Cosmological Bootstrap as Holographic Input

The cosmological bootstrap constrains wavefunction coefficients and correlators using de Sitter symmetry, singularities, factorization, locality assumptions, and cutting identities. These data can test the bulk side of a cosmological holographic proposal. They do not, by themselves, define a boundary Hilbert space, inner product, time evolution, or reconstruction map, and therefore are not already a holographic dual.

Required background. Total-Energy and Factorization Singularities supplies the analytic data; Wavefunction Coefficients versus In-In Correlators fixes the object being constrained.

Helpful background. Use Cosmological Cutting Identities and Unitarity Constraints, Locality, Contact Ambiguities, and Field-Redefinition Equivalence, Cosmological Symmetry and Ward Identities, Contact, Exchange, and Seed Solutions, Dispersion Relations, Subtractions, and Qualified Positivity, Reconstruction from Symmetry, Singularities, and Cuts, and Cosmological Bootstrap: Loops, Initial States, and Limits for the imported results.

Singularities encode selected bulk processes

For an $n$-point Bunch–Davies wavefunction coefficient, define the total energy

$$k_T=\sum_{a=1}^n k_a.$$

In many tree-level local interactions, the leading singularity as $k_T\to0$ has residue proportional to the corresponding flat-space scattering amplitude, after polarization and normalization factors are removed. Exchange diagrams also have partial-energy singularities. If $p=|\mathbf k_1+\mathbf k_2|$, a four-point exchange can develop singularities at

$$E_L=k_1+k_2+p=0, \qquad E_R=k_3+k_4+p=0,$$

whose residues factorize into lower-point data and an internal two-point normalization. Cosmological polytopes make these singularity and factorization structures geometrically explicit for broad classes of diagrams Arkani-Hamed, Benincasa, and Postnikov 2018, §§ 2–5.

These are analytic continuations in energy variables; physical magnitudes $k_a$ remain positive. A pole determines selected nonlocal data, not the entire coefficient.

First application: reconstruct an exchange contribution

Suppose $\psi_3^{(L)}$ and $\psi_3^{(R)}$ are known and an internal scalar has power spectrum $P(p)$. Factorization requires a double residue of the schematic form

$$\operatorname*{Res}_{E_L=0}\operatorname*{Res}_{E_R=0}\psi_4 \propto\frac{\psi_3^{(L)}\psi_3^{(R)}}{P(p)},$$

with the precise conjugation and discontinuity fixed by whether one is bootstrapping $\Psi$, $|\Psi|^2$, or an in-in correlator. de Sitter Ward identities fix the allowed momentum dependence; cutting identities relate discontinuities to products of lower-point quantities. These constraints can identify the exchanged mass and spin and reject a proposed bulk vertex whose residue or phase is wrong.

Now add a local quartic contact interaction. It contributes a homogeneous/contact solution with no exchange factorization pole but compatible symmetries. Subtractions in a dispersion relation encode the same freedom. Singularities and cuts therefore reconstruct the exchange part only after a locality class and contact basis are declared.

Holographic use and its boundary

A proposed dS/CFT dictionary may predict a boundary object whose correlators equal $\psi_n$. Bootstrap output then supplies quantitative tests of dimensions, tensor structures, and factorization. Yet it says nothing by itself about reflection positivity, the boundary inner product, finite-time static-patch access, or nonperturbative state counting. A bulk solution reconstructed from analytic data is likewise not unique under field redefinitions and contact terms.

The controlled claim normally assumes the Bunch–Davies contour, perturbation theory, a specified loop order, and locality below a cutoff. At late time, infrared logs or alternate initial states can change the analytic structure. A heavy-particle signal establishes compatible bulk exchange data, not a complete boundary theory.

Adversarial control: add invisible data

Hold fixed every selected total- and partial-energy residue, then add an allowed contact solution. If the inferred bulk Lagrangian changes, the singularity set did not determine it uniquely. Next replace the initial state by a Bogoliubov state; folded singularities can appear while de Sitter-like kinematics partly survive. A dictionary that omitted the state has overclaimed.

These methods provide powerful reconstruction and consistency tests for specified cosmological coefficients, including locality and unitarity diagnostics. They do not establish a duality, uniqueness, or a normalized observer algebra. The bootstrap derivations are given in cosmological QFT; this page uses their results to test holographic proposals.

The chapter overview compares these assumptions and limitations.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Arkani-Hamed, N., Benincasa, P., and Postnikov, A. (2018). “Cosmological Polytopes and the Wavefunction of the Universe.” Journal of High Energy Physics 2018(4), 105. DOI.