Wands' Duality in Cosmology

• Wands' Duality is a cosmological principle where distinct background evolutions produce the same Mukhanov–Sasaki equation through the invariant z''/z.
• It links slow-roll, ultra-slow-roll, and constant-roll inflation as well as contracting scenarios by preserving the effective potential and the comoving Hubble horizon.
• This duality implies that although phase-space trajectories differ, Gaussian quantum-correlation measures remain identical, underscoring invariant perturbative dynamics.

Wands’ duality is a cosmological duality in which distinct background evolutions generate the same perturbation equation for the canonical scalar variable, so that the degeneracy is controlled by the background combination $z''/z$ or, in $e$-fold time, $H^{-2}z''/z$. In the inflationary literature it relates backgrounds such as slow-roll, constant-roll, and ultra-slow-roll when they yield the same Mukhanov–Sasaki equation, while in contracting cosmologies it appears through an expanding–contracting map $a \leftrightarrow H$ that preserves the comoving Hubble horizon. Recent work further interprets the duality as a local, scale-independent canonical transformation, implying that certain Gaussian quantum-correlation measures are identical for Wands-dual realizations even when the background trajectories differ (Brahma et al., 1 Jul 2026, Karam et al., 2022, Kinney et al., 2010).

1. Perturbative definition and invariant structure

In the formulations considered here, scalar perturbations are encoded in a canonical Mukhanov variable written either as

$u \equiv z\,\mathcal R_c$

or as

$v=z\zeta,$

with mode equation

$$u_k''+\left(k^2-\frac{z''}{z}\right)u_k=0 \qquad \text{or} \qquad v_k'' + \left(k^2 - \frac{z''}{z}\right) v_k = 0.$$

The essential point is that the background enters the perturbation dynamics only through $z''/z$, or equivalently through $H^{-2}z''/z$ when time is measured in $e$-folds. Wands’ duality is therefore a degeneracy of background evolution at the level of this effective potential for perturbations: two different histories can be dual when they induce the same $e$0 (Karam et al., 2022, Brahma et al., 1 Jul 2026).

A central example occurs in quasi-de Sitter form, where

$e$1

so that the mode equation becomes

$e$2

For constant $e$3, the Bunch–Davies solution is

$e$4

In the simplified setting where only $e$5 matters, the Hankel index obeys

$e$6

so a given $e$7 corresponds to two backgrounds,

$e$8

These constitute a Wands-dual pair (Brahma et al., 1 Jul 2026).

This structure makes the duality mathematically precise: the mode equation is invariant even when the background solution is not. A plausible implication is that Wands’ duality is best understood as an equivalence of perturbative kinematics rather than an identity of cosmological histories.

2. Slow-roll, ultra-slow-roll, and constant-roll realizations

The best-known Wands-dual pair in single-field inflation is the slow-roll/ultra-slow-roll pair. In the quasi-de Sitter parametrization, slow-roll has $e$9, ultra-slow-roll has $H^{-2}z''/z$0, and both give $H^{-2}z''/z$1. Consequently,

$H^{-2}z''/z$2

The two backgrounds are therefore degenerate at the level of the Mukhanov–Sasaki configuration variable $H^{-2}z''/z$3 (Brahma et al., 1 Jul 2026).

In the primordial-black-hole literature, the same mechanism is used in a slightly broader form. The ultra-slow-roll phase that amplifies perturbations is dual to a subsequent constant-roll phase, and in the paper’s terminology the SR, USR, T2, and CR phases are organized by the behavior of $H^{-2}z''/z$4. The paper states explicitly that in the USR, T2, and CR phases, $H^{-2}z''/z$5 stays constant, so the USR phase is dual to the CR phase. This is the basis for the statement that the spectrum generated in the later CR phase mirrors the spectrum generated during USR, especially its tilt on either side of the peak (Karam et al., 2022).

A compact derivation follows by expanding the potential around a local maximum $H^{-2}z''/z$6: $H^{-2}z''/z$7 The solution is

$H^{-2}z''/z$8

Then

$H^{-2}z''/z$9

which is independent of the coefficients $a \leftrightarrow H$0. The two branches of the solution correspond to two dual background histories, one USR-like and one CR-like, but they generate the same perturbation equation (Karam et al., 2022).

This duality is not restricted to an exact de Sitter limit. Rather, it organizes a family of piecewise quasi-de Sitter backgrounds whose perturbative output can be solved analytically.

3. Phase space, canonical transformations, and quantum-information invariants

Wands-dual backgrounds are not identical in phase space. Even when $a \leftrightarrow H$1 matches, the conjugate momentum generally differs: $a \leftrightarrow H$2 For this reason, the duals have the same configuration-space dynamics but different phase-space trajectories (Brahma et al., 1 Jul 2026).

Recent work reformulates this observation in continuous-variable Gaussian language. The field and conjugate momentum are assembled into

$a \leftrightarrow H$3

and then coarse-grained over a physical scale $a \leftrightarrow H$4. Because coarse-graining modifies the equal-point commutator, the operators are locally rescaled so that the normalized variables satisfy the canonical relation

$a \leftrightarrow H$5

For two spatially separated regions, one obtains a two-mode covariance matrix $a \leftrightarrow H$6, whose physically meaningful basis-independent data are the symplectic eigenvalues. For the symmetric two-mode case,

$a \leftrightarrow H$7

These determine the one-mode invariant, the inter-patch invariant, and the Gaussian information measures built from them (Brahma et al., 1 Jul 2026).

The key result is that Wands-dual realizations are related by a local, scale-independent canonical transformation,

$a \leftrightarrow H$8

with

$a \leftrightarrow H$9

At the level of the bipartite covariance matrix this is an element of

$u \equiv z\,\mathcal R_c$0

not a generic global $u \equiv z\,\mathcal R_c$1 transformation. Because the transformation is local and $u \equiv z\,\mathcal R_c$2-independent, the momentum-space integral defining the coarse-grained covariance matrix commutes with it, and the symplectic spectrum is unchanged (Brahma et al., 1 Jul 2026).

The consequence is a new “quantum-informatic symmetry.” Although the covariance-matrix entries differ substantially between slow-roll and ultra-slow-roll, the symplectic eigenvalues coincide, and therefore so do the entanglement entropy, mutual information, quantum discord, and log-negativity. For the separated patches considered, the paper finds numerically that $u \equiv z\,\mathcal R_c$3, so $u \equiv z\,\mathcal R_c$4, but still

$u \equiv z\,\mathcal R_c$5

The paper summarizes this by stating that local linear entanglement witnesses cannot distinguish Wands-dual inflationary histories (Brahma et al., 1 Jul 2026).

4. Analytically solvable inflationary models and primordial black holes

Wands’ duality is especially useful in analytically solvable models of single-field inflation for primordial-black-hole production. The background equation is written in terms of $u \equiv z\,\mathcal R_c$6 and

$u \equiv z\,\mathcal R_c$7

with

$u \equiv z\,\mathcal R_c$8

The Hubble slow-roll parameters are

$u \equiv z\,\mathcal R_c$9

Within this framework the SR, T1, USR, T2, and CR phases are classified by the behavior of these parameters, and the duality is encoded by the equality of $v=z\zeta,$0 in the USR and CR phases (Karam et al., 2022).

The paper exploits this by introducing an “instantaneous transition” model in which $v=z\zeta,$1 is piecewise constant with a delta-function dip at the transition. This renders the Mukhanov–Sasaki equation analytically solvable in Bessel functions and produces a power spectrum with three asymptotic regimes: $v=z\zeta,$2 in the momentum intervals stated in the paper. The peak position and height are summarized as

$v=z\zeta,$3

In this description the position of the peak is tied mainly to the transition scale $v=z\zeta,$4, while the height is controlled by the duration of USR through $v=z\zeta,$5 (Karam et al., 2022).

The duality is then realized by a simple inflaton potential of two joined concave parabolas,

$v=z\zeta,$6

with $v=z\zeta,$7, $v=z\zeta,$8, and $v=z\zeta,$9. The resulting scenarios can satisfy the CMB constraints

$$u_k''+\left(k^2-\frac{z''}{z}\right)u_k=0 \qquad \text{or} \qquad v_k'' + \left(k^2 - \frac{z''}{z}\right) v_k = 0.$$0

while producing primordial black holes of arbitrary mass. The paper gives two explicit examples: model $$u_k''+\left(k^2-\frac{z''}{z}\right)u_k=0 \qquad \text{or} \qquad v_k'' + \left(k^2 - \frac{z''}{z}\right) v_k = 0.$$1 with PBHs around $$u_k''+\left(k^2-\frac{z''}{z}\right)u_k=0 \qquad \text{or} \qquad v_k'' + \left(k^2 - \frac{z''}{z}\right) v_k = 0.$$2, and model $$u_k''+\left(k^2-\frac{z''}{z}\right)u_k=0 \qquad \text{or} \qquad v_k'' + \left(k^2 - \frac{z''}{z}\right) v_k = 0.$$3 with PBHs around $$u_k''+\left(k^2-\frac{z''}{z}\right)u_k=0 \qquad \text{or} \qquad v_k'' + \left(k^2 - \frac{z''}{z}\right) v_k = 0.$$4 (Karam et al., 2022).

The same analysis also isolates the limitations of the idealized construction. Sharp transitions generate oscillations around the peak, described in the paper as likely unphysical artifacts of the idealization, whereas a smoother “box” model spreads the transition over a finite interval $$u_k''+\left(k^2-\frac{z''}{z}\right)u_k=0 \qquad \text{or} \qquad v_k'' + \left(k^2 - \frac{z''}{z}\right) v_k = 0.$$5 and damps the oscillations. The asymptotic slopes in SR and CR remain those of the instantaneous model, but the peak becomes smoother as $$u_k''+\left(k^2-\frac{z''}{z}\right)u_k=0 \qquad \text{or} \qquad v_k'' + \left(k^2 - \frac{z''}{z}\right) v_k = 0.$$6 increases. Using the resulting peak shapes, the paper concludes that COBE/FIRAS bounds

$$u_k''+\left(k^2-\frac{z''}{z}\right)u_k=0 \qquad \text{or} \qquad v_k'' + \left(k^2 - \frac{z''}{z}\right) v_k = 0.$$7

exclude the formation of PBHs heavier than

$$u_k''+\left(k^2-\frac{z''}{z}\right)u_k=0 \qquad \text{or} \qquad v_k'' + \left(k^2 - \frac{z''}{z}\right) v_k = 0.$$8

in single-field inflation (Karam et al., 2022).

5. Expanding–contracting duality and the cyclic-cosmology flow hierarchy

A second major use of Wands-style duality appears in contracting cosmologies. In a flat FRW universe,

$$u_k''+\left(k^2-\frac{z''}{z}\right)u_k=0 \qquad \text{or} \qquad v_k'' + \left(k^2 - \frac{z''}{z}\right) v_k = 0.$$9

and the Boyle–Lidsey map

$z''/z$0

preserves the comoving horizon because

$z''/z$1

Thus inflationary and contracting descriptions can generate perturbations by shrinking the same comoving horizon (Kinney et al., 2010).

Under this duality, the equation of state transforms as

$z''/z$2

An inflationary phase with $z''/z$3 therefore maps to a contracting dual with $z''/z$4. The paper’s main formal claim is that the inflationary flow hierarchy is invariant under this map. The standard Hubble flow parameters

$z''/z$5

are replaced by the dual quantities

$z''/z$6

and the dual flow equations have exactly the same form as the inflationary ones (Kinney et al., 2010).

This permits a “dual slow-roll” approximation for contracting universes: $z''/z$7 In that limit the background is quasi-static, $z''/z$8, and the potential is negative,

$z''/z$9

The paper emphasizes that this is the dual of slow-roll, not simply the limit $H^{-2}z''/z$0 (Kinney et al., 2010).

For perturbations, the contracting case differs conceptually from inflation because the growing mode typically resides in the Newtonian potential $H^{-2}z''/z$1, not in $H^{-2}z''/z$2. The paper therefore tracks both scalar variables and derives, in the dual-slow-roll regime,

$H^{-2}z''/z$3

Within this framework two special cases are recovered. The first is Wands’ matter-dominated contracting solution, corresponding to

$H^{-2}z''/z$4

which gives a scale-invariant spectrum for the growing $H^{-2}z''/z$5-mode. The second is adiabatic ekpyrosis, obtained for

$H^{-2}z''/z$6

for which

$H^{-2}z''/z$7

These appear as special points in the dual flow space rather than isolated exceptions (Kinney et al., 2010).

6. Interpretation, limitations, and recurring misconceptions

The most common misconception is to read Wands’ duality as a statement that the dual backgrounds are physically identical. The recent quasi-de Sitter analysis explicitly rejects that interpretation. The two backgrounds are degenerate at the level of the Mukhanov–Sasaki configuration variable, but they are not identical in phase space, and the curvature perturbation $H^{-2}z''/z$8 still differs because

$H^{-2}z''/z$9

with $e$0 background-dependent. Slow-roll and ultra-slow-roll therefore remain different cosmological histories, especially in their superhorizon evolution of $e$1 (Brahma et al., 1 Jul 2026).

A second misconception is that the duality fixes the full observable power spectrum. In the PBH constructions it fixes the asymptotic tails through the equality of $e$2, but the detailed peak shape depends on the SR-to-USR transition. The paper distinguishes an instantaneous-transition model, which produces oscillations around the peak, from smoother realizations in which the oscillations are damped (Karam et al., 2022).

A third misconception is that the inflationary intuition for $e$3 automatically carries over to contracting cosmologies. The cyclic-cosmology analysis stresses the opposite: in contraction one must track both $e$4 and $e$5, because the growing mode typically sits in $e$6, and the observable outcome depends on how the bounce transfers modes to the expanding phase (Kinney et al., 2010).

The three principal realizations discussed in the literature surveyed here can be summarized as follows.

Context	Dual variables or branches	Invariant object
Quasi-de Sitter inflation	$e$7	Mukhanov–Sasaki equation for $e$8
PBH-oriented single-field inflation	USR branch and CR branch	$e$9
Cyclic/ekpyrotic cosmology	$e$00	Comoving horizon $e$01

Taken together, these results suggest a unified interpretation. Wands’ duality is a perturbative degeneracy principle that can be expressed either as equality of $e$02 in inflationary backgrounds or as horizon-preserving exchange $e$03 in expanding/contracting cosmologies. A plausible implication is that its real scope lies in the invariant structures it preserves—mode equations, horizon flow, and local symplectic data—rather than in a background-level equivalence of spacetimes.