No Quartic Graviton Dispersion from the Local Covariant Four-Derivative Truncation

Core contributions

• A posited kernel, stated as such: the results have the form if the local Lorentzian transverse-traceless kernel is $\Box(c_{\mathrm{EH}} + \beta\Box)$, then the conclusions follow. They are not statements about $S_\Pi$ itself, nor about the full renormalized kernel.
• Direct-sum solution structure: for $c_{\mathrm{EH}} \neq 0$ the two factors are coprime and $\ker\mathcal{O}_{\mathrm{TT}} = \ker\Box \oplus \ker(c_{\mathrm{EH}} + \beta\Box)$, a direct sum and not a union.
• Undeformed massless sector: the massless summand obeys $\omega^2 = |\vec k|^2$ exactly, with no correction at any order in $|\vec k|$.
• No quartic coefficient, hence no scale bound through $\alpha = 4$: $A_4 = 0$ on the massless branch, so no mapping of a quartic coefficient onto $\ell_\chi$ exists, and any bound on $\ell_\chi$ resting on that identification does not hold.
• Pole content: for $\beta \neq 0$ a helicity-$2$ pole at $m_2^2 = -c_{\mathrm{EH}}/\beta$ carries a residue opposite to the massless one — a ghost character independent of the sign of $c_{\mathrm{EH}}/\beta$.
• A boundary, explicitly drawn: the finite $c_{\mathrm{EH}}^{\mathrm{ren}}$ and $\beta^{\mathrm{ren}}$, the signed mass squared, and the fate of an $R^2$ scalar are matching data supplied from outside the truncation.

The operator considered

Assume that, linearized about Minkowski spacetime and restricted to the transverse-traceless sector, the local Lorentzian quadratic kernel truncated at four derivatives is $$\mathcal{O}_{\mathrm{TT}} = c_{\mathrm{EH}}\Box + \beta\Box^2 ,$$ with $c_{\mathrm{EH}}$ and $\beta$ real constants, non-local form factors and all terms of order $\Box^3$ and higher being discarded. The difference between this truncation and the full renormalized kernel matters wherever the discarded terms are not negligible — in particular in the ultraviolet.

Conventions. Signature $(-,+,+,+)$ and units with $c = 1$. For a plane wave $e^{ik\cdot x}$ with $k^\mu = (\omega, \vec k)$ one has $\Box \to -k^2$, where $k^2 = -\omega^2 + |\vec k|^2$.

Solution structure: a direct sum

Write $P = \Box$ and $Q = c_{\mathrm{EH}} + \beta\Box$, so that $\mathcal{O}_{\mathrm{TT}} = PQ = QP$. Let $\mathcal{V}$ be any complex vector space of transverse-traceless tensor fields on which $\Box$ acts and which is stable under it — smooth fields, or tempered distributions, indifferently. The argument is algebraic and uses no property of $\mathcal{V}$ beyond stability.

For $c_{\mathrm{EH}} \neq 0$ the two factors are coprime, by the Bézout identity $$Q - \beta P = c_{\mathrm{EH}} ,$$ a nonzero constant. The solution space is then the direct sum $$\ker\mathcal{O}_{\mathrm{TT}} = \ker\Box \,\oplus\, \ker(c_{\mathrm{EH}} + \beta\Box) ,$$ and the decomposition of any $h^{\mathrm{TT}}$ as $h^{\mathrm{TT}} = h_0 + h_m$ with $h_0 \in \ker P$ and $h_m \in \ker Q$ is unique. If in addition $\beta \neq 0$, then $\ker Q$ is characterized by $\Box h_m = -(c_{\mathrm{EH}}/\beta)\,h_m$.

The kernel is not a union. Let $c_{\mathrm{EH}} \neq 0$, let $\beta \neq 0$, and suppose that both $\ker P \neq \{0\}$ and $\ker Q \neq \{0\}$ in $\mathcal{V}$. These are hypotheses on $\mathcal{V}$, not consequences of $c_{\mathrm{EH}} \neq 0$ and $\beta \neq 0$: a stable space may contain no solution of either factor equation. They hold in the tempered distributions, where $\ker P$ carries the plane waves with $k^2 = 0$ and $\ker Q$ those with $k^2 = c_{\mathrm{EH}}/\beta$. Then the kernel is strictly larger than $\ker P \cup \ker Q$: given nonzero $h_0 \in \ker P$ and $h_m \in \ker Q$, the sum $h_0 + h_m$ solves $\mathcal{O}_{\mathrm{TT}} h = 0$ while satisfying neither factor equation, since $\Box(h_0+h_m) = -(c_{\mathrm{EH}}/\beta)h_m \neq 0$ and $(c_{\mathrm{EH}} + \beta\Box)(h_0+h_m) = c_{\mathrm{EH}} h_0 \neq 0$. A generic solution is a superposition of the two sectors, not a member of either.

Neither hypothesis follows from the other, and each fails on an admissible space with $\beta \neq 0$ fixed. Both $\ker P$ and $\ker Q$ are stable under $\Box$, so each is an admissible choice of $\mathcal{V}$, and on each the operator degenerates: on $\mathcal{V} = \ker P$ the operator $\Box$ acts as $0$, so $Q$ acts as $c_{\mathrm{EH}} I$ and is invertible, giving $\ker Q = \{0\}$; on $\mathcal{V} = \ker Q$ the operator $\Box$ acts as $-(c_{\mathrm{EH}}/\beta) I$ and is invertible, giving $\ker P = \{0\}$. In both cases the union coincides with the kernel and strictness fails. Strictness is thus a statement about a space containing both sectors, not about the operator alone.

Degenerate case. If $c_{\mathrm{EH}} = 0$ the Bézout identity fails and the decomposition is unavailable: $\mathcal{O}_{\mathrm{TT}} = \beta\Box^2$, and $\ker\Box^2$ contains generalized solutions annihilated by $\Box^2$ but not by $\Box$.

Absence of a quartic graviton dispersion

For $c_{\mathrm{EH}} \neq 0$ the massless summand lies in $\ker\Box$ by construction, and $\ker\Box$ does not depend on $\beta$. Hence for plane waves $h_0 \sim e^{ik\cdot x}$, $$\omega^2 = |\vec k|^2 ,$$ with no correction at any order in $|\vec k|$.

A relation deforming the massless dispersion at quartic order would describe a single branch. The kernel has no such branch. Expanding $(c_{\mathrm{EH}} + \beta z)^{-1}$ in powers of $z$ about $z = 0$ generates a series of terms analytic at the origin — contact terms, in position space — added to the massless pole; it does not move that pole, whose location is the zero of $\Box$ and is independent of $\beta$. Reading the resulting series as a deformed dispersion relation mistakes an analytic background for a shift of the pole.

Gravitational-wave catalogues test modified dispersion through the phenomenological relation $$E^2 = p^2c^2 + A_\alpha\,p^\alpha c^\alpha ,$$ constraining $A_\alpha$ for a range of exponents $\alpha$. The case $\alpha = 4$ is what a quartic deformation of the massless branch would populate; the case $\alpha = 0$ corresponds instead to a mass term, so the parametrization is not blind to massive branches. Under the truncation with $c_{\mathrm{EH}} \neq 0$, the undeformed massless sector gives $A_4 = 0$ on the massless branch. There is therefore no mapping of a quartic coefficient onto the pre-geometric scale $\ell_\chi$, and any bound on $\ell_\chi$ obtained by identifying an interferometric limit on $A_4$ with a quartic spectral coefficient does not hold.

What is not claimed. This does not say that these data cannot constrain this framework. The $\alpha = 0$ sector bears on massive branches, and the pole of the next section is one. Turning that into an actual constraint would require its mass, its coupling to the detector, and its excitation in the source — none of which is supplied here. The negative result concerns the $A_4$ route only.

No front-velocity claim. The relation $\omega^2 = |\vec k|^2$ is a statement about the truncation, not about signal fronts. A front velocity is governed by the large-$k$ behaviour of the full kernel, precisely where the discarded non-local and $\mathcal{O}(\Box^3)$ terms are not negligible. No statement about front propagation is made here, in either direction.

Pole content

Writing $z$ for the momentum-space image of $\Box$, the transverse-traceless propagator is, for $c_{\mathrm{EH}} \neq 0$ and $\beta \neq 0$ — the partial fractions below divide by $\beta$ — $$\frac{1}{z(c_{\mathrm{EH}} + \beta z)} = \frac{1}{c_{\mathrm{EH}}\,z} \;-\; \frac{1}{c_{\mathrm{EH}}\,(z + c_{\mathrm{EH}}/\beta)} .$$ Two facts follow, and they are independent of one another. First, ghost character: the two residues have opposite signs, whatever the signs of $c_{\mathrm{EH}}$ and $\beta$, so the second pole is a ghost relative to the massless one — this does not depend on $c_{\mathrm{EH}}/\beta$. Second, tachyonic character: the second pole sits at $z = -c_{\mathrm{EH}}/\beta$, so the sign of $c_{\mathrm{EH}}/\beta$ fixes the sign of the corresponding mass squared, and with it whether the mode is tachyonic. This is a separate question, with a separate answer.

For $\beta \neq 0$ the second pole sits at $k^2 = c_{\mathrm{EH}}/\beta$, that is at $$m_2^2 = -\,\frac{c_{\mathrm{EH}}}{\beta} ,$$ in the conventions above. This resembles the pole content of quadratic gravity, where a massless graviton is accompanied by a massive spin-$2$ ghost (Stelle 1977; Calmet–Capozziello–Pryer 2017). The resemblance is a comparison and not an identification: the transverse-traceless calculation establishes a helicity-$2$ pole with opposite residue, whereas the massive spin-$2$ multiplet is a statement about all five polarizations. The scalar and vector sectors are not treated here, so the full multiplet structure is not established.

The scalar sector is not settled. The minimal $a_4$ carries an $R^2$ sector alongside $C^2$. In the effective action at second order in curvature this generically supports a massive scalar in addition to the massive spin-$2$ mode. Excluding it requires a matching condition on the $R^2$ coefficient, so the mode content of the truncation is not fully determined here.

What is matching data

The Seeley–DeWitt coefficient $a_4$ controls a logarithmic contribution to the running of the four-derivative couplings. It does not fix the finite value of $\beta^{\mathrm{ren}}$.

The normalization and sign of $\tfrac12\log\det{}'A_g$ are not at issue: they are fixed by the definition of the functional. What leaves $\beta^{\mathrm{ren}}$ undetermined is the finite parts of the counterterms, the complete operator content, the continuation to Lorentzian signature, and the renormalization condition. An identification $\beta = -2\alpha_C$ read off the Weyl-squared part of $a_4$ supplies none of these, and is not a determination of $\beta^{\mathrm{ren}}$. The same holds for $c_{\mathrm{EH}}^{\mathrm{ren}}$. Consequently the signed mass squared $$m_2^2 = -\,\frac{c_{\mathrm{EH}}^{\mathrm{ren}}}{\beta^{\mathrm{ren}}}$$ — equivalently the pole scale $|c_{\mathrm{EH}}^{\mathrm{ren}}/\beta^{\mathrm{ren}}|$ together with its sign — is a matching datum, and is not derivably tied to $\ell_\chi$.

Open problems

• The in-in construction. A Schwinger–Keldysh effective action is built from doubled field configurations on a closed time path, with the branches coupled by the boundary condition at the turning point. Substituting a retarded Green operator for $A_g^{-1}$ in an ordinary single-field Hessian is not that construction and does not follow from it. Whether the projective spectral action admits a closed-time-path formulation, and whether the quadratic form it would yield agrees with the truncation, is open.
• The vector sector. The linearized vector equation carries a $\Box\bar h_{0i}$ term, hence a second time derivative that harmonic gauge does not remove. Only under a stationarity assumption does it reduce to an elliptic constraint. A derivation from the full linear system with spin projectors, and the fate of the four-derivative correction within it, are open.
• Beyond the truncation. The truncation discards non-local form factors and $\mathcal{O}(\Box^3)$ terms. Whether the absence of a quartic graviton dispersion survives their inclusion is not addressed here.

Interpretive status

In its transverse-traceless sector, the posited kernel resembles quadratic gravity. It is tempting to read that resemblance as support for the spectral construction; the reading is available but weak, since any construction generating $C^2$ and $R^2$ sectors would resemble it equally well — and the resemblance is established here only for one helicity-$2$ pole, not for the full mode content. The defensible statement is narrower: the posited kernel is compatible with the known infrared structure and predicts nothing beyond it that is presently observable.

What this note contributes is a boundary. Several inputs are missing before gravitational waves can test this framework at all, and they are not interchangeable: a Lorentzian construction from which the kernel actually follows, rather than being posited; a renormalization condition fixing $m_2^2$; the coupling of that mode to a detector and its excitation in a source; and a treatment of the scalar and vector sectors sufficient to settle the mode content. Naming them is what makes them addressable, and none of them is supplied here.

Relation to the Cosmochrony program

This article is paper III of the spectral gravity sub-programme. It connects to the broader Cosmochrony program through the pre-geometric scale $\ell_\chi$ and the projective spectral action, and it removes a putative interferometric handle on $\ell_\chi$ while isolating what a genuine one would require — beginning with a Lorentzian construction from which the kernel actually follows.

References

Jérôme Beau. No Quartic Graviton Dispersion from the Local Covariant Four-Derivative Truncation. Preprint. 10.5281/zenodo.18826644