No-Boundary Instantons in Quantum Cosmology

• No-boundary instantons are complex saddle-point solutions from the Hartle–Hawking proposal, derived through a Euclidean path integral over compact geometries.
• They use analytic continuation along complex time contours to transition from Euclidean to Lorentzian regimes, ensuring classicalization via inflationary, ekpyrotic, or other attractor mechanisms.
• Their semiclassical weighting, expressed as P ∼ e^(–2Re(S_E)), modulates the initial conditions across diverse settings, including multi-field inflation, cyclic models, and modified gravity scenarios.

Searching arXiv for recent and foundational papers on no-boundary instantons to support the article. No-boundary instantons are saddle-point geometries of the Hartle–Hawking no-boundary wave function in which the Euclidean path integral over compact, regular geometries with only a final boundary is approximated by regular $O(4)$-symmetric or more general solutions that typically become complex-valued in the interior and are continued to a Lorentzian regime. In the modern literature these saddles are often “fuzzy instantons”: the metric and matter fields are complex along an appropriate contour in complex time, while the final boundary data are taken to be real and classical. Across inflationary, ekpyrotic, anti-de Sitter, loop-quantum-cosmology, bigravity, anisotropic, and wormhole settings, the central technical questions are the existence of such saddles, the conditions under which they classicalize, and the semiclassical weighting $P\sim e^{-2\,\mathrm{Re}(S_E)}$ that they induce (Yeom, 2021).

1. Euclidean path integral and saddle-point definition

In the no-boundary proposal, the wave function is defined by a Euclidean path integral over compact, regular geometries with no initial boundary and specified final boundary data. A representative form used in inflationary and ekpyrotic models is

$$\Psi[h_{\mu\nu},\chi] = \int \mathcal{D}g_{\mu\nu}\,\mathcal{D}\Phi \; e^{-S_E[g_{\mu\nu},\Phi]},$$

or, in minisuperspace,

$\Psi(b,\chi)\sim \sum e^{-S_E(b,\chi)},$

with the sum evaluated by steepest descent over saddle points (Hwang et al., 2014). In the standard closed-FRW ansatz,

$ds^2=d\tau^2+a^2(\tau)d\Omega_3^2,$

the instanton is regular at the “South Pole” $\tau=0$, where the geometry closes off smoothly and the usual regularity conditions are

$a(0)=0,\qquad a'(0)=1,\qquad \phi'(0)=0$

for a single scalar, or

$a(0)=0,\qquad \dot a(0)=1,\qquad \dot\phi_i(0)=0$

in multi-field notation (Battarra et al., 2014).

In minisuperspace the action may also be recast in superspace variables $q^A$ with Hamiltonian constraint

$H=\frac12 G^{AB}p_Ap_B+\mathcal V(q^A)=0,$

which becomes the Wheeler–DeWitt equation upon quantization in the two-field inflationary analysis (Hwang et al., 2014). In this formulation, a no-boundary instanton is not merely a Euclidean solution of the field equations but a semiclassical saddle contributing to the wave function of the universe.

A recurring distinction in the literature is between ordinary real instantons and complex no-boundary instantons. For generic boundary data there is no real solution satisfying both regularity at the South Pole and specified final values; accordingly the relevant saddles are typically complex, with $P\sim e^{-2\,\mathrm{Re}(S_E)}$0 and the matter fields promoted to complex functions and the action evaluated on a contour in complexified time (Battarra et al., 2014). The review literature treats this complexification as structural rather than exceptional: if instantons are dynamical, then instantons are necessarily complexified, and “fuzzy instantons” provide the operative saddle-point realization of the Hartle–Hawking state (Yeom, 2021).

2. Complex contours, classicality, and the emergence of Lorentzian histories

The defining physical requirement is not only regularity of the Euclidean saddle but also the existence of a real Lorentzian classical history after analytic continuation. In the inflationary two-field analysis, the contour is taken as $P\sim e^{-2\,\mathrm{Re}(S_E)}$1, with a Euclidean segment from the South Pole to a turning point and then a Lorentzian segment; the complex phases of the South Pole field values and the turning point are tuned so that the late-time history classicalizes (Hwang et al., 2014). Ekpyrotic instantons are constructed on an L-shaped contour consisting of a vertical segment from the South Pole, a horizontal complex segment, and a final vertical Lorentzian segment (Battarra et al., 2014).

Classicality is expressed in WKB form by requiring the phase of the wave function to vary much more rapidly than its amplitude. Equivalent formulations used in the literature are

$P\sim e^{-2\,\mathrm{Re}(S_E)}$2

or, for explicit minisuperspace variables,

$P\sim e^{-2\,\mathrm{Re}(S_E)}$3

Operationally, this means that the imaginary parts of the metric and fields become negligible compared with the real parts at late Lorentzian time, while $P\sim e^{-2\,\mathrm{Re}(S_E)}$4 approaches a constant (Battarra et al., 2014).

Different cosmological mechanisms can supply the required classicalization. In inflationary models, the relevant attractor is slow-roll inflation, and the two-field analysis emphasizes that a history needs a period of slow-roll inflation after the Euclidean-to-Lorentzian transition in order to classicalize (Hwang et al., 2014). In ekpyrotic models, by contrast, the classicalization mechanism is the ekpyrotic attractor. For the steep negative exponential potential

$P\sim e^{-2\,\mathrm{Re}(S_E)}$5

the asymptotic corrections decay only if

$P\sim e^{-2\,\mathrm{Re}(S_E)}$6

and under that condition the imaginary parts of $P\sim e^{-2\,\mathrm{Re}(S_E)}$7 and $P\sim e^{-2\,\mathrm{Re}(S_E)}$8 die away so that a real Lorentzian contracting history emerges (Battarra et al., 2014).

The literature also stresses that classicality is model-dependent and may fail in settings where a finite semiclassical weight can nevertheless be assigned. Near a tachyonic top in anti-de Sitter space, the no-boundary regulator yields finite nonzero probabilities for generic complex instantons, but the scalar and metric may remain significantly complex after Wick rotation. This leads to the explicit contrast that semi-classical observers in de Sitter are classicalized individually, whereas a semi-classical boundary observer in anti-de Sitter may notice the dispersion of quantum fields as a kind of uncertainty (Lee et al., 2014).

3. Inflationary no-boundary instantons

In inflationary minisuperspace models, no-boundary instantons are usually studied in Einstein gravity coupled to one or more scalars with regular South Pole data. A canonical two-field example uses

$P\sim e^{-2\,\mathrm{Re}(S_E)}$9

together with an $$\Psi[h_{\mu\nu},\chi] = \int \mathcal{D}g_{\mu\nu}\,\mathcal{D}\Phi \; e^{-S_E[g_{\mu\nu},\Phi]},$$0-symmetric closed universe ansatz. After rescaling by $$\Psi[h_{\mu\nu},\chi] = \int \mathcal{D}g_{\mu\nu}\,\mathcal{D}\Phi \; e^{-S_E[g_{\mu\nu},\Phi]},$$1, the problem depends only on the ratio $$\Psi[h_{\mu\nu},\chi] = \int \mathcal{D}g_{\mu\nu}\,\mathcal{D}\Phi \; e^{-S_E[g_{\mu\nu},\Phi]},$$2, and the saddle-point equations are solved numerically by tuning the complex South Pole phases and the turning point so that the late-time history becomes classical (Hwang et al., 2014).

The central physical result of that analysis is that a relatively massive direction can increase the expected number of $$\Psi[h_{\mu\nu},\chi] = \int \mathcal{D}g_{\mu\nu}\,\mathcal{D}\Phi \; e^{-S_E[g_{\mu\nu},\Phi]},$$3-foldings. If $$\Psi[h_{\mu\nu},\chi] = \int \mathcal{D}g_{\mu\nu}\,\mathcal{D}\Phi \; e^{-S_E[g_{\mu\nu},\Phi]},$$4 is the slow direction and $$\Psi[h_{\mu\nu},\chi] = \int \mathcal{D}g_{\mu\nu}\,\mathcal{D}\Phi \; e^{-S_E[g_{\mu\nu},\Phi]},$$5 is heavier, then the heavy field is harder to classicalize because in Euclidean signature the effective instability is enhanced by its larger mass. To classicalize the heavy field, the instanton must begin from a configuration with larger vacuum energy in the slow direction,

$$\Psi[h_{\mu\nu},\chi] = \int \mathcal{D}g_{\mu\nu}\,\mathcal{D}\Phi \; e^{-S_E[g_{\mu\nu},\Phi]},$$6

so the existence of the massive direction implies the increase of expected $$\Psi[h_{\mu\nu},\chi] = \int \mathcal{D}g_{\mu\nu}\,\mathcal{D}\Phi \; e^{-S_E[g_{\mu\nu},\Phi]},$$7-foldings (Hwang et al., 2014). In the $$\Psi[h_{\mu\nu},\chi] = \int \mathcal{D}g_{\mu\nu}\,\mathcal{D}\Phi \; e^{-S_E[g_{\mu\nu},\Phi]},$$8 limit, the paper derives

$$\Psi[h_{\mu\nu},\chi] = \int \mathcal{D}g_{\mu\nu}\,\mathcal{D}\Phi \; e^{-S_E[g_{\mu\nu},\Phi]},$$9

with the specific estimate

$\Psi(b,\chi)\sim \sum e^{-S_E(b,\chi)},$0

and states that with sufficient mass hierarchy the no-boundary wave function can reasonably explain large inflationary expansion, including more than $\Psi(b,\chi)\sim \sum e^{-S_E(b,\chi)},$1 $\Psi(b,\chi)\sim \sum e^{-S_E(b,\chi)},$2-foldings (Hwang et al., 2014).

This multi-field result is presented in the broader review literature as a natural response to the familiar single-field tension: in simple slow-roll models the no-boundary wave function tends to favor lower potential energy and therefore too little inflation, whereas with two fields and $\Psi(b,\chi)\sim \sum e^{-S_E(b,\chi)},$3 the heavier field helps enforce classicality over a larger range of the inflaton direction and can push the preferred initial condition to much larger $\Psi(b,\chi)\sim \sum e^{-S_E(b,\chi)},$4, yielding many more $\Psi(b,\chi)\sim \sum e^{-S_E(b,\chi)},$5-foldings (Yeom, 2021).

No-boundary instantons have also been generalized beyond isotropic FRW inflation. In anisotropic Bianchi IX minisuperspace, the boundary data include not only the scale factor and inflaton but also anisotropy variables $\Psi(b,\chi)\sim \sum e^{-S_E(b,\chi)},$6, and regular complex no-boundary saddles still exist for arbitrarily large anisotropies (Bramberger et al., 2017). These instantons differ qualitatively from Hawking’s original round Euclidean cap: increasing anisotropy introduces additional singularities and branch structure in the complex $\Psi(b,\chi)\sim \sum e^{-S_E(b,\chi)},$7-plane, so the usual Hawking contour is often not valid and must be replaced by a contour that avoids the singularities (Bramberger et al., 2017). Every classical history found in that analysis still reaches an inflationary phase, with anisotropies decaying away, but the wave function becomes classical more slowly in the anisotropic directions, with

$\Psi(b,\chi)\sim \sum e^{-S_E(b,\chi)},$8

rather than the isotropic scaling

$\Psi(b,\chi)\sim \sum e^{-S_E(b,\chi)},$9

(Bramberger et al., 2017).

4. Ekpyrotic and cyclic no-boundary instantons

In 2014, papers on ekpyrotic cosmology showed that complex no-boundary instantons also exist for steep negative potentials. For

$ds^2=d\tau^2+a^2(\tau)d\Omega_3^2,$0

with $ds^2=d\tau^2+a^2(\tau)d\Omega_3^2,$1, there exists a family of ekpyrotic instantons satisfying no-boundary conditions and evolving into a real contracting universe (Battarra et al., 2014). The decisive mechanism is the attractor structure: near the crunch the solution approaches the ekpyrotic scaling form, and the complex corrections decay if and only if $ds^2=d\tau^2+a^2(\tau)d\Omega_3^2,$2, which is exactly the ekpyrotic attractor condition (Battarra et al., 2014).

These saddles differ sharply from inflationary and de Sitter-type instantons. Inflationary instantons classicalize because the scalar approaches an inflationary attractor and the Lorentzian phase is expanding; ekpyrotic instantons classicalize during contraction, are driven by a steep negative potential, and become real because of the ekpyrotic attractor rather than slow-roll inflation (Battarra et al., 2014). The scalar field is also highly dynamical as classicality is reached, whereas in inflation it becomes slowly rolling and nearly frozen (Battarra et al., 2014).

The semiclassical weighting of ekpyrotic instantons is likewise distinctive. Because the exponential potential has a shift symmetry, the action rescales so that, at fixed steepness $ds^2=d\tau^2+a^2(\tau)d\Omega_3^2,$3,

$ds^2=d\tau^2+a^2(\tau)d\Omega_3^2,$4

and the probability can be written as

$ds^2=d\tau^2+a^2(\tau)d\Omega_3^2,$5

with $ds^2=d\tau^2+a^2(\tau)d\Omega_3^2,$6 in the numerical examples (Battarra et al., 2014). Thus smaller $ds^2=d\tau^2+a^2(\tau)d\Omega_3^2,$7 gives higher probability, favoring long ekpyrotic phases. In mixed potential landscapes containing both inflationary and ekpyrotic sectors, the papers conclude that ekpyrotic and cyclic initial conditions are vastly more probable than inflationary ones (Battarra et al., 2014).

In cyclic cosmologies with

$ds^2=d\tau^2+a^2(\tau)d\Omega_3^2,$8

the literature finds two distinct classes of no-boundary instantons: de Sitter-like instantons emerging in the positive dark-energy plateau and ekpyrotic instantons that remain complex longer and only become real near the approach to the crunch (Battarra et al., 2014). The same starting $ds^2=d\tau^2+a^2(\tau)d\Omega_3^2,$9 can therefore admit two distinct saddles differing in $\tau=0$0, with the numerical analysis indicating that the ekpyrotic-type branch generally has much smaller action and therefore much higher probability (Battarra et al., 2014). An important caveat, stressed already in the original ekpyrotic construction, is that the models as studied do not include NEC violation and therefore do not include a bounce; all classical ekpyrotic histories end in a crunch unless an additional bounce mechanism is incorporated (Battarra et al., 2014).

5. Extensions, deformations, and related Euclidean saddles

No-boundary instantons have been generalized well beyond standard Einstein-scalar inflation. In loop quantum cosmology, the effective Euclidean equations acquire holonomy and inverse-triad corrections, and the geometry near the would-be South Pole changes qualitatively. For a closed FLRW universe with a positive cosmological constant, the small-$\tau=0$1 behavior is governed by inverse-triad effects so that

$\tau=0$2

and the scale factor approaches zero only asymptotically,

$\tau=0$3

The resulting no-boundary instanton therefore has an infinitely stretched Euclidean tail rather than a finite Euclidean cap, while still yielding a finite nucleation probability (Brahma et al., 2018). The same LQC program also argues, in a Lorentzian path-integral formulation, that off-shell instantons experience dynamical signature change with an effective line element

$\tau=0$4

and that this mechanism moves the dangerous branch cuts responsible for perturbative instabilities away from the relevant contour, thereby stabilizing no-boundary perturbations even at sub-Planckian density (Bojowald et al., 2018). A subsequent analysis derives the same stabilization mechanism for off-shell instantons more systematically and identifies $\tau=0$5 as especially favorable for producing an early Euclidean-like region and removing the upside-down Gaussian instability (Bojowald et al., 2020).

Bigravity provides another deformation. In ghost-free bigravity, homogeneous Hawking–Moss instantons become the bigravity analogues of compact no-boundary saddles and split into two branches determined by the consistency condition for the ratio $\tau=0$6 of the two scale factors (Zhang et al., 2014). Branch I behaves essentially like “two copies of GR,” whereas Branch II carries genuine bigravity corrections and is identified with the self-accelerating branch. In the dRGT massive-gravity limit, Branch II develops a solution-dependent positive divergence that cannot be removed by universal renormalization, so its tunneling probability vanishes, while Branch I reduces smoothly to the GR-like result (Zhang et al., 2014). The same analysis argues that only Branch II changes the Hartle–Hawking measure in a way that can make sufficiently large inflationary expansion non-negligible in probability (Zhang et al., 2014).

Anti-de Sitter settings require additional care because the naive Euclidean action can diverge. The “no-boundary regulator” complexifies the contour, metric, and scalar field so that a regulated action

$\tau=0$7

can be assigned to generic instantons near a tachyonic top, without the fine-tuning otherwise needed to remove the slow-decay mode (Lee et al., 2014). The resulting saddles are physically interpreted not as standard decay bounces that destroy the tachyonic top, but as quantum or thermal fluctuations around the top (Lee et al., 2014).

Recent work has also placed no-boundary instantons in a common family with Euclidean wineglass wormholes. For scalar matter plus axionic or magnetic charge, a charged Euclidean wineglass wormhole can interpolate from an asymptotically AdS region to a “mouth” where the geometry is analytically continued into an expanding Lorentzian universe (Lavrelashvili et al., 11 Mar 2026). As the charge is taken to zero, the throat shrinks and pinches off,

$\tau=0$8

leaving a disconnected no-boundary instanton that decouples from the asymptotic AdS region (Lavrelashvili et al., 11 Mar 2026). A follow-up study describes this explicitly as a topology-changing transition in which the wormhole stem pinches off and the solution splits into a background spacetime plus a disconnected no-boundary bowl; small-charge solutions are less suppressed and seed longer inflationary phases, but the zero-charge no-boundary limit carries the largest weighting in the family (Lavrelashvili et al., 11 May 2026).

A nearby but conceptually distinct development is the “Boundary Proposal,” which replaces the smooth $\tau=0$9 no-boundary instanton by a sphere with two holes bounded by end-of-the-world branes. The paper presents this construction as being on equal footing with the no-boundary process and shows that its action interpolates between $a(0)=0,\qquad a'(0)=1,\qquad \phi'(0)=0$0 at $a(0)=0,\qquad a'(0)=1,\qquad \phi'(0)=0$1 and the usual no-boundary action in the large negative-tension limit (Friedrich et al., 2024). This comparison does not alter the definition of a no-boundary instanton, but it sharpens the role played by the absence of an initial boundary in the Hartle–Hawking construction.

6. Probability measures, controversies, and open directions

Across the literature, the semiclassical weighting of a no-boundary instanton is controlled by the real part of the Euclidean action. In its most common form,

$a(0)=0,\qquad a'(0)=1,\qquad \phi'(0)=0$2

or equivalently $a(0)=0,\qquad a'(0)=1,\qquad \phi'(0)=0$3, so lower $a(0)=0,\qquad a'(0)=1,\qquad \phi'(0)=0$4 means higher relative probability (Hwang et al., 2014). This common formula, however, produces different cosmological preferences depending on the model. In single-field slow-roll settings it typically favors smaller positive potential energy and therefore too little inflation; in the two-field model a mass hierarchy reshapes the classicalization boundary and can make large inflation a natural prediction; in mixed landscapes the ekpyrotic and cyclic branches are argued to dominate overwhelmingly; and in wineglass wormhole families the zero-charge no-boundary endpoint dominates the probability distribution overall (Hwang et al., 2014).

Several controversies recur. One concerns the status of complexification itself. The review literature treats complex-valued fuzzy instantons as unavoidable once the instanton is dynamical, but alternative path-integral analyses have emphasized contour dependence and saddle selection; the wormhole literature correspondingly notes unresolved questions about which saddles truly belong in the gravitational path integral and how contour choices and allowability criteria affect the result (Yeom, 2021). A second controversy concerns perturbative stability. Lorentzian path-integral treatments identified unstable Gaussian weights for perturbations around no-boundary saddles, while loop-quantum-cosmology analyses argue that dynamical signature change in off-shell instantons removes the relevant branch cuts and restores stability (Bojowald et al., 2018).

A third issue is the meaning of classicality. Inflationary and ekpyrotic models use attractor dynamics to classicalize the solution, but anti-de Sitter instantons regulated by the no-boundary prescription may remain nonclassical after Wick rotation (Lee et al., 2014). This suggests that “existence of a finite no-boundary saddle” and “emergence of a classical Lorentzian history” are not equivalent statements.

The main limitations are also consistent across the literature. Many analyses are restricted to minisuperspace, often with homogeneous $a(0)=0,\qquad a'(0)=1,\qquad \phi'(0)=0$5-symmetric ansätze and simple scalar potentials such as quadratic or exponential forms (Hwang et al., 2014). The ekpyrotic and cyclic models considered in 2014 do not include a bounce (Battarra et al., 2014). The loop-quantum-cosmology studies restrict matter content to a cosmological constant or very flat potential and use effective semiclassical dynamics (Brahma et al., 2018). The wormhole/no-boundary connection is established for explicit axionic and magnetic models, but broader measure questions, disconnected backgrounds, and possible implications for factorization remain open (Lavrelashvili et al., 11 May 2026).

Taken together, these results suggest a precise but evolving picture. No-boundary instantons are regular Euclidean or complexified saddle points that prepare cosmological states without an initial boundary; their concrete form depends strongly on the matter sector, modified-gravity corrections, and the attractor mechanism responsible for classicalization. Their modern study has expanded from half-$a(0)=0,\qquad a'(0)=1,\qquad \phi'(0)=0$6 de Sitter caps to multi-field inflationary saddles, ekpyrotic contracting histories, anisotropic complex geometries, regulated anti-de Sitter instantons, loop-quantum-cosmology tails and signature-changing caps, and zero-charge limits of Euclidean wormholes. The common structural question remains the same throughout: which complex saddles are regular, classicalizable, and semiclassically dominant, and what initial conditions for cosmology do they thereby select.