A Cosmological PDE

In cosmologies driven by a perfect fluid with constant pressure-to-energy ration w, the following wave equation appears:

(1) \left( \partial^2 - \frac{\kappa(\kappa+1)}{\eta^2} \right) \psi[\eta,\vec{x}] = J[\eta,\vec{x}].

The real parameter \kappa depends on both w and the dimension of spacetime d; whereas J is the source of the waves \psi, We are working in the conformal time coordinate system, where \eta \geq 0 for 0 \leq w \leq 1 and \eta \leq 0 for w=-1.

Solution without Fourier Transform: Nariai’s Ansatz

In such a context, it may be tempting to first transform the equation to Fourier space,

\ddot{\psi}[\eta,\vec{k}] + \left( k^2 - \frac{\kappa(\kappa+1)}{\eta^2} \right) \psi[\eta,\vec{k}] = J[\eta,\vec{k}] .

But, if the goal of the investigation is to understand how the signal propagates in the cosmological spacetime — what travels on the null cone, and what travels inside of it — then I would assert, in this case, that there is a better way to arrive at the answer directly in real spacetime. Specifically, I am going to describe how to obtain via Nariai’s ansatz the retarded Green’s function of the wave operator in eq. (1), obeying

(2) \left( \partial_{\eta,\vec{x}}^2 - \frac{\kappa(\kappa+1)}{\eta^2} \right) G[x,x'] = \delta^{(d)}[x-x'] = \left( \partial_{\eta',\vec{x}'}^2 - \frac{\kappa(\kappa+1)}{\eta'^2} \right) G[x,x'].

Even though the \kappa(\kappa+1)/\eta^2 term breaks time translation symmetry, space translation symmetry is still retained. Hence, we expect that the even dimensional Green’s functions G_{2+2n} can be obtained from the (1+1)D one; and the odd dimensional Green’s functions G_{3+2n} can be derived from the (2+1)D one.

(3.Recursion) G_{d+2}[\eta,\eta';R \equiv |\vec{x}-\vec{x}'|] = -\frac{1}{2\pi R} \frac{\partial}{\partial R} G_d[\eta,\eta';R].

Nariai’s ansatz amounts to multiplying the flat spacetime massless scalar Green’s function (i.e., the \kappa=0 answer \overline{G}_d[x-x']) by a function that depends on spacetime solely through the object

(3.CFT) s \equiv \bar{\sigma}/(\eta \eta') ;

where \bar{\sigma} \equiv (1/2)( (\eta-\eta')^2 - (\vec{x}-\vec{x}')^2 ) is the square of the geodesic distance between (t,\vec{x}) and (t',\vec{x}') in flat spacetime. Namely, we postulate

(4) G_d[x,x'] = \overline{G}_d[x-x'] \cdot F_d[s].

Keeping in mind \partial^2 \overline{G}_d[x-x'] = \delta^{(d)}[x-x'] and the retarded solutions

\overline{G}_2[\bar{\sigma}] = \Theta[t-t'] \frac{\Theta[\bar{\sigma}]}{2}, \\ \overline{G}_3[\bar{\sigma}] = \Theta[t-t'] \frac{\Theta[\bar{\sigma}]}{2\pi \sqrt{2 \bar{\sigma}}};

by inserting Nariai’s ansatz in eq. (4) into its PDE with respect to (\eta,\vec{x}) in eq. (2) yields in (1+1)D

(5.2D) F_2[s] \delta^{(2)}[x-x'] + \Theta[t-t'] \left( \frac{1}{\eta\eta'} + \frac{1}{\eta^2} \right) \bar{\sigma} \cdot \delta[\bar{\sigma}] F_2'[s] \\ \qquad\qquad + \frac{\overline{G}_2[\bar{\sigma}]}{\eta^2} \left\{ s (s+2) F_2''[s] + 2 (s+1) F_2'[s] - \kappa(\kappa+1) F_2[s] \right\} = \delta^{(2)}[x-x'];

and in (2+1)D

(5.3D) F_3[s] \delta^{(3)}[x-x'] + \Theta[t-t'] \frac{\sqrt{\bar{\sigma}}}{\pi \sqrt{2}} \left(\frac{1}{\eta\eta'} + \frac{1}{\eta^2}\right) \delta[\bar{\sigma}] F_3'[s] \\ \qquad\qquad + \frac{\overline{G}_3[\bar{\sigma}]}{\eta^2} \left\{ s (s+2) F_3''[s] + (2s+1) F_3'[s] - \kappa(\kappa+1) F_3[s] \right\} = \delta^{(2)}[x-x'].

F[0]=1 now follows from demanding the coefficients of the \delta^{(d)}[x-x'] on the left-hand-sides of equations (5.2D) and (5.3D) to be unity. The \bar{\sigma} \delta[\bar{\sigma}] term is zero. For the remaining term proportional to \overline{G}[\bar{\sigma}] to vanish, F needs to obey the homogeneous equations

s (s+2) F_2''[s] + 2 (s+1) F_2'[s] - \kappa(\kappa+1) F_2[s] = 0

and

s (s+2) F_3''[s] + (2s+1) F_3'[s] - \kappa(\kappa+1) F_3[s] = 0.

Nariai’s ansatz has thus reduced a PDE into an ODE with known analytic solutions.

Results

Altogether, we find the Legendre function F_2[s] = P_\kappa[1+s] and the associated Legendre function F_3[s] = (s/(2+s))^{1/4} P_\kappa^{(1/2)}[1+s]. That the solution is unique is because the other linearly independent solution involves Q_\kappa[1+s] for even d; and Q_{\kappa}^{(1/2)}[1+s] for odd d. Both are singular as s \to 0.

For even dimensions, eq. (3.Recursion) therefore reads

(6.Even) G_2[x,x'] = \Theta[t-t'] \frac{\Theta[\bar{\sigma}]}{2} P_\kappa[1+s] \\ G_{2+2n}[x,x'] = \Theta[t-t'] \left( \frac{1}{2\pi} \frac{\partial}{\partial \bar{\sigma}} \right)^n \left( \frac{\Theta[\bar{\sigma}]}{2} P_\kappa[1+s] \right).

For odd dimensions, eq. (3.Recursion) in turn reads

(6.Odd) G_3[x,x'] = \Theta[t-t'] \frac{ \Theta[\bar{\sigma}] }{ 2\pi \sqrt{2 \bar{\sigma} } } \left( \frac{s}{s+2} \right)^{\frac{1}{4}} P_{\kappa}^{(1/2)}[1+s] \\ G_{3+2n}[x,x'] = \Theta[t-t'] \left( \frac{1}{2\pi} \frac{\partial}{\partial \bar{\sigma} } \right)^n \left( \frac{ \Theta[\bar{\sigma}] }{ 2\pi \sqrt{2 \bar{\sigma} } } \left( \frac{s}{s+2} \right)^{\frac{1}{4}} P_{\kappa}^{(1/2)}[1+s] \right).

Four Dimensions

In (3+1)D we may witness how Nariai’s ansatz yields a clean split between light cone versus tail propagation without doing any Fourier transformation; i.e., eq. (6.Even) readily informs us:

(6.4D) G_4[x,x'] = \frac{\Theta[t-t']}{4\pi} \left( \delta[\bar{\sigma}] + \Theta[\bar{\sigma}] \frac{P'_\kappa[1+s]}{\eta \eta'} \right),

where we have used P_\nu[1] = 1. The \delta[\bar{\sigma}] describes propagation on the Minkowski light cone; in fact, we have the (hopefully familiar) retarded conditon \Theta[t-t'] \delta[\bar{\sigma}] = \delta[t-t'-R]/R. Whereas, the expression involving the derivative of the Legendre function, P'_\kappa[1+s]/(\eta\eta'), describes the detailed inside-the-null-cone propagation of the waves.

Speculations

Is there a deeper meaning to Nariai’s ansatz? Note that the object s in eq. (3.CFT) is conformally invariant from a higher dimensional point-of-view. Namely, by viewing the d-coordinates (\eta,\vec{x}) and (\eta',\vec{x}') as residing on a de Sitter-like hyperboloid in one higher dimensional flat spacetime, one may in fact derive s as a Lorentz ‘dot product’ between the two points.

Reference

  • Y.Z. Chu, “More On Cosmological Gravitational Waves And Their Memories,” Class. Quant. Grav. 34, no.19, 194001 (2017) arXiv:1611.00018 [gr-qc]].
  • H. Nariai, “On the Green’s Function in an Expanding Universe and Its Role in the Problem of Mach’s Principle,” Prog. Theor. Phys. 40, 49 (1968).

Author: Yi-Zen Chu

I am a theoretical physicist, with research interests spanning gravitation and field theory, particle cosmology and Mathematica software development.

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