EFT Validity and Coarse-Graining Sketch

1. Effective Field Theory (EFT) Validity of the Non-Local Operator $K(\square)$

The non-local operator $K(\square)$ was introduced to formalise the quantum penetration mechanism:

\mathcal{K}(\square) = \frac{1}{\square + m^2} \left[ 1 + \frac{\square}{\Lambda^2} \right]^{-2}

Where $\Lambda$ is the scale of non-locality, related to the quantum penetration length $\lambda_q$ .

1.1. EFT Cutoff and UV Behaviour

The EME theory, as formulated, is an Effective Field Theory (EFT) valid up to the energy scale $\Lambda$ . The non-local term acts as a regulator, ensuring the high-momentum (UV) behaviour of the EME field propagator is suppressed.

Cutoff Scale: The EFT is valid for energies $E \ll \Lambda$ . The scale $\Lambda$ is related to the inverse of the non-local length scale $\lambda_q$ : $\Lambda \sim 1/\lambda_q$ . Since $\lambda_q \approx 10^{-9} \text{ m}$ , the cutoff scale $\Lambda$ is in the $\text{GeV}$ range, which is significantly higher than the energy scales of the phenomena the EME theory is designed to explain (e.g., gravitational interactions).
Renormalisability: The non-local nature of the operator, specifically the negative power of the d'Alembertian in the denominator, means the theory is technically non-renormalisable in the traditional sense. However, as an EFT, this is acceptable. The non-local term is a manifestation of integrating out heavier, unknown degrees of freedom (e.g., the full quantum gravity theory or a deeper QED/QCD effect) that become active at the scale $\Lambda$ . The choice of $n=2$ ensures the UV suppression is strong enough to control divergences in loop calculations up to the cutoff $\Lambda$ .

2. Sketch of the Cosmological Coarse-Graining Derivation

The cosmological extension requires averaging the microscopic EME energy-momentum tensor $T_{\mu\nu}^{\text{EME}}$ to obtain the effective fluid $T_{\mu\nu}^{\text{EME(cosmo)}}$ .

2.1. Formal Averaging Procedure

The effective cosmological tensor is defined by the volume average:

T_{\mu\nu}^{\text{EME(cosmo)}} = \langle T_{\mu\nu}^{\text{EME}} \rangle = \frac{1}{V} \int_V T_{\mu\nu}^{\text{EME}} dV

The averaging volume $V$ must satisfy the scale hierarchy: $\lambda_c^3 \ll V \ll H^{-3}$ .

2.2. Averaging the EME Lagrangian Terms

The microscopic $T_{\mu\nu}^{\text{EME}}$ is derived from the EME Lagrangian:

T_{\mu\nu}^{\text{EME}} = F_{\mu\lambda}^{EME} F_{\nu}^{\lambda, EME} - \frac{1}{4} g_{\mu\nu} F_{\alpha\beta}^{EME} F^{\alpha\beta}_{EME} + \nabla_\mu \phi \nabla_\nu \phi - \frac{1}{2} g_{\mu\nu} (\nabla_\alpha \phi) (\nabla^\alpha \phi) - g_{\mu\nu} V(\phi) - \kappa \phi T_{\mu\nu}^{M}

The averaging process simplifies this significantly:

Vector Field Terms: The microscopic EME vector field $A_\mu^{EME}$ is highly oscillatory and sourced by local currents. Over a large volume $V$ , the average of the quadratic terms $\langle F_{\mu\lambda}^{EME} F_{\nu}^{\lambda, EME} \rangle$ is negligible compared to the scalar field terms, effectively averaging to zero.
Scalar Field Terms: The scalar field $\phi$ is the primary source of the long-range EME effect. The average of the kinetic terms $\langle \nabla_\mu \phi \nabla_\nu \phi \rangle$ and the potential $\langle V(\phi) \rangle$ yields the effective density and pressure.
Interaction Term: The crucial term is the average of the matter-field coupling $\langle \kappa \phi T_{\mu\nu}^{M} \rangle$ . Since the matter energy-momentum $T_{\mu\nu}^{M}$ is dominated by the matter density $\rho_M$ , this term is proportional to the average matter density $\langle \rho_M \rangle$ , which is the standard $\rho_b$ in the Friedmann equations. The effective EME density $\rho_{\text{EME}}$ is thus directly linked to the baryonic matter density $\rho_b$ via the scalar field $\phi$ .

2.3. Resulting Effective Fluid

The coarse-graining procedure results in an effective fluid with:

\rho_{\text{EME}} \approx \langle \frac{1}{2} \dot{\phi}^2 \rangle + \langle V(\phi) \rangle - \langle \kappa \phi \rho_M \rangle

The effective pressure $p_{\text{EME}}$ is derived from the averaged terms, leading to the scale-dependent sound speed $c_s^2(k, a)$ that distinguishes the EME dark matter analogue from $\Lambda$ CDM. This sketch justifies the use of the effective fluid approximation for cosmological calculations.

3. Conclusion

These refinements provide the necessary theoretical depth to address the final scrutiny points. The non-local operator is placed within the context of EFT, and the cosmological coarse-graining is justified by a formal averaging procedure over the microscopic EME Lagrangian.

Refinement: EFT Validity and Coarse-Graining Sketch