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 Incoming links: Effective sound-speed counterterm in large-scale structure

P_{13}

is one ordering of

⟨ δ^{(1)} δ^{(3)} ⟩

. Then

P (k, t) = D^{2} P_{lin} (k) + P_{22} (k, t) + 2 P_{13} (k, t) + \dots,

P_{22} = 2 D^{4} \int \frac{d ^{3} q}{( 2 π ) ^{3}} F_{2, s} (q, k - q)^{2} P_{lin} (q) P_{lin} (∣ k - q ∣)

P_{13} = 3 D^{4} P_{lin} (k) \int \frac{d ^{3} q}{( 2 π ) ^{3}} F_{3, s} (k, q, - q) P_{lin} (q) .

The factors two and three count Wick contractions. The mixed

P_{12}

contains an odd number of zero-mean Gaussian fields, so $P_{12} = 0$ .

For

q ≫ k

F_{2, s} = O (k^{2} / q^{2})

, so

P_{22}^{UV} = O (D^{4} k^{4} \int \frac{d ^{3} q}{( 2 π ) ^{3}} \frac{P _{lin} ( q ) ^{2}}{q ^{4}}) .

The third-order kernel has

F_{3, s} (k, q, - q) = O (k^{2} / q^{2})

2 P_{13}^{UV} = O (D^{4} k^{2} P_{lin} (k) \int \frac{d ^{3} q}{( 2 π ) ^{3}} \frac{P _{lin} ( q )}{q ^{2}}) .

These integrals expose the sensitivity to unresolved short-scale dynamics. The effective sound-speed counterterm in large-scale structure has exactly the required deterministic shape:

P_{ct} (k, t) = - 2 c_{eff}^{2} (t) k^{2} P_{L} (k, t), P_{L} = D^{2} P_{lin} .

Here

c_{eff}^{2}

absorbs the time and normalization conventions and has dimensions of length squared. A stochastic correction begins at

k^{4}

and can absorb the corresponding short-scale sensitivity of

P_{22}

P_{EFT} (k) = P_{11} + P_{22} + 2 P_{13} - 2 c_{eff}^{2} k^{2} P_{11} (k) + P_{stoch} (k) + \dots,

where the normalization scale may be absorbed into

c_{eff}^{2}

P_{stoch} = O (k^{4})

at small

k

P_{13}^{UV} (k) \propto P_{L} (k) \int^{Λ} \frac{d ^{3} q}{( 2 π ) ^{3}} \frac{k ^{2}}{q ^{2}} P_{L} (q) = k^{2} P_{L} (k) \frac{1}{2 π ^{2}} \int^{Λ} d q P_{L} (q) .

Its cutoff dependence has exactly the

k^{2} P_{L} (k)

c_{eff}^{2} (Λ)

cancels it. The stochastic and higher-derivative counterterms similarly absorb the allowed analytic ultraviolet dependence of

P_{22}

and higher orders.

For internal momentum

q ≫ k

k^{4}

k^{2} P (k)

. The latter has the same momentum dependence as the effective sound-speed counterterm in large-scale structure. Short-scale stochastic contributions start at

k^{4}