Adiabatic Particle Number

An adiabatic particle number is a deliberately approximate, time-dependent particle concept for a smooth expanding universe. It compares an exact mode with a finite-order WKB basis whose frequency is built locally from the scale factor and its derivatives. The answer becomes robust in an asymptotically static region or in a controlled high-frequency adiabatic regime; at finite time and finite order it retains a quantified basis dependence.

Required background. Partial Differential Equations and Green Functions supplies normalized mode evolution and its conserved Wronskian. Geodesics and Jacobi Fields provides the comoving geodesics whose proper time is the cosmic time used in the expansion; no geodesic-deviation estimate is invoked.

Helpful background. Flat-Spacetime Quantum Field Theory supplies oscillator number operators. Causality and Global Structure provides the meaning of an asymptotically stationary region. Quantum Fields in Curved Spacetime defines the prescribed geometry, state, and renormalized-observable boundary.

Exact FLRW modes as time-dependent oscillators

Take a smooth spatially flat FLRW metric in cosmic time,

$$ds^2=-dt^2+a(t)^2d\mathbf x^2,$$

and a real scalar satisfying $(\Box-m^2-\xi R)\phi=0$. A Fourier mode $\phi_k$ obeys

$$\ddot\phi_k+3H\dot\phi_k +\left(\frac{k^2}{a^2}+m^2+\xi R\right)\phi_k=0, \qquad H=\frac{\dot a}{a}.$$

With $h_k=a^{3/2}\phi_k$, the friction term disappears:

$$\ddot h_k+\Omega_k^2h_k=0,$$ $$\Omega_k^2 =\frac{k^2}{a^2}+m^2+\xi R -\frac94H^2-\frac32\dot H.$$

Choose the normalization $h_k\dot{\bar h}_k-\dot h_k\bar h_k=i$ up to the fixed overall convention inherited from the field expansion. This Wronskian is the mode form of the conserved Klein–Gordon product. The construction below applies on intervals where the chosen approximate frequency remains real and positive; a turning point $\Omega_k^2=0$ requires matching or a uniform approximation, not blind WKB iteration.

Finite-order WKB bases

A normalized WKB trial mode is

$$f_k^{(n)}(t)=\frac{1}{\sqrt{2W_k^{(n)}(t)}} \exp\left[-i\int^t W_k^{(n)}(s)\,ds\right].$$

Substitution into the exact oscillator equation gives the nonlinear frequency equation

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

Adiabatic order counts time derivatives of $a$: start with $W_k^{(0)}=\omega_k=\sqrt{k^2/a^2+m^2}$, insert the lower-order result on the right, and retain terms only through the requested order. Curvature and the terms involving $H$ begin at second order. Parker’s construction makes both the cosmic-time measurement prescription and the order-by-order normalization explicit Parker 2012, §§4–8, especially Eqs. (12)–(13) and (22)–(45).

At any chosen time, decompose an exact normalized solution by Wronskian projection,

$$h_k=\alpha_k^{(n)}f_k^{(n)} +\beta_k^{(n)}\bar f_k^{(n)}, \qquad |\alpha_k^{(n)}|^2-|\beta_k^{(n)}|^2=1.$$

Equivalently, $\alpha_k^{(n)}$ and $\beta_k^{(n)}$ are fixed by matching both $h_k$ and $\dot h_k$ to the instantaneous canonical pair $(f_k^{(n)},\dot f_k^{(n)})$. The order-$n$ occupation is

$$N_k^{(n)}(t)=|\beta_k^{(n)}(t)|^2.$$

The projection can be evaluated without fitting oscillatory phases. With the normalization $f_k\dot{\bar f}_k-\dot f_k\bar f_k=i$,

$$\begin{aligned} \alpha_k^{(n)}&=i\left(\bar f_k^{(n)}\dot h_k-\dot{\bar f}_k^{(n)}h_k\right),\\ \beta_k^{(n)}&=-i\left(f_k^{(n)}\dot h_k-\dot f_k^{(n)}h_k\right). \end{aligned}$$

For a real positive WKB frequency $W_k^{(n)}$, this gives

$$N_k^{(n)} =\frac{1}{2W_k^{(n)}} \left|\dot h_k+ \left(iW_k^{(n)}+\frac{\dot W_k^{(n)}}{2W_k^{(n)}}\right)h_k\right|^2.$$

It vanishes when the exact canonical data equal the selected WKB data. A turning point at which the chosen $W_k^{(n)}$ ceases to be positive invalidates this particle basis; it is not a divergence in an invariant particle observable.

It depends on the exact state through $h_k$, but also on $t$, the canonical rescaling, the WKB order, and the measurement prescription. Merely writing $\omega_k(t)$ and diagonalizing an instantaneous Hamiltonian is another basis choice, not an invariant cure.

When different orders agree

The relevant small quantities for an ultraviolet mode are ratios such as

$$\epsilon_{1k}=\frac{H}{\omega_k}, \qquad \epsilon_{2k}=\frac{\dot H}{\omega_k^2}, \qquad \epsilon_{3k}=\frac{\ddot H}{\omega_k^3}.$$

Where these and the corresponding higher derivatives are uniformly small, changing from order $n$ to $n+2$ changes $W_k$ and $N_k$ only beyond the retained order. That is controlled insensitivity, not exact uniqueness. Low-frequency modes, rapid transitions, and WKB turning points can make the difference large.

There is a stronger endpoint statement. If $a(t)$ becomes exactly constant in an early or late region, every derivative correction vanishes there, $W_k^{(n)}=\sqrt{k^2/a^2+m^2}$ for all $n$, and the basis reduces to the ordinary stationary positive-frequency basis. The late occupation relative to that out basis is then independent of which finite WKB order was used to organize the intermediate calculation. Parker’s original expanding-universe analysis establishes the asymptotic particle interpretation and its pair mixing for smooth, statically bounded expansion Parker 1969, complete article, pp. 1057–1068.

Particle diagnostics are not subtraction prescriptions

Adiabatic expansion also organizes ultraviolet counterterms, but two uses must be kept distinct. In four dimensions, removing the ultraviolet divergence in a mode-summed particle number requires terms through second adiabatic order in Parker’s measurement construction. Renormalizing $\langle\phi^2\rangle$ likewise involves lower orders, whereas a conserved stress–energy tensor requires subtraction through fourth adiabatic order. Parker and Fulling’s stress-tensor method uses modewise adiabatic terms together with conservation; it is not evidence that a fourth-order instantaneous particle count is a unique local observable Parker and Fulling 1974, complete article, pp. 341–354.

A finite-order WKB state also need not have the full Hadamard short-distance property. Infinite adiabatic order, under the usual smoothness hypotheses, is the relevant bridge to Hadamard behavior; finite order controls only a corresponding finite portion of the ultraviolet expansion. Thus one must not infer admissibility for every local composite observable from a well-behaved finite-order $N_k^{(n)}$.

Three consistency checks

Static limit. If $\dot a=0$, then every derivative correction vanishes. An exact positive-frequency solution proportional to $e^{-i\omega_k t}$ has $\beta_k^{(n)}=0$ at every order.

Conformal limit. For $m=0$ and $\xi=1/6$ in four dimensions, use conformal time and the rescaled field $a\phi$. Its modes obey the flat oscillator equation, so conformally related in and out vacua give $\beta_k=0$ for any smooth $a$ when no boundary breaks conformal invariance. Expansion alone is not sufficient for creation.

Order comparison. Compute $N_k^{(0)}$, $N_k^{(2)}$, and, where needed, $N_k^{(4)}$ over the same interval. Agreement improving as $k/aH$ grows is the expected ultraviolet check. Persistent disagreement at large physical momentum indicates a normalization or derivative-counting error; disagreement near a turning point announces failure of the approximation, not extra particles.

Check your understanding

Assume $a(t)=a_0$ on a late-time interval and an exact normalized mode there is

$$h_k(t)=\frac{A_ke^{-i\omega_kt}+B_ke^{+i\omega_kt}}{\sqrt{2\omega_k}}, \qquad |A_k|^2-|B_k|^2=1.$$

Determine $N_k^{(n)}$ on that interval for any finite adiabatic order, and explain why the answer is less ambiguous there than during expansion.

Solution

Because every derivative of $a$ vanishes, the iterative equation gives $W_k^{(n)}=\omega_k$ for all orders and the WKB basis is the exact stationary basis. Wronskian projection therefore identifies $\alpha_k^{(n)}=A_k$ and $\beta_k^{(n)}=B_k$ up to a common phase, so $N_k^{(n)}=|B_k|^2$. The result is order-independent because the asymptotic timelike symmetry has selected the positive-frequency split; during expansion, different truncated WKB frequencies differ by higher derivative terms and no such exact selector exists.

References

• Parker, Leonard, “Particle Creation and Particle Number in an Expanding Universe,” Journal of Physics A: Mathematical and Theoretical 45 (2012), 374023, DOI, Open PDF.
• Parker, Leonard, “Quantized Fields and Particle Creation in Expanding Universes. I,” Physical Review 183 (1969), 1057–1068, DOI.
• Parker, Leonard, and Stephen A. Fulling, “Adiabatic Regularization of the Energy-Momentum Tensor of a Quantized Field in Homogeneous Spaces,” Physical Review D 9 (1974), 341–354, DOI.