mu^2q=0$. Its growing exponent is $[-3H+\sqrt{9H^2+4\mu^2}]/2$, so comparing $\mu$ with $H$ is more informative than merely recording the sign of $m_{\mathrm{eff}}^2$.

Jeans instability: physical infrared growth

For a nonrelativistic, self-gravitating fluid of background density $\rho$ and adiabatic sound speed $c_m$, a local static approximation gives the Jeans dispersion relation below. This idealization subtracts the uniform background source from Poisson’s equation and retains only its perturbation: an infinite uniform static Newtonian fluid is not itself a global equilibrium. A cosmological derivation instead keeps expansion consistently. Linearizing the local continuity, Euler, and perturbation Poisson equations gives

$$\omega^2=c_m^2k^2-4\pi G\rho.$$

The Jeans scale and longest-wavelength growth time are

$$k_J^2=\frac{4\pi G\rho}{c_m^2}, \qquad \tau_J=(4\pi G\rho)^{-1/2}.$$

Modes with $k\lt k_J$ collapse because gravity overcomes pressure. This is an infrared instability of a chosen homogeneous matter background and is the mechanism of structure formation, not a UV failure of the gravitational EFT. In an expanding universe, Hubble friction and time-dependent density modify the simple exponential, but the physical comparison remains between growth, pressure support, and background evolution.

The sign follows directly from the coupled equations. Continuity gives $\dot\delta=-\nabla\!\cdot v$, Euler evolution gives $\dot v=-c_m^2\nabla\delta-\nabla\Phi$, and Poisson’s equation gives $\nabla^2\Phi=4\pi G\rho\delta$. Eliminating $v$ and $\Phi$ yields $\ddot\delta-c_m^2\nabla^2\delta-4\pi G\rho\delta=0$. Pressure supplies the positive $k^2$ term, while the attractive gravitational constraint supplies the negative scale-independent term. This derivation also shows why changing the matter sound speed moves the threshold without changing Newton’s constant.

Mathematically, Jeans growth is a tachyonic eigenvalue of the coupled matter–gravity system. Physically, it should not be vetoed by a stability rule designed to exclude arbitrarily rapid high-$k$ growth. The relevant branch, matter model, and boundary conditions must remain visible.

A scale-by-scale decision procedure

First verify that constraints have been solved and $K$ is positive on the chosen branch. Next compute the eigenfrequencies throughout the controlled window rather than only at $k=0$. Classify a growing mode by the scaling of $\Gamma$: proportional to $k$ indicates gradient failure, bounded by a mass scale indicates tachyonic behavior, and tied to $G\rho$ in the coupled density sector indicates Jeans growth. Finally compare $\Gamma^{-1}$ with the background lifetime, the source crossing time, and the EFT resolution scale.

Independent checks include dimensions—every term in $\omega^2$ has units of inverse time squared—and limiting behavior. Sending $G\to0$ removes Jeans growth; sending $m_{\mathrm{eff}}^2\to0$ removes tachyonic growth; taking $k\to\infty$ isolates the gradient eigenvalues. A claimed classification that fails one of these limits has mixed distinct physical effects.

A useful reporting rule is to quote a dimensionless amplification exponent rather than a sign alone. For a mode present for time $T$, compute $\mathcal A(k)=\Gamma(k)T$ over the resolved wavenumber interval. Values $\mathcal A\ll1$ give little fractional amplification of an initially linear perturbation. If its initial amplitude is $\delta_0\ll1$, linearization fails when $\delta_0 e^{\mathcal A}$ becomes order unity, or $\mathcal A\gtrsim\ln(1/\delta_0)$. Plotting or tabulating $\mathcal A(k)$ distinguishes ultraviolet-rising gradient growth from infrared-bounded tachyonic or Jeans growth without silently assuming the same initial amplitude.

Suppose, for example, that a cosmological scalar has $c_s^2\gt0$ at all subcutoff wavenumbers but one negative mass eigenvalue of magnitude $0.1H^2$. The sign alone would fail a strict no-tachyon rule, yet the growing exponent is much smaller than $H$, so the field changes by only order unity over many expansion times. A second branch with $c_s^2=-10^{-2}$ is qualitatively worse: modes near a cutoff hundreds of times larger than $H$ grow within a small fraction of one expansion time. This comparison explains why a single binary “stable/unstable” label loses the scale information needed to judge the intended background.

This page supplies theory-side stability criteria, not observational exclusion regions. Likelihoods and mutable numerical bounds belong to Tests of Gravity; stable quantum-gravity EFT belongs to Quantum Gravity; generic quantum-field calculations receive only the bounded bridge needed to interpret the instability.

Common pitfalls

Treating every negative $\omega^2$ alike. Inspect its dependence on $k$ and the background scales. A high-$k$ gradient catastrophe and slow Jeans growth demand opposite interpretations.

Checking only one wavelength. A stable homogeneous mode does not prove positive sound-speed eigenvalues, while an unstable homogeneous mode does not prove ultraviolet failure.

Ignoring the EFT window. Neither modes beyond the cutoff nor wavelengths comparable to the background scale justify a local WKB dispersion relation.

Exercises

A mode has $c_s^2=-0.04$ at $k_{\mathrm{phys}}=0.2\Lambda$ inside a controlled EFT window. Find its growth rate in units of $\Lambda$ and explain the stability verdict.

Solution

Here $|c_s|=0.2$, so $\Gamma=|c_s|k_{\mathrm{phys}}=0.04\Lambda$ and $\tau=25\Lambda^{-1}$. Because the mode lies below the cutoff and the rate increases with wavenumber, this is a genuine gradient instability of the tested branch. Its finite value at this one wavelength does not make it a harmless slow tachyon.

Derive the Jeans wavenumber from $\omega^2=c_m^2k^2-4\pi G\rho$ and state what happens as $G\to0$.

Solution

The threshold is $\omega^2=0$, giving $k_J=\sqrt{4\pi G\rho}/c_m$. Modes below this wavenumber grow, whereas pressure stabilizes shorter modes. When $G\to0$, $k_J\to0$ and the growth rate vanishes, confirming that the instability comes from self-gravity rather than a wrong-sign matter gradient term.

References

• Gsponer, Rafaela, and Johannes Noller. 2022. “On Tachyonic Stability Priors for Dark Energy.” Physical Review D 105, 064002. DOI; arXiv:2107.01044.
• Kobayashi, Tsutomu. 2019. “Horndeski Theory and Beyond: A Review.” Reports on Progress in Physics 82, 086901. DOI; arXiv:1901.07183.