Building a Boostless Bootstrap
for the Bispectrum

For the convenience of the reader, we summarize in the following the Bootstrap Rules. All the rules are formulated on the boundary in terms of the actual observables, namely the future asymptotic of equal-time correlators. Notice that invariance of the correlators under de Sitter boosts is not assumed.

• •

  Rule 1: Because of homogeneity, isotropy and scale invariance, the bispectrum for fields of any spin can be decomposed into a polarization factor and a trimmed bispectrum $\mathcal{B}$,

  $\displaystyle B$

  $\displaystyle=\sum_{\text{contractions}}\left[\epsilon^{h_{1}}(\mathbf{k}_{1})\epsilon^{h_{2}}(\mathbf{k}_{2})\epsilon^{h_{3}}(\mathbf{k}_{3})\mathbf{k}_{1}^{\alpha_{1}}\mathbf{k}_{2}^{\alpha_{2}}\mathbf{k}_{3}^{\alpha_{3}}\right]\times\mathcal{B}\,,$

  (1.1)

  with index contractions left implicit. For $\alpha_{\text{tot}}=\alpha_{1}+\alpha_{2}+\alpha_{3}$, the trimmed bispectrum must satisfy

  $\displaystyle\mathcal{B}$

  $\displaystyle=\mathcal{B}(k_{1},k_{2},k_{3})\,,$

  $\displaystyle\mathcal{B}(\lambda k_{1},\lambda k_{2},\lambda k_{3})$

  $\displaystyle=\lambda^{-\left(6+\alpha_{\text{tot}}\right)}\mathcal{B}(k_{1},k_{2},k_{3})\,.$

  (1.2)

• •

  Rule 2: At tree level around quasi de Sitter spacetime, for fields with an arbitrary but constant speed of sound $c_{s}$, the trimmed bispectrum $\mathcal{B}$ must be a rational function of the norms $k_{1}$, $k_{2}$ and $k_{3}$ of the momenta¹¹1The exception are potential logarithms, which are discussed in Section 4.3.,

  $\displaystyle\mathcal{B}=\frac{\text{Poly}_{\beta}(k_{1},k_{2},k_{3})}{\text{Poly}_{6+\alpha_{\text{tot}}+\beta}(k_{1},k_{2},k_{3})}\,,$

  (1.3)

  for some non-negative integer $\beta$.

• •

  Rule 3: For any $n$-point correlator $B_{n}$, the residue of the total energy pole $k_{T}\equiv\sum_{a=1}^{n}k_{a}\to 0$ is related to a corresponding amplitude $A_{n}$ by [6, 28, 27]

  $\displaystyle\lim_{k_{T}\to 0}B_{n}=\frac{(-1)^{n}H^{p+n-1}(p-1)!}{2^{n-1}}\times\frac{\real\left(i^{1+n+p}A_{n}\right)}{\left(\prod_{a=1}^{n}k_{a}\right)^{2}k_{T}^{p}}\,.$

  (1.4)

  Here the order of the pole $p$ must be positive (see (2.26) for $p=0$) and is related to the mass dimensions $D_{\alpha}$ of the interactions responsible for the correlator by the expression

  $\displaystyle p=1+\sum_{\alpha}(D_{\alpha}-4)\,.$

  (1.5)

• •

  Rule 4: For the bispectrum of three identical fields, the trimmed bispectrum must be symmetric under permutations if the polarization factor is. Any symmetric polynomial can be written in a unique way in terms of sums and products of Elementary Symmetric Polynomials (ESP). Hence

  $\displaystyle\mathcal{B}_{XXX}=\frac{\text{Poly}_{\beta}(k_{T},e_{2},e_{3})}{\text{Poly}_{6+\alpha_{\text{tot}}+\beta}(k_{T},e_{2},e_{3})}\,,$

  (1.6)

  where

  $\displaystyle e_{1}$

  $\displaystyle\equiv\sum_{a=1}^{3}k_{a}=k_{T}\,,$

  $\displaystyle e_{2}$

  $\displaystyle\equiv\sum_{a<b}^{3}k_{a}k_{b}\,,$

  $\displaystyle e_{3}$

  $\displaystyle\equiv k_{1}k_{2}k_{3}\,.$

  (1.7)

• •

  Rule 5: Locality and the choice of the Bunch-Davies vacuum restrict the denominator of the trimmed bispectrum to take the following form

  $\displaystyle\mathcal{B}_{XYZ}=\frac{\text{Poly}_{3m+p-6-\alpha_{\text{tot}}}(k_{1},k_{2},k_{3})}{k_{T}^{p}e_{3}^{m}}\,.$

  (1.8)

  where $p$ is determined by (1.5) and in general $m\geq 3$. For the symmetry breaking pattern of the Effective Field Theory (EFT) of inflation [29, 30], one has $m=3$. We argue that a necessary condition for locality is simply

  $\displaystyle\lim_{q\to 0}\frac{B_{n}(\mathbf{q},\mathbf{k}_{1},\dots,\mathbf{k}_{n})}{P(q)}<\infty\,,$

  (1.9)

  where $B_{n}$ is an $n$-point correlator and $P(q)$ is the power spectrum of the soft field.

• •

  Rule 6: In single-clock cosmologies, correlators of curvature perturbations and gravitons are defined by the soft theorems they must obey. Explicit expressions for soft scalars and soft gravitons are given in (2.61) and (2.64), respectively.

Using these rules we were able to obtain the following results

• •

  The $\langle\gamma\gamma\gamma\rangle$ and $\langle\gamma\zeta\zeta\rangle$ bispectra in canonical slow-roll inflation follow straightforwardly from the Bootstrap Rules above (quadratic order in derivatives).

• •

  We re-derived the $\langle\zeta\zeta\zeta\rangle$ bispectrum in canonical single-field inflation, discussing how its associated amplitude breaks boosts at $\mathcal{O}(\epsilon)$. In two cases, we were able to fix all but a part of one free coefficient in the bootstrap Ansatz: (i) at finite $\epsilon$ and to leading order in slow roll and (ii) in the limit $\epsilon\to 0$ and to next-to-leading order (NLO) in slow roll. In the more general non-canonical model these are indeed free parameters.

• •

  We derived the $\langle\zeta\zeta\zeta\rangle$ bispectrum to any order in derivatives in the EFT of inflation, (4.44). As an example, we discuss explicitly the case of up to cubic order in derivatives, (4.49).

• •

  We re-derived the $\langle\zeta\gamma\gamma\rangle$ bispectrum in canonical slow-roll inflation. To this end we made use of an ad-hoc model of the flat-space amplitude.

• •

  We emphasize that the residue of the total-energy pole in $\langle\zeta\gamma\gamma\rangle$ and $\langle\zeta\zeta\zeta\rangle$ is a not manifestly local amplitude, and indeed has inverse powers of momenta. We trace this back to the existence of constrained fields in the Lagrangian descriptions, as required by gauge invariance. Ultimately, the presence of these not manifestly local terms is probably required by locality when coupling to massless spinning fields on curved spacetime.

By rotation invariance, all bispectra must be written in terms of contractions of the three momenta and the polarization tensors with the Kronecker delta $\delta_{ij}$. Since the polarization tensors must appear linearly, the most general form is

	$\displaystyle B$	$\displaystyle=\sum_{\text{contractions}}\left[\epsilon^{h_{1}}(\mathbf{k}_{1})\epsilon^{h_{2}}(\mathbf{k}_{2})\epsilon^{h_{3}}(\mathbf{k}_{3})\mathbf{k}_{1}^{\alpha_{1}}\mathbf{k}_{2}^{\alpha_{2}}\mathbf{k}_{3}^{\alpha_{3}}\right]\times\mathcal{B}$		(2.9)
		$\displaystyle=\sum_{\text{contractions}}\left(\text{polarization factor}\right)\times\left(\text{trimmed bispectrum}\right)\,,$		(2.10)

where the contractions of all spatial indices in the polarization factor (the square brackets in the first line) are implicit, the sum is over all relevant contractions with different non-negative integer powers $\alpha_{1,2,3}$ and the trimmed bispectrum $\mathcal{B}$ for each contraction is defined by this expression. For scalars $h_{1,2,3}=\alpha_{1,2,3}=0$ and so $\mathcal{B}$ coincides with the full bispectrum $B$. However this is not the case in the presence of spinning particles.

For a generic $n$-point correlation function, $\mathcal{B}$ must be invariant under rotations and translations and therefore can only depend on $3n-6$ independent, rotation-invariant combinations of the $n$ momenta, for $n\geq 3$. A useful simplification takes place for the bispectrum, $n=3$, for which the three variables can be chosen to be the norms of the momenta, $\{k_{1},k_{2},k_{3}\}$. Conversely, all higher-point correlators, with $n\geq 4$, depend also on some of the angles between different momenta. This feature of $n=3$ makes it particularly easy to impose Bose symmetry, as we will see in Rule 3. Scale invariance implies that $B$ must be a homogeneous function of $\{\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3}\}$ of degree $-6$. In models of inflation, this scaling generally receives small slow-roll corrections, typically of order $\epsilon$ and $\eta$, where

$\displaystyle\epsilon$

$\displaystyle\equiv-\frac{\dot{H}}{H^{2}}\,,$

$\displaystyle\eta$

$\displaystyle\equiv\frac{\dot{\epsilon}}{H\epsilon}\,.$

(2.11)

We will not attempt to recover these small corrections to the scaling exponents, but we will recover all bispectra in which the slow-roll parameters appear as overall prefactors. The reason is twofold. First, these corrections are very small in most models, and might be challenging to detecte observationally (see e.g. [45, 46, 47, 48, 49, 50, 51]), with some exceptions such as for example resonant non-Gaussianity [52, 53, 54, 55, 56, 57, 58, 59]. Second, in single-clock inflation, scale invariance can only be an exact symmetry if $n$-point correlators scale exactly as $k^{-3(n-1)}$. As shown in [21], this can be proven directly at the level of the algebra for any correlator. Alternatively for the bispectrum one can simply notice that, if we denote by $\Delta_{\zeta}$ the scaling dimension of $\zeta$, scale invariance combined with the soft theorems require

$\displaystyle\frac{1}{k^{6-3\Delta_{\zeta}}}\propto B_{\zeta\zeta\zeta}\overset{!}{=}P_{\zeta}^{2}\propto\left(\frac{1}{k^{3-2\Delta_{\zeta}}}\right)^{2}\,,$

(2.12)

for which the only solution is $\Delta_{\zeta}=0$.

Scale invariance can also be understood from a bulk perspective, i.e. considering the evolution of the fields as dictated by some bulk action. From this point of view, scale invariance is a consequence of the existence of an unbroken diagonal combination of time translations and some internal symmetry, both of which are individually broken (for detailed discussion see e.g. [41, 60, 61, 62, 63]). When combined with the assumption that the fields become constant in time in the asymptotic future, this symmetry implies the scale invariance discussed above.

Before concluding it is worth mentioning discrete symmetries. Since we will only be discussing neutral bosons and time is out of the picture, the only relevant discrete symmetry is parity (point inversion), $\mathbf{k}\to-\mathbf{k}$. For the bispectrum of scalars, invariance under parity follows from invariance under rotations. In particular, since all three-vectors in a bispectrum must lie on the same plane by momentum conservation, a rotation $U$ by $180^{\circ}$ around the axis perpendicular to the plane inverts the direction of all momenta, just as parity would:

	$\displaystyle\langle\phi(\mathbf{k}_{1})\phi(\mathbf{k}_{2})\phi(\mathbf{k}_{3})\rangle$	$\displaystyle=\langle UU^{-1}\phi(\mathbf{k}_{1})UU^{-1}\phi(\mathbf{k}_{2})UU^{-1}\phi(\mathbf{k}_{3})UU^{-1}\rangle$		(2.13)
		$\displaystyle=\langle\phi(U\mathbf{k}_{1})\phi(U\mathbf{k}_{2})\phi(U\mathbf{k}_{3})\rangle$		(2.14)
		$\displaystyle=\langle\phi(-\mathbf{k}_{1})\phi(-\mathbf{k}_{2})\phi(-\mathbf{k}_{3})\rangle\,,$		(2.15)

where we used (with abuse of notation) that $U\phi(\mathbf{k})U^{-1}=\phi(U\mathbf{k})$ and that the vacuum is invariant under rotations. The last expression is precisely the parity transformation of the first one, and so scalar bispectra must always be invariant under parity. In passing, notice that since $\phi(-\mathbf{k})=\phi^{\ast}(\mathbf{k})$ the above equation implies that the scalar bispectrum must be real.

Conversely, spinning particles invariance under parity does not follow from rotation invariance, and one can indeed have parity-violating bispectra. For example, for a graviton one finds the following transformations⁴⁴4These transformations follow from the definitions in (2.2) and the fact that the graviton has two spatial indices and therefore under parity $P\gamma_{ij}(\mathbf{k})P=\gamma_{ij}(-\mathbf{k})$, while under rotations one has the standard transformation $U\gamma_{ij}(\mathbf{k})U^{-1}=R_{ii^{\prime}}\gamma_{i^{\prime}j^{\prime}}(R\mathbf{k})(R^{-1})_{j^{\prime}j}$.

	$\displaystyle P:\gamma^{\pm}(\mathbf{k})\to P\gamma^{\pm}(\mathbf{k})P^{-1}$	$\displaystyle=\gamma^{\mp}(-\mathbf{k})$	$\displaystyle\text{(parity)}\,,$		(2.16)
	$\displaystyle U:\gamma^{\pm}(\mathbf{k})\to U\gamma^{\pm}(\mathbf{k})U^{-1}$	$\displaystyle=\gamma^{\pm}(-\mathbf{k})$	$\displaystyle\text{($180^{\circ}$ rotation)}\,,$		(2.17)

under parity or a rotation of $180^{\circ}$ of the plane of the momenta, respectively. All the explicit examples in this work will be invariant under parity.

Summarizing, Rule 1 can be stated at the level of the trimmed bispectrum $\mathcal{B}$ (defined in (2.9))

	$\displaystyle\mathcal{B}$	$\displaystyle=\mathcal{B}(k_{1},k_{2},k_{3})\,,$		(2.18)
	$\displaystyle\mathcal{B}(\lambda k_{1},\lambda k_{2},\lambda k_{3})$	$\displaystyle=\lambda^{-\left(6+\alpha_{\text{tot}}\right)}\mathcal{B}(k_{1},k_{2},k_{3})\,,$		(2.19)

where $\alpha_{\text{tot}}\equiv\alpha_{1}+\alpha_{2}+\alpha_{3}\geq 0$. For example, for three scalars $\alpha_{\text{tot}}=0$.

The bispectrum must be a rational function of rotationally invariant contractions of the momenta $\mathbf{k}_{a}$ and of the possible polarization tensors $\epsilon^{s_{b}}$ with helicity $s_{b}=\pm|s_{b}|$ for $a,b=1,2,3$. Given the definition in (2.9), this means that the trimmed bispectrum $\mathcal{B}$ is simply a rational function of the three norms of the momenta,

$\displaystyle\mathcal{B}=\frac{\text{Poly}_{\beta}(k_{1},k_{2},k_{3})}{\text{Poly}_{6+\alpha_{\text{tot}}+\beta}(k_{1},k_{2},k_{3})}\,,$

(2.20)

where numerator and denominator are two polynomials in the norm of the momenta of degree $\beta$ and $6+\alpha_{\text{tot}}+\beta$, respectively. Rule 2 is not a consequence of Rule 1, which allows for arbitrary functions of scale-invariant combinations such as $k_{a}/k_{b}$. Rather, Rule 2 is tantamount to assuming that (i) the bulk evolution is described by a weakly coupled, local effective field theory (EFT), (ii) the free theory is well-approximated by the de Sitter mode functions for a massless fields with arbitrary but constant speed of sound $c_{s}$, up to an arbitrary normalization⁵⁵5The reference normalization we have choosen corresponds to a Lagrangian $\frac{1}{2}\left[c_{s}^{-2}\dot{\phi}^{2}-(\partial_{i}\phi)^{2}\right]$.,

$\displaystyle X^{h}(\mathbf{k},\tau)\propto\frac{H}{\sqrt{2c_{s}k^{3}}}\left[(1+ic_{s}k\tau)e^{-ic_{s}k\tau}a_{\mathbf{k}}+(1-ic_{s}k\tau)e^{+ic_{s}k\tau}a_{-\mathbf{k}}^{\dagger}\right]\,.$

(2.21)

Note that when $c_{s}\neq 1$, these mode functions are not invariant under de Sitter boosts. Intuitively this is evident in the flat-space limit where a Lorentz boost leaves the light cone invariant, but in general displaces the sound cone, unless they coincide as happens for $c_{s}=1$. When the mode functions deviate appreciably from the de Sitter mode functions, such as for example when the slow-roll approximation breaks down or the speed of sound has a strong time dependence, Rule 2 is generally violated (see e.g. the terms subleading in slow-roll in [64, 65]).

As the saying goes, there’s an exception to every rule. The exception to this rule is a possible logarithm of the sum of the norms, arising even when the mode functions are precisely those in (2.21). This term features several peculiarities. First, it must be symmetric under the full group of de Sitter isometries and therefore results from the triple-$K$ expression [66] for conformally invariant three-point functions, as shown in [13]. Second, this term is not exactly scale invariant. Instead, its variation produces a purely local term, in a way that is analogous to a conformal anomaly in a CFT. In the context of $\zeta$ correlators in single-field inflation, the existence of this term can be thought of as a consequence of Maldacena’s consistency relation when accounting for the small slow-roll deviation from the scaling $k^{-6}$ in the bispectrum [13]. Finally, it’s worth mentioning that unitarity dictates that logarithmic terms are related to the imaginary part of the corresponding wavefunction coefficient [27]. We will discuss these logarithmic terms in Section 4.3.

In the limit in which the sum of the norms of the momenta vanishes⁶⁶6When the fields in the correlators have different speeds of sound, the pole appears at $\sum_{a}c_{s}^{(a)}k_{a}\to 0$. In this work, we will only consider cases where $c_{s}$ is the same for all fields and so this reduces to the stated condition.,

$\displaystyle k_{T}\equiv k_{1}+k_{2}+k_{3}\to 0\,,$

(2.22)

any $n$-point correlator $B_{n}$ is fixed by the UV-limit of a corresponding amplitude (the precise coefficient was derived in [27])

$\displaystyle\lim_{k_{T}\to 0}B_{n}=\frac{(-1)^{n}H^{p+n-1}(p-1)!}{2^{n-1}}\times\frac{\real\left(i^{1+n+p}A_{n}\right)}{\left(\prod_{a=1}^{n}k_{a}\right)^{2}k_{T}^{p}}\,,$

(2.23)

for some interaction-dependent positive integer exponent $p$ to be discussed in the following, and where we assumed canonically normalized fields for concreteness. For the bispectrum, $n=3$, this reduces to

$\displaystyle\lim_{k_{T}\to 0}B_{3}=-\frac{H^{p+2}(p-1)!}{4}\times\frac{\real\left(i^{p}A_{3}\right)}{\left(k_{1}k_{2}k_{3}\right)^{2}k_{T}^{p}}\,,$

(2.24)

which holds for $p$ a positive integer. In the limiting case $p=0$, the correlator evaluated at time $\tau_{0}$ possesses in general an IR logarithmic divergence⁷⁷7For correlators of $\zeta$ this divergence is absent at tree-level and the $\log{\tau_{0}}$ factor is substituted by some reference time around the time when modes exit the Hubble radius. as $\tau_{0}\to 0^{-}$. In this case, the same formula (2.23) holds with the substitution⁸⁸8Note that it is only the coefficient of the $\log\left(-\tau_{0}k_{T}\right)$ term that can be fixed by the amplitude, but not the coefficient of $k_{T}^{0}$. This is because the amplitude is invariant under field redefinitions while the term $k_{T}^{0}$ is not. [27]

$\displaystyle\frac{(p-1)!}{\left(-ik_{T}\right)^{p}}\to\log\left(-\tau_{0}k_{T}\right)\,,$

(2.25)

and so

$\displaystyle\lim_{k_{T}\rightarrow 0}B_{3}=-\frac{H^{2}}{(k_{1}k_{2}k_{3})^{2}}\log(-\tau_{0}k_{T})\real A_{3}^{\prime}\,.$

(2.26)

These stated results deserves some explanation. A relation between cosmological correlators and amplitudes was first observed in [6, 28] and then discussed in [23, 14, 25]. A careful proof in the in-in formalism has recently been presented in [27], where several details of the exact form of the relation were derived that had not previously appeared in the literature. Following⁹⁹9Notice that (2.23) is more naturally written in terms of the coefficients of the wavefunction of the universe, rather than in terms of correlators. In this paper, we avoid discussing the wavefunction of the universe because we are mostly interested in the bispectrum and the relation to the cubic wavefunction coefficient $\psi_{3}$ is very simple $\displaystyle P(k)$ $\displaystyle=\frac{1}{2\text{Re }\psi_{2}^{\prime}(k,k)}\,,$ $\displaystyle B_{3}(\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3})$ $\displaystyle=\frac{2\text{Re}\,\psi_{3}^{\prime}(\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3})}{\prod_{a=1}^{3}2\text{Re }\psi_{2}^{\prime}(k_{a},k_{a})}\,.$ (2.27) [23], we will refer to $k_{T}=0$ as the “total-energy pole”. This nomenclature originates from thinking of $k=|\mathbf{k}|$ as the energy component of a hypothetical light-like four-vector $(k,\mathbf{k})$. From this perspective, $k$ can be thought of as the energy associated to the wavenumber $\mathbf{k}$. In flat spacetime, when computing amplitudes (in time-translation invariant theories), we always find an energy-conserving delta function that forces the sum of all energies to vanish, and arises from a time integral of the form

$\displaystyle\int_{-\infty}^{+\infty}dt\prod_{a}e^{iE_{a}t}=2\pi\,\delta_{D}\left(E_{T}\right)\,,$

(2.28)

where $E_{T}=E_{1}+E_{2}+\dots$. Conversely, cosmological correlators do not conserve energy and are non-vanishing for generic positive values of $k_{T}$. The reason for this is twofold. First, in cosmology we are interested in expanding spacetimes, for which time translations are spontaneously broken. This is evident for example in the form of the mode functions (2.21), which are different from the familiar Minkowski counterpart, $e^{ikt}$. Second, unlike amplitudes, correlators are obtained from the product of fields at some equal time, which also breaks time translations. In fact, even in Minkowski spacetime, equal-time correlators do not feature any energy conserving delta function. Instead the delta function is replaced by a pole in the total energy. For Minkowski correlators computed in the usual interaction-picture formalism, this arises from an integral over the time of the interaction, of the form

$\displaystyle\int_{-\infty}^{0}dt\prod_{a}e^{iE_{a}t}=\frac{1}{E_{T}}\,,$

(2.29)

where we chose to evaluate the correlator at $t=0$ and we left implicit the rotation of the integral into the complex plane that projects onto the vacuum at early times and guarantees convergence. This pole at $E_{T}$ can also be understood in a different way starting from the Lippmann-Schwinger equation, a.k.a. using old-fashioned perturbation theory (see e.g. [67]). It arises when inverting the time-independent Schrödinger equation to obtain the formal solution

$$\ket{\psi}=\ket{\phi}+\frac{1}{E_{0}-H_{0}\pm i\varepsilon}H_{int}\ket{\psi}\,.$$

(2.30)

Using this expression to compute correlators in Minkowski and setting the reference energy to zero, $E_{0}=0$, we find that $H_{0}$ acts on energy eigenstates to give a total energy pole $1/E_{T}$. In cosmology, a total energy pole also arises from an integral over the conformal time $\tau$ of the interaction, but now the order of the pole can be higher than one due to the additional powers of $\tau$ coming from the time dependent interactions,

$\displaystyle\int_{-\infty}^{0}d\tau\tau^{p-1}\prod_{a}e^{ik_{a}\tau}=\frac{1}{k_{T}^{p}}\,,$

(2.31)

where again we evaluated the correlator at $\tau=0$ and left the $i\epsilon$ prescription implicit. The special case $p=0$ gives the logarithmic terms we discussed in Rule 2. It would be nice to find an alternative derivation of this higher-order pole that, as in the Lipmann-Schwinger discussion for Minkowski, does not involve any time integral.

Using dimensional analysis and scale invariance, we can derive a useful expression for the order $p$ of the pole for any $n$-point correlator $B_{n}$, defined analogously to the bispectrum $B$ in (2.8). By dimensional analysis, in $(3+1)$-spacetime dimensions, an $n$-particle amplitude and an $n$-point correlator have mass dimension

$\displaystyle[A_{n}]$

$\displaystyle=4-n\,,$

$\displaystyle[B_{n}]$

$\displaystyle=n-3(n-1)\,,$

(2.32)

respectively, where we assumed that each of the fields in the correlator has mass dimension one, as it is the case for canonically normalized scalars and tensors. Notice that the amplitude is independent of the normalization of the fields, but the correlator is not. Now consider a set of interactions of dimension $D_{\alpha}$

$\displaystyle\mathcal{L}\supset\sum_{\alpha}\frac{\mathcal{O}_{\alpha}}{\Lambda_{\alpha}^{D_{\alpha}-4}}\,,$

(2.33)

where $\Lambda_{\alpha}$ determines the corresponding coupling constant. For an amplitude generated by $V$ vertices the scaling must be (at any loop order)

$\displaystyle A_{n}\sim\frac{E^{4-n+\sum_{\alpha}^{V}(D_{\alpha}-4)}}{\prod_{\alpha^{\prime}}^{V}\Lambda_{\alpha^{\prime}}^{D_{\alpha^{\prime}}}}\,,$

(2.34)

where $E$ collectively represents the energy or spatial momenta of the external particles. On the other hand, scale invariance fixes the scaling with $k$ of all (time-independent) correlators to be¹⁰¹⁰10The missing factors to make up the mass dimension in (2.32) are the coupling constants $\Lambda_{\alpha}$ and factors of the Hubble parameter $H$.

$\displaystyle B_{n}\sim\frac{1}{k^{3n-3}}\,.$

(2.35)

By using the momentum/energy dependence in (2.34) and (2.35) in the relation (2.23), one finds

$\displaystyle E^{4-n+\sum_{\alpha}^{V}D_{\alpha}}=\frac{k^{3-3n+p+2n}}{k_{T}^{p}}\quad\Rightarrow\quad p=1+\sum_{\alpha}(D_{\alpha}-4)\,.$

(2.36)

Specifying this general formula to tree-level contact diagrams we find

$\displaystyle p_{\alpha}=D_{\alpha}-3\quad\quad\text{(contact interaction)}\,,$

(2.37)

for any interaction of mass dimension $D_{\alpha}$. One can use some graph theory to rewrite this formula in an alternative way. First recall that for any graph with $\alpha=1,\dots,V$ vertices of valence $n_{\alpha}$ (i.e. the number legs coming out of each vertex), $I$ internal and $E$ external lines one has

$\displaystyle\sum_{\alpha}n_{\alpha}=2I+E\,.$

(2.38)

Then recall that the number of loops $L$ is the first Betti number of the graph and for connected graphs it is given by

$\displaystyle L=I-V+1\,.$

(2.39)

Using these relations in (2.36) we find the alternative expression

$\displaystyle p=4L+2E-3+\sum_{\alpha=1}^{V}(D_{\alpha}-2n_{\alpha})\,.$

(2.40)

In this paper we are interested in $p$ for the tree level bispectrum, for which (2.36) simply tells us that $p_{\alpha}$ is precisely the total number of time and space derivatives appearing in the interaction

$\displaystyle p$

$\displaystyle=\text{number of derivatives}$

$\displaystyle\text{(bispectrum, $n=3$)}\,.$

(2.41)

Summarizing, we have found that near the total energy pole, $k_{T}\to 0$, any correlator is fixed by a corresponding amplitude as in (2.23), with $p$ given in (2.36). In particular, this means that for contact interactions such as the tree-level bispectrum, a Laurent expansion in powers of $k_{T}$ corresponds directly to the EFT expansion in operators of higher and higher dimension! This observation will prove very powerful in bootstrapping the bispectrum.

Since we will consider only correlators of integer-spin fields, which must obey Bose statistics, the correlators must obey Bose symmetry when identical fields are considered. Let’s begin with the case in which all three fields in the bispectrum are the same scalar. Then, the bispectrum must be symmetric under any permutation of the norms of the momenta, $\{k_{1},k_{2},k_{3}\}$. While of course one can write a symmetric function just by starting with any non-symmetric term and summing over all permutations (“orbits”), it is very useful to make the symmetries of the problem manifest through our choice of variables¹¹¹¹11An analogous point for amplitudes has been recently made in [68, 69]. The author is thankful to W. Haddadin and especially to S. Melville for introducing him to this branch of mathematics and explaining to him some nifty results.. Fortunately, thanks to Rule 2 this is indeed possible.

Recall that by Rule 2 the trimmed bispectrum must be a rational function, so we can use a powerful result in commutative algebra to uniquely write down the polynomials in the numerator and denominator of (2.20). The fundamental theorem of symmetric polynomials says that any symmetric polynomial (for us on the field of real numbers) in a set of $n$ variables can be written uniquely as sum and products of the first $n$ elementary symmetric polynomials (ESPs)¹²¹²12In principle one could choose to use other families of symmetric polynomials, such as power-sum symmetric polynomials, but it turns out that the ESPs are the most convenient choice. and numerical factors. Our interest is on $n=3$, in which case the relevant ESPs are

	$\displaystyle e_{1}$	$\displaystyle\equiv k_{1}+k_{2}+k_{3}=k_{T}\,,$		(2.42)
	$\displaystyle e_{2}$	$\displaystyle\equiv k_{1}k_{2}+k_{2}k_{3}+k_{1}k_{3}\,,$		(2.43)
	$\displaystyle e_{3}$	$\displaystyle\equiv k_{1}k_{2}k_{3}\,.$		(2.44)

Notice that the first ESP is precisely the total energy, and so in the following we will write $k_{T}$ instead of $e_{1}$. We are not the first to use ESPs to write scalar bispectra, and examples have already appeared in the literature, see e.g. [70, 71]. However, it is worth stressing a few additional points:

• •

  The fact that the decomposition in ESPs is unique is extremely useful to bootstrap the bispectrum, as we will see in the next section. If one tried to proceed by brute force writing down all possible monomials and summing over their permutations, one would end up with a very large number of parameters whose degeneracy cannot be constrained by any bootstrap rule.

• •

  ESPs are very convenient to study various limits of the bispectrum. As we discussed, the limit $e_{1}=k_{T}\to 0$, with $e_{2}$ and $e_{3}$ finite is fixed by a corresponding amplitude. Also, we will see later that the squeezed limits, which are constrained by the very powerful soft theorems, can be studied naturally by taking $e_{3}\to 0$ while keeping $k_{T}$ and $e_{2}$ finite¹³¹³13This leads to the natural question of what is the physical interpretation of $e_{2}\to 0$ with $k_{T}$ and $e_{3}$ finite. Unfortunately, we don’t have an interesting answer at the moment..

• •

  The permutation group is particularly simple for the bispectrum, but is more complicated starting with the trispectrum because then some angles between different momenta must be kept as independent variables. Nevertheless, using results from the Hilbert series and the Hironaka decomposition one can find a complete and mutually independent set of basis polynomials, which will be presented elsewhere.

The discussion above generalizes straightforwardly to the case in which one of the three scalars is different from the other two. The situation in which spinning fields are present is instead more interesting. Let’s focus on three identical spinning fields, such as in the graviton bispectrum $B_{\gamma\gamma\gamma}$. Now the bispectrum is given by a sum of terms, each of which is the product of a polarization factor times the trimmed bispectrum $\mathcal{B}$ as in (2.9). When the polarization factor is permutation invariant by itself then so must be $\mathcal{B}$, which can hence again be written uniquely in terms of ESPs. This is the case for example for the bispectrum generated in GR [38]. However, it might also be the case that some polarization factors are not fully permutation invariant, in which case $\mathcal{B}$ is also not permutation invariant and cannot be written in terms of ESPs. This happens for example for some (higher-order) interaction induced by the coupling with the time-dependent inflation, as for the operator in $S_{\Lambda_{3}}$ in [72].

Summarizing, when the polarization factor in (2.9) is invariant under permutations, the trimmed bispectrum of three identical fields can be written in terms of elementary symmetric polynomials as

$\displaystyle\mathcal{B}_{XXX}=\frac{\text{Poly}_{\beta}(k_{T},e_{2},e_{3})}{\text{Poly}_{6+\alpha_{\text{tot}}+\beta}(k_{T},e_{2},e_{3})}\,,$

(2.45)

where $\beta$ is a non-negative integer and $\alpha_{\text{tot}}$ was defined below (2.19).

When combined into a single fraction, the denominator of any bispectrum can only be the monomial $k_{T}^{p}e_{3}^{m}$. This fixes the order of the numerator in terms of the integers $p$ (discussed in Rule 3) and $m$,

$\displaystyle\mathcal{B}_{XYZ}=\frac{\text{Poly}_{3m+p-6-\alpha_{\text{tot}}}(k_{1},k_{2},k_{3})}{k_{T}^{p}e_{3}^{m}}\,.$

(2.46)

This rule follows from locality and the choice of the Bunch-Davies vacuum. Under the most conservative notion of locality one should choose $m=3$, which indeed is what appears in almost all known models. However, under a more open-minded and yet precise definition of locality one should also consider $m>3$, which indeed appears in solid inflation [73], where $m=5$. It would be nice to study that possibility further, especially given that the soft theorems for solid inflation are already known [74, 75]. We leave this for the future. Let’s start discussing the choice of vacuum and move on later to tackle locality.

The rules outlined so far apply to any massless scalar or tensor fields on a (quasi) de Sitter background. However, one specific scalar and one specific tensor field are of particular interest: curvature perturbations on constant-energy hyper-surfaces $\zeta$ and graviton perturbations $\gamma_{ij}$. The former are the seed of the anisotropies in the cosmic microwave background and of the large scale structures in our universe and can therefore be measured from cosmological observations. Graviton perturbations have so far eluded observation but are the target of a major observational effort that will improve the current sensitivity by two orders of magnitude in the coming decades.

When working in the bulk within a certain class of models, we carefully define $\zeta$ and $\gamma_{ij}$ from the fields in the action and then compute their correlators. But from an exclusively boundary perspective, where we don’t assume an action to begin with, how can we know if a given correlator is the correlator of $\zeta$ or of $\gamma_{ij}$ as opposed to the correlator of some other scalar or tensor field? The answer to this question has been recently given in [21]. Within single-clock inflation and without any further assumptions about the particle content, a given correlator is a correlator of $\zeta$ (in some cosmological model) if and only if it obeys all the soft theorems [37, 34, 33, 35] that generalize Maldacena’s consistency relation [38]. This statement is also true for correlators of $\gamma_{ij}$ mutatis mutandis. Focusing on the bispectrum, when the momentum $\mathbf{k}_{l}$ of $\zeta$ is much smaller than the two other short momenta, the bispectrum must obey the soft theorem

$\displaystyle\lim_{k_{l}\to 0}\langle\zeta(\mathbf{k}_{l})X(\mathbf{k}_{s}-\mathbf{k}_{l}/2)Y(-\mathbf{k}_{s}-\mathbf{k}_{l}/2)\rangle^{\prime}=P_{\zeta}(k_{l})\frac{\partial}{\partial\log k_{s}}\left(k^{3}\langle X(\mathbf{k}_{s})Y(-\mathbf{k}_{s})\rangle^{\prime}\right)+\dots\,,$

(2.61)

where the dots stand for terms that are suppressed by at least a factor of $\left(k_{l}/k_{s}\right)^{2}$ [38, 80, 37, 34] and the correct normalization for the $\zeta$ power spectrum is the standard expression

$\displaystyle\langle\zeta(\mathbf{k})\zeta(-\mathbf{k})\rangle^{\prime}=P_{\zeta}(k)=\frac{H^{2}}{4\epsilon c_{s}M_{\mathrm{P}l}^{2}}\frac{1}{k^{3}}\,.$

(2.62)

There is also a soft theorem for soft gravitons, $\gamma_{ij}(\mathbf{k}_{l})$. The leading order (LO) term in the limit $k_{l}\to 0$ is fixed by diff invariance to be [38, 34]

$\displaystyle\langle\gamma^{h}(\mathbf{k}_{l})X(\mathbf{k}_{s}-\mathbf{k}_{l}/2)Y(-\mathbf{k}_{s}-\mathbf{k}_{l}/2)\rangle^{\prime}$

$\displaystyle\to-\frac{1}{2}P_{\gamma}(k_{l})\epsilon_{ij}^{h}(\mathbf{k}_{l})k_{s}^{i}\partial_{k_{s}^{j}}P_{XY}(k_{s})\left[1+\mathcal{O}\left(\frac{k_{l}}{k_{s}}\right)\right]\,.$

(2.63)

In general, diffeomorphisms cannot be used to fix the whole next-to-leading order (NLO) contribution. However, here we notice that the NLO must vanish upon averaging over the angle $\theta$ between $\mathbf{k}_{l}$ and $\mathbf{k}_{s}$, as a consequence of parity in $\mathbf{k}_{l}\to-\mathbf{k}_{l}$.