Primordial bispectrum ($B(k_1,k_2,k_3)$)

Statistical homogeneity defines the primordial bispectrum by

A bispectrum is a Fourier-space third-order statistic; primordial perturbations supply the cosmological field in this instance.

With cosmic-time interaction coefficient $g$, constant $H,ϵ,c_s$, and modes whose positive-frequency part is $H(1+ik{c}_s\tau )e^{-ik{c}_s\tau }/\sqrt{4ϵc_s{k}^3}$ in units $M_p=1$, two Wick contractions per differentiated pair and the late-time scalar gradient vertex integral giveHere $⟨{\zeta }^3{⟩}_c=(2\pi {)}^3{\delta }^3(\sum {\mathbf{k}}_i)B$. The extra factor $a$ in $dt=ad\tau$ must be included. A positive $g$ yields a negative bispectrum in an equilateral bispectrum configuration; changing the sign of the interaction changes the sign of $B$.

For $H_{\mathrm{int}}=-M_{\mathrm{Pl}}^2\int d^{3}x{a}^3{ϵ}^2\zeta {\dot{\zeta }}^2$ and the standard Gaussian de Sitter modes, the tree in-in formalism gives $B=H^4{\sum }_{\mathrm{cyc}}{k}_2^2{k}_3^2(K^{-1}+k_1{K}^{-2})/[16ϵM_{\mathrm{Pl}}^4(k_1{k}_2{k}_3{)}^3]$, where $K={\sum }_i{k}_i$. The result includes six connected Wick contractions and a vacuum contour.

The comoving curvature perturbation is dimensionless. With spatial coordinates and time carrying inverse mass dimensions, the cubic interaction $\int dt{d}^{3}x{a}^3{ϵ}^2\zeta {\dot{\zeta }}^2$ needs a mass-squared coefficient. Its standard gravitational normalization is $M_{\mathrm{Pl}}^2$. Six De Sitter curvature mode functions supply $M_{\mathrm{Pl}}^{-6}$; this vertex then supplies $M_{\mathrm{Pl}}^2$, leaving the physical $M_{\mathrm{Pl}}^{-4}$ curvature primordial bispectrum. The Wick-pairing multiplicity for a zeta zeta-prime-squared vertex is independently six: three choices of undifferentiated external leg times two differentiated-leg assignments.

The undifferentiated soft-leg permutation gives $B(q,k,k)\simeq (ϵ/2)P(q)P(k)$ as $q/k\to 0$. It retains a squeezed $q^{-3}$ enhancement, but corresponds to local amplitude $f_{\mathrm{NL}}^{\mathrm{effective}}=5ϵ/24$. Soft momentum placed only at derivative slots would miss this dominant term.

Three external fields can be joined to the three cubic vertex fields in six connected Wick contractions. There are three choices for the external field at the undifferentiated slot and two exchanges of the differentiated slots. Three cyclic terms therefore require an explicit factor of two. Ordered contractions with an earlier vertex use complex-conjugated mode functions.

A constant primordial bispectrum means a constant bispectrum shape function, not a bispectrum independent of wavenumber. With $P(k)=2{\pi }^2{\Delta }_{\zeta }^2/k^3$ and scale-independent ${\Delta }_{\zeta }^2$, one normalization is $B=f_{\mathrm{NL}}[P(k_1)P(k_2)P(k_3){]}^{2/3}=4{\pi }^4{f}_{\mathrm{NL}}{\Delta }_{\zeta }^4/(k_1{k}_2{k}_3{)}^2$.

A flattened primordial bispectrum configuration lies at the collinear boundary of the allowed momentum-triangle domain. For a cubic time-derivative curvature bispectrum, it is finite rather than singular. Its equal-short-side value $k_1=k_2=k_3/2$ gives $k_1{k}_2{k}_3/(k_1+k_2+k_3{)}^3=1/32$, compared with $1/27$ at an equilateral bispectrum configuration.

An equilateral primordial bispectrum configuration has equal external wavenumbers. For a cubic time-derivative curvature bispectrum, $S\propto k_1{k}_2{k}_3/(k_1+k_2+k_3{)}^3$ is maximized here at fixed perimeter by the arithmetic-geometric mean inequality.

For a scale-invariant primordial bispectrum, the displayed shape function is invariant under common rescaling of all three wavenumbers. It isolates triangle dependence from the overall power spectrum amplitude. It indicates intrinsic shape preference but alone does not determine observational signal-to-noise, which also depends on transfer functions, covariance and survey weights.

Put $C=ϵ+1-c_s^2$. For constant $H,ϵ,c_s>0$ and cubic interaction $H_{\mathrm{int}}=-M_{\mathrm{Pl}}^2\int d^{3}x{a}^3ϵC{\dot{\zeta }}^3/(H{c}_s^2)$, the Bunch-Davies vacuum modes $u_k=H(1+ik{c}_s\tau )e^{-ik{c}_s\tau }/\sqrt{4ϵM_{\mathrm{Pl}}^2{c}_s{k}^3}$ giveTo derive this primordial bispectrum, transform $dt=ad\tau$, $\dot{\zeta }={\zeta }^{\prime }/a$, use the six connected Wick contractions, and integrate ${\int }_{-\infty (1-i0)}^0{\tau }^2{e}^{i{c}_s(k_1+k_2+k_3)\tau }d\tau =2i/[c_s(k_1+k_2+k_3){]}^3$ in the in-in formalism. The normalized equilateral amplitude is $B(k,k,k)/P(k{)}^2=2C/9$; raw inverse powers of $c_s$ must not be confused with normalized primordial non-Gaussianity.

For $H_{\mathrm{int}}=-\int d^{3}x{a}^3ϵ\mathcal{F}{\dot{\zeta }}^3/H$, $a=-1/(H\tau )$ and stipulated modes $u_k=H(1+ik\tau )e^{-ik\tau }/\sqrt{4ϵk^3}$, six connected Wick contractions and ${\int }_{-\infty (1-i0)}^0{\tau }^2{e}^{iK\tau }d\tau =2i/K^3$ yield $B=3H^4\mathcal{F}/(8{ϵ}^2{k}_1{k}_2{k}_3{K}^3)$. Here $⟨{\zeta }^3{⟩}_c=(2\pi {)}^3\delta (\sum {\mathbf{k}}_i)B$. Using unstarred vertex modes with the external-before-vertex prefactor incorrectly reverses the sign. Non-unit sound speed also changes the quadratic modes, so this result is conditional on the specified unit-speed modes.

A primordial bispectrum is squeezed when one external Fourier mode has much smaller wavenumber than the other two. Momentum conservation then makes the two short wavevectors almost opposite. This configuration measures the response of short-scale correlations to a long perturbation.

For a massless scalar in de Sitter spacetime with interaction $S_{\mathrm{int}}=(\lambda /3!)\int dt{d}^{3}x{a}^3{\dot{\varphi }}^2\varphi$, the tree-level late-time primordial bispectrum isIt scales as $B(\mu k_1,\mu k_2,\mu k_3)={\mu }^{-6}B(k_1,k_2,k_3)$.

If a scalar vacuum is invariant under an exact scalar-field shift symmetry, the Ward identity makes the leading soft bispectrum vanish:

For a single-clock adiabatic inflationary background, a long comoving curvature perturbation acts on short modes as a spatial dilation. The squeezed bispectrum therefore obeysat leading order in $q/k$.

In an attractor with one adiabatic clock, the leading squeezed primordial bispectrum is the response to a coordinate dilation. Holding a halo's physical mass and smoothing scale fixed cancels this dilation in its abundance response. Thus the consistency-relation term does not generate the physical $k^{-2}$ scale-dependent halo bias from local non-Gaussianity. For example, a top-hat filter of fixed physical radius has coordinate radius $R_c=e^{-{\zeta }_L}{R}_0$, while the short density contrast is ${\delta }_s(\mathbf{x})={\delta }_s^{(0)}(e^{{\zeta }_L}\mathbf{x})$. The substitution $\mathbf{y}=e^{{\zeta }_L}\mathbf{x}$ shows that its normalized average over $|\mathbf{x}|<R_c$ equals the original average over $|\mathbf{y}|<R_0$. Genuine local effects start at two spatial derivatives of the long mode. Non-attractor backgrounds or nonstandard initial states need not satisfy this argument.