Cosmology, Strong Coupling, and Viability

Homogeneous bimetric cosmology is organized by a factorized Bianchi constraint. One factor fixes an algebraic ratio of scale factors; the other relates the two lapses and permits that ratio to evolve. This branch classification is exact at the background level, but viability is decided only after separate no-ghost, gradient, tachyonic, strong-coupling, and EFT-cutoff checks. No single “Higuchi test” replaces that sequence.

Required background. Variational and Symplectic Geometry supplies constrained perturbation and kinetic-matrix language, and Weak-Field Gravity supplies helicity and linear-stability diagnostics.

Helpful background. Einstein Equations supplies Friedmann dynamics; Gravitational Actions and Boundary Terms supplies the minisuperspace variation; Formulations of General Relativity supplies lapse constraints; and Gravity Beyond General Relativity supplies the cosmological model-selection setting.

Two FLRW metrics and one Bianchi factor

Use the same interaction convention as elsewhere in the chapter,

$$S=\sqrt{g^{-1}f},\qquad \alpha=\frac{m_f}{m_g}, \qquad \mu^2=\frac{m^4}{m_g^2},$$

with interaction $-2m^4\sqrt{-g}\sum_{n=0}^4\beta_ne_n(S)$. The Einstein terms have coefficients $m_g^2$ and $m_f^2$, without a factor $1/2$, so the corresponding reduced Planck masses obey $M_g^2=2m_g^2$ and $M_f^2=2m_f^2$. Matter couples only to $g$, and $\rho=T_{\mu\nu}u^\mu u^\nu$ uses the site’s stress tensor $T_{\mu\nu}=-2(\sqrt{-g})^{-1}\delta S_m/\delta g^{\mu\nu}$. For equal spatial curvature $k$, choose cosmic time for the matter-coupled $g$ metric:

$$ds_g^2=-dt^2+a(t)^2d\Sigma_k^2, \qquad ds_f^2=-X(t)^2dt^2+b(t)^2d\Sigma_k^2, \qquad y=\frac{b}{a}.$$

On a real positive square-root branch, $S^\mu{}_\nu=\operatorname{diag}(X,y,y,y)$. The $g$ Friedmann equation is

$$3\left(H^2+\frac{k}{a^2}\right) =\frac{\rho}{2m_g^2} +\mu^2\left(\beta_0+3\beta_1y+3\beta_2y^2+\beta_3y^3\right),$$

and covariance of the interaction gives the factorized Bianchi condition

$$\left(\beta_1+2\beta_2y+\beta_3y^2\right) \left(\dot b-X\dot a\right)=0.$$

The ansatz and factorization are derived in Schmidt-May and von Strauss 2016, §6.3.1, Eqs. (6.26)–(6.32), PDF. Their Eq. (5.24) defines the source stress tensor without the factor two used here, so $T^{\mathrm{src}}_{\mu\nu}=T_{\mu\nu}/2$. Translating that definition explains the factor $1/(2m_g^2)$ above. As an independent check, varying $m_g^2\int\sqrt{-g}R+S_m$ with the interaction removed gives $G_{\mu\nu}=T_{\mu\nu}/(2m_g^2)$, or $3H^2=\rho/M_g^2$ on the flat branch.

Algebraic and dynamical branches

On the algebraic branch,

$$\beta_1+2\beta_2y+\beta_3y^2=0,$$

so a discrete root fixes $y$ to a constant, assuming the polynomial is not identically zero. The case $\beta_1=\beta_2=\beta_3=0$ instead removes all interactions coupling the two metrics and must be treated separately. The interaction in the $g$ background equation then acts as a cosmological constant. This background-level resemblance to general relativity is deceptive: scalar and vector modes can lose their quadratic kinetic terms, making linear perturbation theory strongly coupled, and additional nonperturbative or tensor instabilities can occur.

On the dynamical branch,

$$X=\frac{\dot b}{\dot a}.$$

For $\dot a\ne0$, the relation implies $H_f=\dot b/(Xb)=H/y$. The source-free $f$ Friedmann equation consequently gives

$$3\left(H^2+\frac{k}{a^2}\right) =\frac{\mu^2}{\alpha^2} \left(\frac{\beta_1}{y}+3\beta_2+3\beta_3y+\beta_4y^2\right).$$

Equate this with the $g$ equation and multiply by $y/\mu^2$. Defining $\varrho=\rho/(2m^4)$ gives the algebraic equation

$$\begin{aligned} 0={}&\beta_3y^4+\left(3\beta_2-\frac{\beta_4}{\alpha^2}\right)y^3 +3\left(\beta_1-\frac{\beta_3}{\alpha^2}\right)y^2\\ &+\left(\beta_0+\varrho-\frac{3\beta_2}{\alpha^2}\right)y -\frac{\beta_1}{\alpha^2}. \end{aligned}$$

Different real roots define distinct histories and cannot be joined freely through a point where the lapse, square root, or kinetic matrix becomes singular. The conventional dynamical subclasses are:

• Infinite branch: $y$ diverges in the early-time limit and decreases toward a later finite value.
• Finite branch: $y$ approaches a finite early-time value—often zero in commonly studied matter-era solutions—and evolves toward a finite late-time root.
• Exotic branches: the remaining roots, often associated with a bounce, turnaround, lapse sign change, or an asymptotically static epoch.

This classification is a statement about background roots, not a stability ranking. Its derivation and qualifications appear in Schmidt-May and von Strauss 2016, § 6.3.2, pp. 68–69.

Five distinct consistency tests

After solving the lapse and nondynamical scalar/vector variables, write a schematic physical mode action as

$$S^{(2)}=\frac12\int dt\,d^3k\,a^3 \left[K(t)|\dot q|^2-G(t)\frac{k^2}{a^2}|q|^2-M^2(t)|q|^2\right].$$

The following failures are different:

1. Higuchi/helicity-0 ghost. On exact de Sitter, a genuinely massive Fierz–Pauli field avoids the helicity-0 negative-norm range only for $m_{\mathrm{FP}}^2\ge2H^2$; $m_{\mathrm{FP}}=0$ is the separate gauge-invariant massless theory, not a point inside the five-polarization massive family Higuchi 1987, pp. 397–436. On evolving bimetric backgrounds the relevant statement is positivity of the reduced scalar kinetic matrix; inserting a time-dependent mass into the de Sitter formula is not a derivation.
2. Kinetic ghost. A negative eigenvalue of $K$ gives a wrong-sign mode even if the background Friedmann equation is regular.
3. Gradient instability. With $K\gt0$, a negative high-frequency sound-speed eigenvalue, schematically $G/K\lt0$, produces rapid short-wavelength growth. Passing a Higuchi-type kinetic bound does not rule this out.
4. Strong coupling. If a kinetic eigenvalue vanishes, canonical normalization drives interactions to a low scale. “No mode at quadratic order” may therefore mean loss of perturbative control rather than a new gauge symmetry.
5. EFT cutoff. The strong-coupling scale, often organized around a background-dressed version of $\Lambda_3=(m_{\mathrm{eff}}^2M_{\mathrm{eff}})^{1/3}$, need not equal the ultraviolet cutoff. Here $m_{\mathrm{eff}}$ and $M_{\mathrm{eff}}$ denote the mass and canonical interaction scale of the selected massive sector. Predictions require $H$, physical wave number, time-variation scales, and characteristic inverse curvature radii below the applicable cutoff and omitted-operator scale.

The distinction between Higuchi and gradient conditions in bimetric cosmology is explicit in de Rham 2014, § 12.4.2, eqs. (12.50)–(12.55). The distinction between a strong-coupling scale and a cutoff is emphasized in de Rham 2014, § 10.5, p. 105.

Results and open questions

The theoretical status summarized here is fixed at 2026-08-19. It concerns branch consistency, not a fit to a mutable cosmological data release.

Background branch/result	Evidence status at the cutoff	Scope and limitation
Fixed-reference dRGT with Minkowski reference metric	No nontrivial spatially flat or closed FLRW branch in the standard homogeneous ansatz; open branches exist de Rham 2014, § 12.2, eqs. (12.13)–(12.21)	The no-go is ansatz- and reference-metric-specific; open self-accelerating solutions can have vanishing quadratic kinetics or other instabilities
Algebraic bimetric branch	Exact constant-$y$ background branch, often GR-like in the $g$ Friedmann equation Schmidt-May and von Strauss 2016, § 6.3.2, pp. 68–69	Missing kinetic terms and reported nonperturbative/tensor problems prevent background agreement from establishing predictivity
Infinite dynamical branch	Exists as a root family; stability is parameter-dependent, with special submodels behaving better than generic choices Schmidt-May and von Strauss 2016, §§ 6.3.2–6.3.4, pp. 68–71	Ghost and gradient conclusions cannot be transferred between parameter subspaces; nonlinear completion of a failing linear epoch is unresolved
Finite dynamical branch	Contains extensively studied backgrounds; sizable regions are no-ghost but develop an early scalar gradient instability Schmidt-May and von Strauss 2016, § 6.3.4, pp. 70–71	A nonlinear-screening interpretation identifies its onset with $H\sim m_{\mathrm{FP}}$, but a complete bimetric nonlinear perturbation calculation and EFT hierarchy remain required Lüben, Schmidt-May, and Smirnov 2020, §§ IV.B and VI
Exotic dynamical branches	Background bounces, turnarounds, or static limits occur for selected roots Schmidt-May and von Strauss 2016, §§ 6.3.2 and 6.3.4, pp. 68–71	Perturbations and branch crossings are less completely classified; lapse, square-root, and kinetic degeneracies require case-by-case checks

For the fixed-reference distinction, see de Rham 2014, § 12.2, eqs. (12.13)–(12.21). For bimetric algebraic/finite/infinite/exotic branches and their perturbative status, see Schmidt-May and von Strauss 2016, §§ 6.3.2–6.3.4, pp. 68–71.

There are two recurring competing readings of a problematic linear epoch. One treats it as exclusion of that background because the mode grows within the nominal low-energy regime. The other proposes that nonlinear Vainshtein-type effects intervene before linear growth becomes predictive. A concrete version of the second interpretation identifies $H\sim m_{\mathrm{FP}}$ as the cosmological Vainshtein scale and relates the finite-branch WKB instability to the breakdown of the linear Stückelberg description Lüben, Schmidt-May, and Smirnov 2020, § IV.B. That paper also makes its early-time phenomenology conditional on integrating out the nonlinear massive mode and calls for a fully bimetric nonlinear perturbation calculation Lüben, Schmidt-May, and Smirnov 2020, § VI. It therefore supplies evidence for the screening interpretation without closing the hierarchy among growth, screening, strong coupling, and the EFT cutoff. Similarly, satisfying a generalized Higuchi condition establishes only a no-ghost statement for the analyzed scalar sector; it is not a certificate of gradient stability or ultraviolet completion.

Open questions. The conclusions could change when a nonlinear evolution follows a finite- or infinite-branch cosmology through the nominal gradient-unstable epoch with a demonstrated EFT hierarchy; when a full perturbation analysis changes the status of an exotic or algebraic branch; when a real-square-root or lapse-singularity theorem changes the allowed root continuations; or when the cited branch/stability results are corrected, superseded, or withdrawn. Updated observational fits can change empirical conclusions without resolving these theoretical distinctions.

Exercises

1. Diagnose the two Bianchi factors. Take $(\beta_1,\beta_2,\beta_3)=(1,-1,0)$, $a(t)=t^{2/3}$, and $b(t)=y_0a(t)$. Find the algebraic root and the lapse required on the dynamical branch. What is $S$ for positive lapse?

Solution

The algebraic polynomial is $1-2y=0$, so $y_0=1/2$. On the dynamical branch,

$$X=\frac{\dot b}{\dot a}=y_0=\frac12.$$

Since $g^{-1}f=\operatorname{diag}(X^2,y_0^2,y_0^2,y_0^2)$, the positive real branch is

$$S=\operatorname{diag}\left(\frac12,\frac12,\frac12,\frac12\right).$$

At this particular point both Bianchi factors vanish. The Bianchi identity alone therefore does not decide which local branch continuation is taken; the remaining field equations and the nondegeneracy of the lapse/root evolution must do so.

2. Separate four failure modes. In the schematic quadratic action, classify: (a) $K\lt0$; (b) $K\gt0$ and $G\lt0$; (c) $K\to0$ with finite cubic interactions; and (d) $K,G\gt0$ but $M^2\lt0$.

Solution

(a) is a kinetic ghost. (b) is a gradient instability whose rate grows with physical wave number inside the EFT regime. (c) is strong coupling: canonical normalization $q_c\sim\sqrt Kq$ enhances the interactions, so quadratic mode counting loses predictive force. (d) is a tachyonic or Jeans-like long-wavelength instability; its physical importance depends on whether its growth time is shorter than the background evolution time. None of these diagnoses, by itself, states where the ultraviolet cutoff lies.

References

• de Rham, Claudia. “Massive Gravity.” Living Reviews in Relativity 17 (2014): 7. doi:10.12942/lrr-2014-7; Open PDF.
• Higuchi, Atsushi. “Forbidden Mass Range for Spin-2 Field Theory in de Sitter Spacetime.” Nuclear Physics B 282 (1987): 397–436. DOI.
• Lüben, Marvin, Angnis Schmidt-May, and Juri Smirnov. “Vainshtein Screening in Bimetric Cosmology.” Physical Review D 102 (2020): 123529. DOI; Open PDF.
• Schmidt-May, Angnis, and Mikael von Strauss. “Recent Developments in Bimetric Theory.” Journal of Physics A: Mathematical and Theoretical 49 (2016): 183001. doi:10.1088/1751-8113/49/18/183001; Open PDF.