Calculating the EFT likelihood via saddle-point expansion

The large-scale structure (LSS) of the universe encodes valuable information about galaxy formation, structure growth, and the physics of dark matter and dark energy. To extract this information, we need to fit our theoretical predictions to the measurements from observations. Conventionally, the galaxies are regarded as the tracers of dark matter distribution, and their $n$-point correlation function is utilized to compare with data to provide constraints on cosmological parameters (see [1] for a comprehensive review).

In recent years, Bayesian forward modeling [2, 3, 4, 5, 6, 7] has emerged as another powerful approach for studying such problems, which aims to directly predict the present-day galaxy distribution given different initial conditions, galaxy bias models, and cosmological parameters, then iteratively compare the resulting galaxy distribution with observations until convergence. The key advantage of this method is that it fully exploits all available information without the need for summary statistics. Consequently, Bayesian forward modeling has been widely applied to field-level analyses of galaxy clustering [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18].

A necessary basis in applying the Bayesian method is to calculate the conditional likelihood $\mathcal{P}[\delta_{g}|\delta]$, which is the functional probability of observing the galaxy overdensity $\delta_{g}$ given the matter overdensity $\delta$ (see [19, 20] for more details). To achieve this, we need to factorize $\mathcal{P}[\delta_{g}|\delta]$ into two components, $\mathcal{P}[\delta]$ and $\mathcal{P}[\delta_{g},\delta]$, which are referred to as the matter likelihood and joint likelihood, respectively. Thanks to the recently developed path integral approach for the LSS [21, 19], both likelihoods can be naturally expressed as the functional Fourier transform of the partition functions $Z[\delta]$ and $Z[\delta_{g},\delta]$. Meanwhile, the effective field theory of the LSS (EFTofLSS) [22, 23, 21, 24, 25, 26, 27, 28, 29] postulates that the contribution of short-wavelength (small-scale) fluctuations to the long-wavelength (large-scale) universe can be considered as a sequence of non-ideal fluid effects which can be absorbed into the corresponding terms of the partition functions, such as the pressure perturbation and the viscosity. It is therefore important to establish a framework to derive the EFT likelihood (hereinafter referred to as likelihood) from a general partition function. However, for general $Z[\delta]$ and $Z[\delta_{g},\delta]$, it is usually unfeasible to evaluate $\mathcal{P}[\delta]$ and $\mathcal{P}[\delta_{g},\delta]$ through immediate calculation because of the complicated functional integral formula. In Ref. [19], a functional approach for computing the likelihood through the saddle-point approximation is proposed, yielding results that are nearly consistent with the traditional EFT estimate, except for additional higher-order terms [20]. Furthermore, these higher-order terms are identified as precise corrections to the likelihood expression.

The present paper aims to calculate the likelihood more precisely. Since the saddle-point contribution in the exponent provides a good approximation to the full likelihood, it is natural to extend this approach to higher orders around the saddle points, i.e. the saddle-point expansion. This semiclassical method was first proposed in quantum mechanics and quantum field theory to calculate the decay rates and leads to reliable results that are consistent with the quantum-mechanical estimates of tunneling rates [30, 31]. More recently, the saddle-point expansion method has also been applied to Yang-Mills theory, leading to the CP conservation in the strong interactions [32, 33]. In this paper, we expect to apply this method within the path-integral formulation for LSS to achieve a more accurate prediction of the conditional likelihood.

In the process of utilizing the saddle-point expansion, there is a notorious problem corresponding to the negative eigenvalues of the second derivative of the classical action at the saddle point. If we substitute the saddle-point solution into the integral, this mode will change the sign on the exponent of the path integral, thus making the integral ill-defined. The resolution to the problem is to apply the Picard-Lefschetz theory [34, 35], deforming the integral contour into the combination of the steepest descent contours which connect different saddle points, and terminate at the convergent regions of the integral. In this way, the imaginary part of different “sub-contours” (often referred to as the Lefschetz thimbles) will cancel each other out to give the real result [36]. In this paper, we will show that we also encounter negative modes, and even complex modes in the calculation of the likelihood, and we can make use of the steepest descent contour method to guarantee the result real. This is the kernel of our work because the result has the physical meaning of probability.

One of our main results is that the saddle-point expansion method used in the calculation of decay rates can be perfectly applied to the calculation of the likelihood. What we need to do is just rewriting the likelihood expression into the following form

$$\mathcal{P}[\delta_{g},\delta]=\int\mathcal{D}\boldsymbol{\phi}_{g}\,e^{-S_{g}[\boldsymbol{\phi}_{g}]}\,,$$

(1.1)

where $\boldsymbol{\phi}_{g}$ is the field that absorbs all the integral parameters and $S_{g}[\boldsymbol{\phi}_{g}]$ is the “modified action”. Then we can use the same arguments as in quantum mechanics and quantum field theory to solve the expression of the likelihood analytically. At the same time, a problem that is not the same as the situation in quantum field theory is that the action is often not real because its expression often contains the imaginary unit $i$. However, in Sec. 4 we will prove that although $S_{g}[\boldsymbol{\phi}_{g}]$ and the eigenvalues of $S_{g}^{\prime\prime}[\bar{\boldsymbol{\phi}}_{g}]$ may be complex, we can still guarantee the integral result given by the sum of different steepest descent contours is real.

Also, we will illustrate how to apply the saddle-point expansion method to compute the likelihood for a given $S_{g}[\boldsymbol{\phi}_{g}]$ by a concrete example, i.e. the EFT likelihood with Gaussian distribution. We will provide a detailed introduction to how to use the gradient flow equation to find the steepest descent contours in the field space. We emphasize here although we have only studied a specific model, our discussion of this method and the treatment of the negative (complex) modes do not depend on the expressions of the action. Our approach is general and can be applied to more complicated situations.

The organization of this paper is as follows. In Sec. 2, we review the saddle-point expansion method in quantum mechanics and quantum field theory to calculate the decay rates. The analogy between quantum field theory and LSS path-integral case will be drawn in Sec. 3. Sec. 4 contains an example of applying this method to calculate the likelihood. Then we will draw our conclusion and discuss future directions in Sec. 5. To make our work complete, we review how to calculate the contribution of the saddle points to the likelihood in Appendix A, and Appendix B contains some discussions on how to solve the eigenvalues of $S^{\prime\prime}_{g}[\bar{\boldsymbol{\phi}}_{g}]$.

Throughout this paper, the notion and conventions used in the path-integral approach of the LSS mainly come from [19].

In this section, we will review the application of saddle-point expansion in quantum mechanics and quantum field theory, inspired by the calculation of decay rates [30, 31, 36, 35]. We will see that the existence of the negative modes of the second derivative of the action at the saddle points compels us to complexify the integral path and deform the integral contour. We need to approximate the integral contour by the sum of the steepest descent contours connecting different saddle points, which is referred to as the combination of different Lefschetz thimbles in the Picard-Lefschetz theory [34, 37]. Our review contains only the most central parts of the saddle-point expansion method, while more detailed and rigorous reviews can be found in [36, 35].

We now begin to explore the possibility of implementing the saddle-point expansion method to the path integral approach to the LSS. Our starting point is the special relationship between the EFT likelihoods $\mathcal{P}[\delta]$, $\mathcal{P}[\delta_{g},\delta]$ and the partition functions $Z[J]$, $Z[J_{g},J]$. Building on the previous work [19], we would like to extend the discussion to second-order terms near the saddle points, with a focus on the treatment of the negative modes. We aim to provide a general approach for calculating the EFT likelihood, taking into account various effects, such as the galaxy stochasticity [21, 45] and the primordial non-Gaussianity [46].

Upon establishing the appropriate theoretical framework, it can be applied to specific models. In this section, we focus on the model presented in [19], which involves the computation of the conditional likelihood with Gaussian distribution. We then demonstrate the application of the steepest descent contour through concrete examples.

As outlined in Sec. 3.1, there are three stochastic terms for matter and galaxy, $P_{\epsilon_{g}}$, $P_{\epsilon_{g}\epsilon_{m}}$, $P_{\epsilon_{m}}$. In the case where only $P_{\epsilon_{g}}$ exists, the path integral can be evaluated exactly, yielding rigorous results. This example has been previously investigated in [20, 19]. Hence, we consider the more intricate case where both $P_{\epsilon_{g}}$ and $P_{\epsilon_{g}\epsilon_{m}}$ are non-vanishing, and demonstrate that the saddle-point expansion yields identical conditional likelihood.

We begin by examining the structure of the actions. After incorporating the aforementioned approximations, the actions now contain only terms up to the second order in $J$ and $J_{g}$, i.e. up to $\boldsymbol{\phi}^{2}$ and $\boldsymbol{\phi}_{g}^{2}$ order. The expression for the “matter action” is

$$S[\boldsymbol{\phi}]=\int_{\boldsymbol{k}}\boldsymbol{\phi}^{a}(\boldsymbol{k})\mathcal{J}^{a}(-\boldsymbol{k})+\frac{1}{2}\int_{\boldsymbol{k},\boldsymbol{k}^{\prime}}\mathcal{M}^{ab}(\boldsymbol{k},\boldsymbol{k}^{\prime})\boldsymbol{\phi}^{a}(\boldsymbol{k})\boldsymbol{\phi}^{b}(\boldsymbol{k}^{\prime})\,,$$

(4.1)

and for the “joint action”

$$S_{g}[\boldsymbol{\phi}_{g}]=\int_{\boldsymbol{k}}\boldsymbol{\phi}^{a}_{g}(\boldsymbol{k})\mathcal{J}^{a}_{g}(-\boldsymbol{k})+\frac{1}{2}\int_{\boldsymbol{k},\boldsymbol{k}^{\prime}}\mathcal{M}^{ab}_{g}(\boldsymbol{k},\boldsymbol{k}^{\prime})\boldsymbol{\phi}^{a}_{g}(\boldsymbol{k})\boldsymbol{\phi}^{b}_{g}(\boldsymbol{k}^{\prime})\,.$$

(4.2)

We can read off the formula of $\mathcal{M}^{ab}$ and $\mathcal{M}_{g}^{ab}$ from the expression of the likelihood, i.e. Eq. (3.5) and Eq. (3.6)

$$\mathcal{M}(\boldsymbol{k},\boldsymbol{k}^{\prime})=(2\pi)^{3}\delta_{D}^{(3)}(\boldsymbol{k}+\boldsymbol{k}^{\prime})\begin{pmatrix}0&iK_{1}(k)D_{1}\\ iK_{1}(k)D_{1}&P^{-1}_{\rm in}(k)\end{pmatrix}\,,$$

(4.3)

and

$${\cal M}_{g}(\boldsymbol{k},\boldsymbol{k}^{\prime})=(2\pi)^{3}\delta^{(3)}_{D}(\boldsymbol{k}+\boldsymbol{k}^{\prime})\begin{pmatrix}P_{\epsilon_{g}}(k)&P_{\epsilon_{g}\epsilon_{m}}(k)&iK_{g,1}(k)D_{1}\\ P_{\epsilon_{g}\epsilon_{m}}(k)&0&iK_{1}(k)D_{1}\\ iK_{g,1}(k)D_{1}&iK_{1}(k)D_{1}&P^{-1}_{\rm in}(k)\end{pmatrix}\,\,.$$

(4.4)

At zeroth order, we can take $K_{1}=1$ and $K_{g,1}=b_{1}$. We briefly review how to calculate the contribution of the saddle points, with the detailed results provided in Appendix A, and here we focus on the issues related to the negative modes.

We first derive the equations of motion for the matter and galaxy fields. By taking the functional derivatives with respect to the fields $\boldsymbol{\phi}$ and $\boldsymbol{\phi_{g}}$, we obtain

	$\displaystyle\frac{\delta S}{\delta\boldsymbol{\phi}}\bigg{|}_{\boldsymbol{\phi}=\bar{\boldsymbol{\phi}}}=\mathcal{J}^{a}+\mathcal{M}^{ab}\bar{\boldsymbol{\phi}}=0\,,$		(4.5)
	$\displaystyle\frac{\delta S_{g}}{\delta\boldsymbol{\phi}_{g}}\bigg{|}_{\boldsymbol{\phi}_{g}=\bar{\boldsymbol{\phi}}_{g}}=\mathcal{J}^{a}_{g}+\mathcal{M}^{ab}_{g}\bar{\boldsymbol{\phi}}_{g}=0\,.$		(4.6)

Note that due to the presence of the Dirac delta function in the expressions for $\mathcal{M}$ and $\mathcal{M}_{g}$, we need not consider the case when $\boldsymbol{k}\neq\boldsymbol{k}^{\prime}$. The expressions of the solutions of these two equations and $-S[\bar{\boldsymbol{\phi}}]$, $-S_{g}[\bar{\boldsymbol{\phi}}_{g}]$ can be found in Appendix A. We now consider the additional terms arising from the saddle-point expansion. For both the matter and galaxy case, $S^{\prime\prime}[\bar{\boldsymbol{\phi}}]$ and $S^{\prime\prime}_{g}[\bar{\boldsymbol{\phi}}_{g}]$ are simply the matrices corresponding to the second-order terms in the fields, $\mathcal{M}^{ab}$ and $\mathcal{M}^{ab}_{g}$. For the matter likelihood, the eigenvalues of $S^{\prime\prime}[\bar{\boldsymbol{\phi}}]$ can be simply calculated,

$$m_{1}=\frac{1}{2}(P_{\rm in}^{-1}-\sqrt{-4D_{1}^{2}+P_{\rm in}^{-2}})\,,\,\,\,\,\,\,m_{2}=\frac{1}{2}(P_{\rm in}^{-1}+\sqrt{-4D_{1}^{2}+P_{\rm in}^{-2}})\,.$$

(4.7)

It is evident that provided the two eigenvalues are real (the case of the complex eigenvalues will be addressed below), they must be positive because of the positivity of $P_{\rm in}$ and $D_{1}$, as shown by the equation $P_{\rm in}^{-1}-\sqrt{-4D_{1}^{2}+P_{\rm in}^{-2}}=\sqrt{P_{\rm in}^{-2}}-\sqrt{-4D_{1}^{2}+P_{\rm in}^{-2}}$. Therefore, for the matter likelihood, there are no issues related to the negative modes, and we can directly proceed with Eq. (3.11) to obtain

$$\mathcal{P}[\delta]\approx e^{-S[\bar{\boldsymbol{\phi}}]}({\rm det}S^{\prime\prime}[\bar{\boldsymbol{\phi}}])^{-1/2}=\sqrt{\pi}\,(\prod_{i}m_{i})^{-1/2}\,{\rm e}^{-S[\bar{\boldsymbol{\phi}}]}=\frac{\sqrt{\pi}}{2}D_{1}\,e^{-S[\bar{\boldsymbol{\phi}}]}.$$

(4.8)

It can be seen that after incorporating the second-order expansion of the saddle points, the resulting expression is equivalent to the original one with the addition of an extra factor, as anticipated. Analogously, the calculation of joint likelihood should be obtained in the same manner, however, we will encounter certain intractable problems now: For a general three-dimensional matrix $\mathcal{M}_{g}$, there are no simple analytical expressions for its eigenvalues. Even when we obtain them through complicated derivations, the solutions are usually not positive, and even not real because $\mathcal{M}_{g}$ is neither Hermitian nor anti-Hermitian. At this time, there are no shortcuts, and we need to follow the previous discussions regarding the negative modes. We need to complexify the field $\boldsymbol{\phi}_{g}$ and then seek the steepest descent contour that is homologous to the original contour. At the same time, there is no zero mode corresponding to $S^{\prime\prime}_{g}[\bar{\boldsymbol{\phi}}_{g}]$, so we do not need to be concerned about the issues related to the collective coordinates.

We will present a general method for solving this type of problem, and the following discussion is derived by analogy with [35]. We start from the complexified $\boldsymbol{\phi}$ space. We introduce the real parameter $u$ to label different paths passing through the saddle points in the complex plane, and impose the boundary condition $\boldsymbol{\phi}_{g}(\boldsymbol{k},u\rightarrow\infty)=\bar{\boldsymbol{\phi}}_{g}$, i.e. under this parametrization, in the limit of $u\rightarrow\infty$, we recover the saddle points. Since our goal is to identify the steepest descent contours, the holomorphic function we consider should be the negative value of the action $\mathcal{I}[\boldsymbol{\phi}_{g}]=-S_{g}[\boldsymbol{\phi}_{g}]$, and then define the Morse function $f[\boldsymbol{\phi}_{g}]={\rm Re}(\mathcal{I}[\boldsymbol{\phi}_{g}])$. Thus, for the saddle point $\boldsymbol{\phi}_{g}$, the steepest descent path is given by the gradient flow equation [34]

$$\frac{\partial\boldsymbol{\phi_{g}}(\boldsymbol{k},u)}{\partial u}=-\overline{\left(\frac{\delta\mathcal{I}[\boldsymbol{\phi}_{g}(\boldsymbol{k},u)]}{\delta\boldsymbol{\phi}_{g}(\boldsymbol{k},u)}\right)},\ \ \ \ {\frac{\partial\overline{\boldsymbol{\phi_{g}}(\boldsymbol{k},u)}}{\partial u}}=-\left(\frac{\delta\mathcal{I}[\boldsymbol{\phi}_{g}(\boldsymbol{k},u)]}{\delta\boldsymbol{\phi}_{g}(\boldsymbol{k},u)}\right),$$

(4.9)

where the overline represents the complex conjugate. We can easily check that

$$\frac{\partial f}{\partial u}=\frac{1}{2}(\frac{\delta\mathcal{I}}{\delta\boldsymbol{\phi}_{g}}\frac{\partial\boldsymbol{\phi}_{g}}{\partial u}+\frac{\delta\bar{\mathcal{I}}}{\delta\bar{\boldsymbol{\phi}}_{g}}\frac{\partial\bar{\boldsymbol{\phi}}_{g}}{\partial u})=-\bigg{|}\frac{\partial\boldsymbol{\phi}_{g}(\boldsymbol{k},u)}{\partial u}\bigg{|}^{2}\leq 0\,,$$

(4.10)

this implies that the real part of $\mathcal{I}[\boldsymbol{\phi}_{g}]$ is decreasing along the contour as it moves away from the saddle point, fulfilling the condition of the steepest descent contour. Upon plugging the expression for the action into Eq. (4.9), we can get

$$\frac{\partial\boldsymbol{\phi}_{g}(\boldsymbol{k},u)}{\partial u}=-{\mathcal{J}}^{*}_{g}(\boldsymbol{k})+{\mathcal{M}}^{*}_{g}{\boldsymbol{\phi}}^{*}_{g}\,.$$

(4.11)

This equation corresponds to the steepest descent contour in this context. Next, we decompose the field as $\boldsymbol{\phi}_{g}=\bar{\boldsymbol{\phi}}_{g}+\boldsymbol{\varphi}_{g}$, and apply the complex conjugate of the saddle-point equation: $\mathcal{J}^{*}_{g}+\mathcal{M}^{*}_{g}\bar{\boldsymbol{\phi}}_{g}^{*}=0$. Consequently, the equation above can be rewritten as

$$\frac{\partial\boldsymbol{\varphi}_{g}(\boldsymbol{k},u)}{\partial u}=\mathcal{M}^{*}_{g}\varphi^{*}_{g}\,.$$

(4.12)

We can employ the method of separation of variables to reduce this equation by making the following ansatz: $\boldsymbol{\phi}_{g}(\boldsymbol{k},u)=\sum_{n}g_{n}(u)\chi_{n}(\boldsymbol{k})$, where $g_{n}(u)\in\mathbb{R}$ and the subscript “$n$” distinguishes different directions. Substituting this ansatz into the above equation, we obtain the following result

$$\mathcal{M}_{g}^{*}\,g_{n}(u)\chi_{n}^{*}(\boldsymbol{k})=\chi_{n}(\boldsymbol{k})\frac{{\rm d}g_{n}(u)}{{\rm d}u}\,,$$

(4.13)

then we have

$$\mathcal{M}_{g}^{*}\,\chi_{n}^{*}(\boldsymbol{k})/\chi_{n}(\boldsymbol{k})=m_{n}=\frac{1}{g_{n}(u)}\frac{{\rm d}g_{n}(u)}{{\rm d}u}\,,$$

(4.14)

where we have introduced the real eigenvalue $m_{n}$, as all the terms in the RHS of the equation are real (a rigorous proof of this will be provided below). Thus the eigenequation can be expressed as

$$\mathcal{M}_{g}^{*}\,\chi_{n}^{*}(\boldsymbol{k})=m_{n}\,\chi_{n}(\boldsymbol{k})\,.$$

(4.15)

For a general three-dimensional matrix $\mathcal{M}_{g}^{*}$, obtaining analytical solutions for the eigenvalues is generally challenging. We can use numerical methods to approximate the eigenvalues after obtaining its explicit formula. Another issue occurs during the saddle-point expansion is that what we need to calculate is $S^{\prime\prime}[\bar{\boldsymbol{\phi}}_{g}]\boldsymbol{\varphi}_{g}\sim\mathcal{M}_{g}\boldsymbol{\varphi}_{g}$, therefore the relevant equation is not the one presented above, but rather its complex conjugate $\mathcal{M}_{g}\,\chi_{n}(\boldsymbol{k})=m_{n}\,\chi_{n}^{*}(\boldsymbol{k})$⁴⁴4In fact, for each eigenvalue $m_{n}$ there is always an accompanying eigenvalue $-m_{n}$, corresponding to the eigenstate ${i\,\chi_{n}(\boldsymbol{k})}$. However, as we will demonstrate in the following discussion, all $m_{n}$ are real and positive, allowing us to disregard all the $-m_{n}$.. Furthermore, in Eq. (4.15), the matrix $\mathcal{M}^{*}_{g}$ is neither Hermitian nor are the eigenfunctions real, which precludes the imposition of the standard normalization condition on the eigenstates. To work out this, we can construct a Hermitian operator by combining Eq. (4.15) with its complex conjugate

$$\begin{pmatrix}\boldsymbol{0}&\mathcal{M}^{*}_{g}\\ \mathcal{M}_{g}&\boldsymbol{0}\end{pmatrix}\begin{pmatrix}\chi_{n}(\boldsymbol{k})\\ {\chi}_{n}^{*}(\boldsymbol{k})\end{pmatrix}=m_{n}\begin{pmatrix}\chi_{n}(\boldsymbol{k})\\ {\chi}_{n}^{*}(\boldsymbol{k})\end{pmatrix}\,,$$

(4.16)

thus the operator on the left-hand side is now Hermitian, and the eigenvalues $m_{n}$ should be real. However, this equation remains intractable for an exact solution, and the steps for obtaining an approximate solution are provided in Appendix B. In this case, we can also impose the orthonormalization to the eigenstates

$$\int_{\boldsymbol{k}}{\chi}_{n}^{*}(\boldsymbol{k})\,\chi_{n}(\boldsymbol{k})=\delta_{\rm mn}\,.$$

(4.17)

On the other hand, for the equation on the RHS of Eq. (4.14), the solution is expected to take the form $g_{n}(u)\sim a_{n}\,{\rm exp}\,(m_{n}u)$, where $a_{n}\in\mathbb{R}$. Using the boundary condition $g(u\rightarrow\infty)=0$, it follows that $m_{n}>0$, i.e. the eigenvalues are all positive and real.

Now we can calculate the joint likelihood

	$\displaystyle\mathcal{P}[\delta_{g},\delta]$	$\displaystyle\approx\int_{C}\mathcal{D}\boldsymbol{\phi}_{g}\,e^{-S_{g}[\bar{\boldsymbol{\phi}}_{g}]-\frac{1}{2}\boldsymbol{\varphi}_{g}S^{\prime\prime}_{g}[\bar{\boldsymbol{\phi}}_{g}]\boldsymbol{\varphi}_{g}}$
		$\displaystyle=\int_{C}\mathcal{D}\boldsymbol{\varphi}_{g}\,e^{-S_{g}[\bar{\boldsymbol{\phi}}_{g}]-\boldsymbol{\varphi}_{g}M_{g}\boldsymbol{\varphi}_{g}}$
		$\displaystyle=\int\mathcal{D}\boldsymbol{\varphi}_{g}\,e^{-S_{g}[\bar{\boldsymbol{\phi}}_{g}]-\sum_{n}m_{n}|\boldsymbol{\varphi}_{g}|^{2}}$
		$\displaystyle=e^{-S_{g}[\bar{\boldsymbol{\phi}}_{g}]}\frac{\pi}{\sum_{n}m_{n}}\,$
		$\displaystyle\approx e^{-S_{g}[\bar{\boldsymbol{\phi}}_{g}]}\frac{\pi}{P_{\epsilon_{g}}-P_{\rm in}^{-1}},$		(4.18)

where from the second to the third line we have employed the eigenequation of $\mathcal{M}_{g}$, and in the subsequent line the Gaussian integral formula for complex variables has been applied. In the final step, we took advantage of the expression of the sum of $m_{n}$, i.e. Eq. (B.9).

We make some comments on the process and results of the calculation as follows:

• •

  It has been demonstrated that the saddle-point expansion method in false vacuum decay theory can be seamlessly applied to the path-integral approach of LSS after some appropriate modifications and re-derivations. Although we have met the negative modes, and even complex modes of $S^{\prime\prime}_{g}[\bar{\boldsymbol{\phi}}_{g}]$, because all the eigenvalues of the gradient flow eigenequations Eq. (4.14) are real and positive ($m_{n}\in\mathbb{R},m_{n}>0$), the result is ensured to be real through the path integral, consistent with the argument as presented in Sec. 2.1. This aspect of the result is very crucial, as it directly relates to its physical interpretation as a probability.

• •

  Although we have examined only one specific model, the rationale for applying the gradient flow equation to determine the steepest descent contour, as we discussed below, is general and can be applied to any kind of more complicated models. After this, we can compute the arbitrary likelihood of the form $\int\mathcal{D}\phi\,{\rm e}^{-S[\phi]}$ up to the second-order near the saddle points. This approach could be instrumental in providing more accurate predictions for the large-scale structure of the universe.

Next, we define the logarithm of the two likelihoods (matter and joint) as $\wp[\delta],\wp[\delta_{g},\delta]$, thus we have

	$\displaystyle\wp[\delta]$	$\displaystyle={\rm ln}\,\mathcal{P}[\delta]=-{\rm ln}\,\left(\frac{\sqrt{\pi}D_{1}}{2}\right)S[\bar{\boldsymbol{\phi}}]\,,$		(4.19)
	$\displaystyle\wp[\delta_{g},\delta]$	$\displaystyle={\rm ln}\,\mathcal{P}[\delta_{g},\delta]=-{\rm ln}\left(\frac{\pi}{\sum_{n}m_{n}}\right)S_{g}[\bar{\boldsymbol{\phi}}_{g}]\approx-{\rm ln}\left(\frac{\pi}{P_{\epsilon_{g}}-P_{\rm in}^{-1}}\right)S_{g}[\bar{\boldsymbol{\phi}}_{g}]\,.$		(4.20)

Since neither the matrices $\mathcal{M}$ nor $\mathcal{M}_{g}$ depend on the overdensity fields $\delta$ and $\delta_{g}$, the eigenvalues will be field-independent. We can also see this in the expression of $m_{n}$ in Appendix B, where we provide a set of approximate solutions for them. Thus the logarithmic terms in Eq. (4.19) and Eq.(4.20) will be absorbed into the normalization factors. Consequently, when defining the conditional likelihood of them, the terms appearing in the conditional likelihood will only include the saddle-point contribution of the two actions, which is

$$\wp[\delta_{g}|\delta]={\rm ln}\,\mathcal{P}[\delta_{g}|\delta]={\rm ln}\,\frac{\mathcal{P}[\delta_{g},\delta]}{\mathcal{P}[\delta]}=-S[\bar{\boldsymbol{\phi}}]+S_{g}[\bar{\boldsymbol{\phi}}_{g}]\,.$$

(4.21)

Upon substituting the explicit expressions for $S[\bar{\boldsymbol{\phi}}]$ and $S_{g}[\bar{\boldsymbol{\phi}}_{g}]$, it becomes evident that our result is consistent with those presented in [19]. One notable observation is that the result following the saddle-point expansion appears to contribute only a factor that can be absorbed into the normalization factor, i.e. ${\rm e}^{-S[\bar{\phi}]}\int\mathcal{D}\varphi\,{\rm e}^{-\frac{1}{2}S^{\prime\prime}[\bar{\phi}]\varphi^{2}}\sim\mathcal{N}{\rm e}^{-S[\bar{\phi}]}$. However, this occurs solely because in this model $S^{\prime\prime}[\bar{\boldsymbol{\phi}}]$ does not exhibit dependence on $\delta_{g}$ and $\delta$. Suppose now we compute a more complex model, for instance, one where the action depends on $\boldsymbol{\phi}^{3}$. In this case the saddle-point solutions for the actions $\bar{\boldsymbol{\phi}}$ and $\bar{\boldsymbol{\phi}}_{g}$ will depend on $\delta_{g}$ and $\delta$, and $S^{\prime\prime}[\bar{\boldsymbol{\phi}}]$ will also exhibit dependence on these variables. In this way, the result of the saddle-point expansion introduces a factor that cannot be absorbed by normalization, and this factor will contribute to the final conditional likelihood. More complicated models will be left for future work to explore.

The path-integral approach of the EFTofLSS offers a novel method for encapsulating cosmological effects into a compact analytical framework, enabling the prediction of cosmological observables. In this work, we aim to apply the saddle-point expansion technique, traditionally employed in quantum field theory for calculating the decay rates, to this context to improve the precision of EFT likelihood calculations. As a result, after appropriately reformulating the likelihood expressions, this task reduces to computing the path integral over the field $\boldsymbol{\phi}$ in three-dimensional Euclidean space. Consequently, by applying the same reasoning as in quantum field theory, we can calculate the likelihood, with the $\phi$ space replaced by the $\boldsymbol{\phi}$ space.

A key challenge in the saddle-point expansion method is the treatment of the negative eigenvalues of the second derivative of the action at the saddle point. In this work, we present a systematic discussion of this aspect in three distinct cases. As an illustrative example, we compute the matter and joint likelihood under the assumption of Gaussian distribution, retaining terms up to $J^{2}$ and $J_{g}^{2}$ order. For the matter likelihood, since there are no negative eigenvalues associated with $S^{\prime\prime}[\boldsymbol{\phi}]$, we can directly apply the Gaussian integral formula to obtain the result. However, when calculating the joint likelihood, the eigenvalues of $S^{\prime\prime}[\boldsymbol{\phi}_{g}]$ are often non-positive and even complex, necessitating an approximation of the original integral contour with a sum of the steepest descent contours in the field space. The treatment of these complex modes goes beyond the scope of the false vacuum decay theory discussion, primarily because the second derivative of the action in the path-integral approach of LSS often takes a specific form, resulting in a general non-Hermitian three-dimensional matrix. Even so, we still demonstrate in Sec. 4 that for this case, the Picard-Lefschetz theory remains applicable in this context by complexifying the action and utilizing the gradient flow equation in the field space. We emphasize here again that although the integrals along each contour may yield complex results, their collective interference leads to a real final result.

Another point is that, as shown in Eq. (4.8) and Eq. (4.18), the result of the saddle-point expansion is essentially equivalent to the saddle-point approximation, differing only by a multiplicative factor. In the example provided in Sec. 4, this factor is independent of $\delta_{g}$ and $\delta$, therefore it will be absorbed into the normalization constant and resulting in the same outcome as in [19], as illustrated in Eq. (4.21). However, this does not imply that the result of the saddle-point expansion is trivial, but rather that the action we have chosen in this example is truncated at $\boldsymbol{\phi}^{2}$ order. Furthermore, the method we present in this paper is general, and can be used to cope with more complicated situations.

Several unresolved issues remain in this work. For instance, we have not obtained an exact analytic solution of Eq.(4.16). The reason is that although the combination of the matrix $\mathcal{M}_{g}$ with its complex conjugate $\mathcal{M}_{g}^{*}$ is Hermitian, as illustrated in Eq. (4.16), solving for this eigenequation is equivalent to solving a cubic equation, which does not have a simple analytic solution. Another important aspect concerns the treatment of higher-order terms. In the theory of false vacuum decay, the Lagrangian is typically truncated at the $\phi^{4}$ order due to the non-renormalizable nature of higher-order terms, which are strongly suppressed by the energy scale of new physics, $\mu$. This raises the question of whether a similar suppression applies to higher-order terms in our case. In the literature [47] and [45], the authors computed the renormalization group equations for the partition function up to $J^{2}$ order and arbitrary $J^{n}$ order ($n\rightarrow\infty$). They found that, given the same initial conditions, the running of the bias parameter $b$ exhibits similar behavior. Does this imply when calculating the EFT likelihood, we can neglect the terms higher than $J^{2}$ (analogous to the non-renormalizable terms in quantum field theory)? In the literature [45] the authors perform a dimension analysis on the terms of all orders of $J$, however, do not draw an analogy to the case of quantum field theory. Then for the higher-order terms of $J$, are they also suppressed by the hard momentum cutoff $\Lambda$ introduced during the regularization process? e.g. consider the term $\int_{\boldsymbol{x}}J^{2}(\boldsymbol{x})O[\delta_{\Lambda}(\boldsymbol{x})]$, should we need to express this as $\frac{C_{*}}{\Lambda^{3+d_{O}}}\int_{\boldsymbol{x}}J^{2}(\boldsymbol{x})O[\delta_{\Lambda}(\boldsymbol{x})]$ such that this term will be suppressed by $\Lambda$, where $d_{O}$ is the dimension of the arbitrary operator $O$ and $C_{*}$ is the dimensionless “coupling constant”? If similar reasoning as in quantum field theory can be applied, then these higher-order terms would contribute little to the partition function. Unfortunately, we have not found relevant discussions in any literature, so we refrain from drawing a definitive conclusion at this stage. Additionally, our calculation assumes a Gaussian initial condition. If the primordial probability distribution deviates from the Gaussian form, then the effective action will include an additional interaction term [46]. For the standard approach [52, 53], this term introduces a cubic interaction term in the field $\boldsymbol{\phi}$ when calculating the likelihood. The saddle-point expansion method would then require a careful treatment of this term. We leave a more detailed investigation of this issue to future work.

JYK thanks Ying-Zhuo Li for his useful discussions. We acknowledge the support by the National Science Foundation of China (No. 12147217, 12347163), the China Postdoctoral Science Foundation (No. 2024M761110), and the Natural Science Foundation of Jilin Province, China (No. 20180101228JC).