Adiabatic Particle Number in Cosmology

Adiabatic particle number compares an exact mode with a finite-order WKB basis. It becomes a controlled late-time diagnostic when the frequency varies slowly and different orders agree within their remainder; at intermediate times it is basis, canonical-variable, and order dependent.

Required background. FLRW mode quantization fixes the Wronskian; adiabatic states fixes the WKB hierarchy; particle observables fixes the operational caveat; and Stokes production fixes nonperturbative switching. Helpful background. Review time-dependent particle creation.

WKB bases and their remainder

For $v_k''+\Omega_k^2v_k=0$, use

$$w_k^{(r)}(\eta)=\frac{1}{\sqrt{2W_k^{(r)}}} \exp\!\left[-i\int^\eta W_k^{(r)}d\eta'\right],$$

where iterative substitution truncates

$$W_k^2=\Omega_k^2-\frac12\frac{W_k''}{W_k} +\frac34\left(\frac{W_k'}{W_k}\right)^2$$

at adiabatic order $r$. Decompose the exact solution as

$$v_k=\alpha_k^{(r)}w_k^{(r)} +\beta_k^{(r)}w_k^{(r)*}, \qquad |\alpha_k^{(r)}|^2-|\beta_k^{(r)}|^2=1.$$

Then $n_k^{(r)}=|\beta_k^{(r)}|^2$. Its meaning is strongest in asymptotic regions where all retained derivatives of $\Omega_k$ vanish or are uniformly small. Comparing $r$ and $r+2$ estimates the truncation error; close agreement at one instant without a small adiabatic parameter is not convergence.

First application: a smooth frequency pulse

Use the weak exactly integrable perturbation

$$\Omega_k^2(\eta)=\omega_k^2+\delta V_k\,\operatorname{sech}^2(\eta/\tau), \qquad |\delta V_k|\ll\omega_k^2.$$

First-order oscillator scattering gives

$$\beta_k^{\rm out} \simeq-\frac{i}{2\omega_k} \int_{-\infty}^{\infty}d\eta\, \delta V_k\operatorname{sech}^2(\eta/\tau)e^{-2i\omega_k\eta} =-\frac{i\pi\delta V_k\tau^2}{\sinh(\pi\omega_k\tau)},$$

and hence

$$n_k^{\rm out}\simeq \frac{\pi^2\delta V_k^2\tau^4} {\sinh^2(\pi\omega_k\tau)}.$$

The exponential suppression for $\omega_k\tau\gg1$ is the adiabatic limit. Numerically evolve the exact mode and project onto zeroth-, second-, and fourth-order bases after the pulse. All give the same late occupation up to the perturbative and WKB remainders; during the pulse they need not. Parker’s original construction already identifies the asymptotic basis as the controlled particle comparison Parker 1969, §§III–IV, pp. 1062–1067.

Projection and uncertainty

The coefficients should be obtained from the conserved symplectic product rather than from amplitudes alone. For a normalized comparison mode $w_k$,

$$\alpha_k=-i\left(v_kw_k^{*\prime}-v_k'w_k^*\right), \qquad \beta_k=i\left(v_kw_k'-v_k'w_k\right).$$

These expressions use both field and momentum data and automatically test $\lvert\alpha_k\rvert^2-\lvert\beta_k\rvert^2=1$ when the exact and comparison modes have Wronskian $i$. Projecting only $v_k$ at one instant leaves its phase-space direction undetermined and can manufacture an occupation.

There are three distinct errors. The numerical error is measured by convergence of the exact mode and its Wronskian. The adiabatic truncation error is estimated from the difference between successive sensible WKB orders in a region where the derivative hierarchy is small. The physical-model error comes from uncertainty in $a(\eta)$, $m$, or other background couplings. They should be varied separately and then propagated to the integrated density.

The WKB series is asymptotic rather than generally convergent at arbitrarily high order. Near a turning point $\Omega_k^2=0$, individual derivative terms diverge and the local basis fails even though the exact solution is regular. Uniform approximations or Stokes analysis then connect controlled regions. A rapidly oscillating intermediate $n_k^{(r)}$ near such a point is not an observable production history. The durable information is the connection coefficient between specified adiabatic regions, or a response of a declared detector.

For a spectrum, convergence must also be uniform enough under the momentum integral. Agreement of low-$k$ curves does not control an ultraviolet tail that dominates $n$ or $\rho$. Check the projected tail against the differentiability of the background and vary $k_{\max}$ independently of time resolution.

The structure map locates this number on the particle branch, separate from local subtraction.

Adiabatic particle number is a controlled asymptotic comparison when basis orders converge; its intermediate-time value is conventional. Schematic; not to scale.

Domain and failure conditions

Use the chapter’s canonical domain table. The pulse result assumes weak $\delta V_k$, well-defined in/out regions, and modes normalized with the same symplectic form.

Adversarial test. Perform two time-dependent canonical rescalings that coincide asymptotically. Their instantaneous diagonalizations generally give different $n_k(\eta)$ during the pulse. Translate the variables and evolve to the shared out region: the final Bogoliubov coefficient agrees. If an intermediate discrepancy is larger than the estimated adiabatic remainder, no basis-independent particle claim is licensed.

The failure map sends a nonconvergent particle diagnostic to a detector or local-observable calculation.

Basis disagreement is expected at intermediate time; only asymptotic agreement or an operational detector definition supports a particle interpretation. Schematic; not to scale.

References

• Parker, L., “Quantized Fields and Particle Creation in Expanding Universes. I,” Physical Review 183, 1057–1068 (1969), doi:10.1103/PhysRev.183.1057.