Refinement: EFT Validity and Coarse-Graining Sketch

Revision note (v13.0). The polynomial regulator $[1+\square/\Lambda^2]^{-2}$ is removed, because it conflicts with the exponential regulator in the causality note. The cutoff is now stated as a free scale $\Lambda_{\rm EFT}$. The v12 statement that $\hbar c/(1\ {\rm nm})$ lies in the "GeV range" is withdrawn: it is 197 eV. The cosmological coarse-graining in Section 2 is flagged as a sketch. It demonstrates no sound speed and no effective-fluid equation of state. The matter-field coupling term that v12 added to the scalar stress tensor is removed. See the architecture section of the main document.

1. EFT Validity

1.1. The effective field theory and its cutoff

The Level 1 theory (the metric, one scalar $\phi$ with potential $V(\phi)$, and matter coupled through $A_i^2(\phi)\,g_{\mu\nu}$) is an effective field theory with a cutoff $\Lambda_{\rm EFT}$. It is valid for energies $E\ll\Lambda_{\rm EFT}$. The coupling $\beta_i/M_{\rm Pl}$ of the scalar to matter has mass dimension $-1$, so the scalar–matter interaction is non-renormalisable. The theory is therefore usable only below a cutoff, and the value of the cutoff is not determined by the theory.

The cutoff is a free scale. $\Lambda_{\rm EFT}$ is not tied to the scalar mass, to the environment-dependent range $\lambda(\rho)$, or to the non-locality scale $\Lambda_{\rm NL}$ of the optional non-local operator. The conditions below must hold for any chosen value.

• All processes considered have $E\ll\Lambda_{\rm EFT}$. This includes the effective scalar mass $m_{\rm eff}(\rho)$ in the densest environment used in an experiment, since $m_{\rm eff}$ rises with density.
• The potential scale $\Lambda$ and the field values reached, $\phi_{\min}(\rho)$, are consistent with the cutoff. This has not been checked for the benchmark potential (not yet computed).
• Radiative corrections to $V(\phi)$ and to the couplings $\beta_i$ are small enough to leave the benchmark form stable. This has not been analysed (Open). It belongs to deliverable (ii) of the UV completion roadmap.

1.2. Withdrawn: the v12 cutoff estimate

The v12 text identified the cutoff with the inverse of a "quantum penetration length", $\Lambda\sim1/\lambda_q$, took $\lambda_q\approx10^{-9}$ m and concluded that the cutoff lies in the GeV range. The arithmetic is wrong:

A length of 1 nm corresponds to 197 eV, not to GeV. The identification of $\Lambda_{\rm EFT}$ with $\hbar c/\lambda_q$ is withdrawn together with the quantum penetration length $\lambda_q$.

1.3. Withdrawn: the polynomial regulator and the choice $n=2$

The v12 note wrote the non-local operator as

and stated that the choice $n=2$ makes the ultraviolet suppression strong enough to control loop divergences. This form is removed. It has a double pole at $p^2=\Lambda^2$, and it conflicts with the exponential entire-function regulator used in the causality note. The statement about $n=2$ is removed with it. Whether any non-local regulator is needed is an open question, because the Level 1 action is local and is treated as an effective field theory below $\Lambda_{\rm EFT}$. The status of the exponential regulator, which is conditional on expert review, is set out in the causality note.

1.4. Withdrawn: the renormalisability argument

The v12 text attributed the non-renormalisability of the theory to "the negative power of the d'Alembertian in the denominator" and described the non-local term as a manifestation of heavier degrees of freedom. The first statement is withdrawn. The non-renormalisability follows from the dimension of the coupling $\beta_i/M_{\rm Pl}$, as stated in Section 1.1. Whether heavier degrees of freedom that would generate a non-local term exist is not established.

2. Cosmological Coarse-Graining: A Sketch

Status: sketch. This section does not derive an effective fluid. It does not demonstrate a sound speed, an equation of state or a dark-matter analogue. Those statements in v12 are withdrawn. The section records the correct starting point and what would have to be done.

2.1. Averaging scale

The cosmological extension averages the microscopic stress tensor over a volume $V$ of linear size $L$. The v12 hierarchy $\lambda_c^3\ll V\ll H^{-3}$ used the fixed $\lambda_c=1\ \mu$m, which is withdrawn. The scale hierarchy at v13.0 is

where $\lambda(\rho)$ is the environment-dependent range of the scalar and $H^{-1}$ is the Hubble radius. The averaging procedure must also treat the non-linear field equation in a medium whose density varies on scales below $L$. That treatment is not given here.

2.2. Stress tensor of the scalar sector

The scalar has the standard stress tensor

and for a homogeneous field $\rho_\phi=\tfrac12\dot\phi^2+V(\phi)$ and $p_\phi=\tfrac12\dot\phi^2-V(\phi)$. Matter couples to $\phi$ through $A^2(\phi)\,g_{\mu\nu}$. For pressureless matter with $A=e^{\beta\phi/M_{\rm Pl}}$ this gives $\rho_m\propto a^{-3}e^{\beta\phi/M_{\rm Pl}}$, so the coupling appears as an exchange of energy between $\phi$ and matter. It does not appear as an extra term in the stress tensor of the scalar.

Two parts of the v12 sketch are removed.

• The term $-\kappa\phi\,T^{M}_{\mu\nu}$ added to the stress tensor, and the corresponding $-\langle\kappa\phi\rho_M\rangle$ in the effective density. The interaction is energy exchange, as above, and not a contribution to $\rho_\phi$.
• The electromagnetic-type vector terms $F^{\rm EME}_{\mu\nu}$. The vector is not part of the Level 1 core, and the v12 argument that its average vanishes is therefore not needed.

2.3. No sound speed is demonstrated

The v12 sketch stated that the averaged terms lead to a "scale-dependent sound speed $c_s^2(k,a)$ that distinguishes the EME dark matter analogue from $\Lambda$CDM". No such sound speed was derived. For a canonical scalar field the sound speed of the field perturbations in the field rest frame is $c_s^2=1$. A different effective value for a matter-like fluid would need an explicit averaging calculation that is not done here. The statement that the sketch "justifies the use of the effective fluid approximation" is withdrawn.

2.4. What replaces the effective fluid

Linear growth of matter perturbations in a scalar-tensor theory with a screened scalar is described by a scale-dependent effective gravitational coupling,

This is a standard scalar-tensor result, the same formula as in the screened scalar document. For real $\beta$, $G_{\rm eff}\ge G$, so the scalar force enhances linear growth on scales below the scalar range. The v12 claim of a suppression of $P(k)$ is withdrawn. The formula has not been evaluated for MCE parameters, and the forecasts for large-scale-structure surveys in the v12 documents are withdrawn. The forecast is not yet computed. See the screened scalar sector and the cosmological extension.

3. Conclusion

The Level 1 theory is an effective field theory below a free cutoff $\Lambda_{\rm EFT}$, with a non-renormalisable scalar–matter coupling. The v12 cutoff estimate, the polynomial regulator and the renormalisability argument are withdrawn. The cosmological coarse-graining is a sketch only: it demonstrates no sound speed, and the quantitative route to cosmological observables is the growth formula for $G_{\rm eff}(k,a)$. That formula is an enhancement of linear growth, and it remains to be evaluated.