With the upcoming precision measurements of the large scale structure, accurate theoretical modeling is essential to interpret the observational data. It requires a huge number of mock catalogs and \(N\)-body simulations are too numerically expensive to be done. Fortunately, it seems that observable quantities at the quasi-linear scales might be accurately modeled semi-analytically. The Lagrangian perturbation theory (LPT) has been widely used to investigate this [2–6]. Also, the initial condition for the N-body simulation are generated using LPT [7–9].

In LPT, the fundamental object is the Lagrangian displacement vector \(\mathbf {S}\), which displaces the particle from its initial position \(\mathbf {q}\) to the final Eulerian position \(\mathbf {x}\),

$$\begin{aligned} \mathbf {x}(\mathbf {q},t) = \mathbf {q}+ \mathbf {S}(\mathbf {q},t) . \end{aligned}$$
(1)

The first order LPT solution is the Zel’dovich approximation [10] and higher order solutions have been obtained [11–15]. From the mass conservation, the matter density perturbation \(\delta \) can be described by a function of \(\mathbf {S}\),

$$\begin{aligned} \delta (\mathbf {x},t) = \int d^3 q \delta _{{D}}\Bigl (\mathbf {x}- \mathbf {q}- \mathbf {S}(\mathbf {q},t)\Bigr ) - 1 . \end{aligned}$$
(2)

One can expand the displacement vector \(\mathbf {S}\) according to the Lagrangian perturbative prescription

$$\begin{aligned} \mathbf {S}(\mathbf {q},t)&\equiv \sum _{n=1} \mathbf {S}^{(n)}(\mathbf {q},t) = \sum _{n=1} D_{(n)}(t) \mathbf {S}^{(n)}(\mathbf {q})\nonumber \\&= \sum _{n=1} I_{n}(t) D^{n}(t) \mathbf {S}^{(n)}(\mathbf {q}) \nonumber \\&\equiv D(t) \mathbf {S}^{(1)}(\mathbf {q}) + E(t) \mathbf {S}^{(2)}(\mathbf {q}) \nonumber \\&+ F_{a}(t) \mathbf {S}^{(3a)}(\mathbf {q}) + F_{b}(t) \mathbf {S}^{(3b)}(\mathbf {q}) + \cdots . \end{aligned}$$
(3)

This explicit separation with respect to the spatial and temporal coordinates for each order (i.e. \(I_{n}\) is a constant) is known to be a property of the perturbative Lagrangian description for an Einstein–de Sitter (EdS) universe [13]. However, the solution at each order can be a separable function of \(t\) and \(\mathbf {q}\) even for general dark energy models by using \(D_{(n)}(t)\) instead of \(D^{(n)}(t)\). After one includes the time dependence of \(I_{n}\) in the each kernel, one can find the exact solution for each order. One can use \(I_{1} = 1\), \(D_{1} = D\) where \(D_1(t)\) is the linear growth factor, and the \(D_{(n)}(t) = I_{n}(t)D^{n}(t)\) are specified as

$$\begin{aligned} D_{(2)}(t)&\equiv E(t) = I_{2}(t) D^{2}(t) , \end{aligned}$$
(4)
$$\begin{aligned} D_{(3a)}(t)&\equiv F_{a}(t) = I_{3a}(t) D^{3}(t) , \end{aligned}$$
(5)
$$\begin{aligned} D_{(3b)}(t)&\equiv F_{b}(t) = I_{3b}(t) D^{3}(t) . \end{aligned}$$
(6)

From (4)–(6), one can obtain the Lagrangian Poisson equation order by order (from the linear to the irrotational third orders),

$$\begin{aligned} \ddot{D} + 2H \dot{D} - 4 \pi G \rho _{m} D&= 0, \end{aligned}$$
(7)
$$\begin{aligned} \ddot{E} + 2H \dot{E} - 4 \pi G \rho _{m} E&= - 4 \pi G \rho _{m} D^2, \nonumber \\&\mathrm{if} \; \mu _{1}(\mathbf {S}^{(2)} ) = \mu _{2}(\mathbf {S}^{(1)},\mathbf {S}^{(1)}) , \end{aligned}$$
(8)
$$\begin{aligned} \ddot{F}_{a} + 2H \dot{F}_{a} - 4 \pi G \rho _{m} F_{a}&= - 8 \pi G \rho _{m} D^3, \nonumber \\&\mathrm{if} \; \mu _{1}(\mathbf {S}^{(3a)} ) = \mu _{3}(\mathbf {S}^{(1)}) , \end{aligned}$$
(9)
$$\begin{aligned} \ddot{F}_{b} + 2H \dot{F}_{b} - 4 \pi G \rho _{m} F_{b}&= - 8 \pi G \rho _{m} D ( E - D^2 ),\nonumber \\&\mathrm{if} \; \mu _{1}(\mathbf {S}^{(3b)}) = \mu _{2}(\mathbf {S}^{(1)},\mathbf {S}^{(2)}) , \nonumber \\ \end{aligned}$$
(10)

where dots represent the derivatives with respect to the cosmic time \(t\) and \(\mu _{2}(S^{(1)},S^{(2)}) = \mu _{2}(S^{(2)},S^{(1)})\) is satisfied for any tensor [14]. \(\mu _{a}(\mathbf {S}^{(n)})\) are defined as

$$\begin{aligned}&\mu _{1}(\mathbf {S}^{(n)}) \equiv S_{ii}^{(n)} , \end{aligned}$$
(11)
$$\begin{aligned}&\mu _{2}(\mathbf {S}^{(n)},\mathbf {S}^{(m)}) \equiv \frac{1}{2} \Bigl ( S_{ii}^{(n)} S_{jj}^{(m)} - S_{ij}^{(n)} S_{ji}^{(m)} \Bigr ) , \end{aligned}$$
(12)
$$\begin{aligned}&\mu _{3}(\mathbf {S}^{(n)}) \equiv \mathrm{det} S_{ij}^{(n)} . \end{aligned}$$
(13)

One can rewrite Eq. (3) in Fourier space, represented by using the linear matter density contrast, \(\widetilde{\delta }_{L}(p)\),

$$\begin{aligned} \widetilde{\mathbf {S}}^{(n)}(\mathbf {k},t)&= - i D_{(n)}(t) \int \frac{d^3 p_1}{(2 \pi )^3} \ldots \frac{d^3 p_n}{(2 \pi )^3} (2\pi )^3 \delta _{{D}}\nonumber \\&\times (\mathbf {p}_{1 \ldots n} - \mathbf {k}) \mathbf {F}^{(n)}(\mathbf {p}_1, \ldots , \mathbf {p}_n) \widetilde{\delta }_{L} (p_1) \ldots \widetilde{\delta }_{L} (p_n) \nonumber \\&= - i \frac{D^{n}(t)}{n !} \int \frac{d^3 p_1}{(2 \pi )^3} \ldots \frac{d^3 p_n}{(2 \pi )^3} (2\pi )^3 \delta _{{D}}\nonumber \\&\times (\mathbf {p}_{1 \ldots n} - \mathbf {k}) n ! I_{n}(t) \mathbf {F}^{(n)}(\mathbf {p}_1, \ldots , \mathbf {p}_n) \widetilde{\delta }_{L} (p_1) \ldots \widetilde{\delta }_{L} (p_n) \nonumber \\&\equiv - i \frac{D^{n}(t)}{n !} \int \frac{d^3 p_1}{(2 \pi )^3} \ldots \frac{d^3 p_n}{(2 \pi )^3} (2\pi )^3 \delta _{{D}}\nonumber \\&\times (\mathbf {p}_{1 \ldots n} - \mathbf {k}) \L ^{(n)} (t, \mathbf {p}_1, \ldots , \mathbf {p}_n) \widetilde{\delta }_{L} (p_1) \ldots \widetilde{\delta }_{L} (p_n) .\nonumber \\ \end{aligned}$$
(14)

In the second equality, we adopt the same notation as in [2–4]. We also use \(\mathbf {p}_{1 \ldots n} = \mathbf {p}_1 + \cdots + \mathbf {p}_n\), \(\mathbf {F}^{(n)} = (-1)^{n} \frac{\mathbf {p}_{1 \ldots n}}{p_{ 1\ldots n}^2} \frac{\kappa ^{(n)}(\mathbf {p}_1, \ldots , \mathbf {p}_n)}{p_1^2 \ldots p_{n}^2} \), and the \(n\)th order kernels \(\kappa ^{(n)}\) are the same as given in [13, 14]. We neglect the transverse terms in \(\mathbf {F}^{(n)}\), which do not appear in a one-loop correction. \(I_{n}(a)\) should be obtained numerically from Eqs. (8)–(10) by using the EdS initial conditions given by Eqs. (31)–(33). In the literature, one uses the coefficients for the EdS solutions,

$$\begin{aligned} I_2 = -\frac{3}{7}, \quad I_{3a} = -\frac{1}{3}, \quad I_{3b} = \frac{10}{21}. \end{aligned}$$
(15)

However, these values are approximate ones using the EdS assumption and we will use the exact values of them.

From (14), one can obtain the perturbative kernels in LPT up to the third order

$$\begin{aligned}&L^{(1)}(\mathbf {p}_1)= \frac{\mathbf {k}}{k^2}, \end{aligned}$$
(16)
$$\begin{aligned}&L^{(2)}(a,\mathbf {p}_1,\mathbf {p}_2) = I_2(a) \frac{\mathbf {k}}{k^2} \Biggl [1 - \Bigl (\frac{\mathbf {p}_1 \cdot \mathbf {p}_2}{p_1p_2} \Bigr )^2 \Biggr ] , \end{aligned}$$
(17)
$$\begin{aligned}&L^{(3a)}(a,\mathbf {p}_1,\mathbf {p}_2,\mathbf {p}_3) = I_{3a}(a) \frac{\mathbf {k}}{k^2} \Biggl [1 - 3 \Bigl (\frac{\mathbf {p}_1 \cdot \mathbf {p}_2}{p_1p_2} \Bigr )^2 \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad + 2 \frac{(\mathbf {p}_1\cdot \mathbf {p}_2)(\mathbf {p}_2\cdot \mathbf {p}_3)(\mathbf {p}_3\cdot \mathbf {p}_1)}{p_1^2\,p_2^2\,p_3^2} \Biggr ], \nonumber \\\end{aligned}$$
(18)
$$\begin{aligned}&L^{(3b)}(a,\mathbf {p}_1,\mathbf {p}_2,\mathbf {p}_3) = I_{3b}(a) \frac{\mathbf {k}}{k^2} \Biggl [1 - \Bigl (\frac{\mathbf {p}_1 \cdot \mathbf {p}_2}{p_1p_2} \Bigr )^2 \Biggr ] \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \times \Biggl [1 - \Bigl (\frac{(\mathbf {p}_1+\mathbf {p}_2) \cdot \mathbf {p}_3}{|p_1+p_2|p_3} \Bigr )^2 \Biggr ] . \end{aligned}$$
(19)

The above kernels are identical to those of EdS given in [2, 14] when \(I_{2}\)–\(I_{3b}\) are given by Eq. (15).

From the above consideration, one can obtain the nonlinear power spectrum with one-loop correction by using a resummation scheme known as integrated perturbation theory [2, 3],

$$\begin{aligned} P(k)&= \exp \Biggl [-k_ik_j \int \frac{d^3 p}{(2\pi )^3} C_{ij}(\mathbf {p}) \Biggr ] \nonumber \\&\times \Biggl [ k_ik_j C_{ij}(\mathbf {k}) + k_ik_jk_k \int \!\!\frac{d^3 p}{(2\pi )^3} C_{ijk} (\mathbf {k},-\mathbf {p},\mathbf {p}\!-\!\mathbf {k}) \nonumber \\&+ \frac{1}{2} k_ik_jk_kk_l \int \frac{d^3 p}{(2\pi )^3} C_{ij}(\mathbf {p}) C_{kl}(\mathbf {k}-\mathbf {p}) \Biggr ] \end{aligned}$$
(20)

where the mixed polyspectrum of the linear density field and the displacement field is defined as

$$\begin{aligned}&\Bigl \langle \widetilde{\delta }_{L}(\mathbf {k}_1)\ldots \widetilde{\delta }_{L}(\mathbf {k}_l)\widetilde{S}_{i_1}(\mathbf {p}_1)\ldots \widetilde{S}_{i_m}(\mathbf {p}_m) \Bigr \rangle _{c} \nonumber \\&\quad = (2\pi )^3\delta _{{D}}(\mathbf {k}_{1 \ldots l}+\mathbf {p}_{1 \ldots m})(- \imath )^m C_{i_i\ldots i_m}\nonumber \\&\quad \quad \times (\mathbf {k}_1,\ldots ,\mathbf {k}_l;\mathbf {p}_1,\ldots ,\mathbf {p}_m) . \end{aligned}$$
(21)

Also the mixed polyspectrum of each order in perturbations is given by

$$\begin{aligned}&\Bigl \langle \widetilde{\delta }_{L}(\mathbf {k}_1)\ldots \widetilde{\delta }_{L}(\mathbf {k}_l)\widetilde{S}^{(n_1)}_{i_1}(\mathbf {p}_1)\ldots \widetilde{S}^{(n_m)}_{i_m}(\mathbf {p}_m) \Bigr \rangle _{c} \nonumber \\&\quad = (2\pi )^3\delta _{{D}}(\mathbf {k}_{1 \ldots l}+\mathbf {p}_{1 \ldots m})(- \imath )^m C^{(n_1\ldots n_m)}_{i_i\ldots i_m}\nonumber \\&\quad \quad \times (\mathbf {k}_1,\ldots ,\mathbf {k}_l;\mathbf {p}_1,\ldots ,\mathbf {p}_m) . \end{aligned}$$
(22)

From Eqs. (20) and (22), one can obtain the matter power spectrum with one-loop correction,

$$\begin{aligned} P(k)&= \exp \Bigl [-k_i k_j \int \frac{d^3 p}{(2\pi )^3} C_{ij}^{(11)}(\mathbf {p}) \Bigr ] \nonumber \\&\times \Bigl ( k_i k_j \Bigl [ C_{ij}^{(11)}(\mathbf {k}) \!+\! C_{ij}^{(22)}(\mathbf {k}) \!+\!C_{ij}^{(13)}(\mathbf {k}) \!+\! C_{ij}^{(31)}(\mathbf {k}) \!\Bigr ]\nonumber \\&+ k_i k_j k_k \int \frac{d^3 p}{(2\pi )^3} \Bigl [C_{ijk}^{(112)}(\mathbf {k},-\mathbf {p},\mathbf {p}-\mathbf {k})\nonumber \\&+ C_{ijk}^{(121)}(\mathbf {k},-\mathbf {p},\mathbf {p}-\mathbf {k}) + C_{ijk}^{(211)}(\mathbf {k},-\mathbf {p},\mathbf {p}-\mathbf {k}) \Bigr ]\nonumber \\&+ \frac{1}{2} k_i k_j k_k k_l \int \frac{d^3 p}{(2\pi )^3} C_{ij}^{(11)}(\mathbf {p}) C_{ij}^{(11)}(\mathbf {k}-\mathbf {p}) \Bigr ) . \end{aligned}$$
(23)

After the analytic angular integration of Eq. (23), one obtains

$$\begin{aligned} P(k)&= \exp \Bigl [-\frac{k^2}{6 \pi ^2} \int d p P_{L}(p) \Bigr ] \nonumber \\&\times \Biggl [ P_{L}(k) + \frac{(2 \pi )^{-2} k^3}{2} \int _{0}^{\infty } \mathrm{d}r P_{L}(kr)\nonumber \\&\times \int _{-1}^{1} \mathrm{d}x P_{L} \Bigl ( k\sqrt{1+r^2-2rx} \Bigr )\nonumber \\&\times \Biggl [ \frac{-I_2 r + x -(1 - I_2) r x^2}{(1 + r^2 - 2rx)} \Biggr ]^2 \nonumber \\&+ \frac{(2 \pi )^{-2} k^3}{48} P_{L}(k) \int _{0}^{\infty } \mathrm{d}r P_{L}(kr)\nonumber \\&\times \Biggl ( -6(2I_2+I_{3b}) r^{-2} + 2(10I_2+11I_{3b}) \nonumber \\&+ 2(-10I_2\!+\!11I_{3b}) r^2 \!+\! 6(2I_2\!-\!I_{3b}) r^4 \!+\! \frac{3}{r^3} (r^2\!-\!1)^3\nonumber \\&\times \Bigl ( (-2I_2+I_{3b}) r^2 -(2I_2+I_{3b}) \Bigr )\ln \Bigl |\frac{1+r}{1-r} \Bigr | \Biggr ) \Biggr ] \nonumber \\&\equiv \exp \Bigl [-\frac{k^2}{6 \pi ^2} \!\int \!d p P_{L}(p) \Bigr ] \Biggl [\! P_{L}(k) \!+\! P_{22}(k) \!+\! P_{13}(k) \!\Biggr ] \nonumber \\&\equiv \exp \Bigl [-\frac{k^2}{6 \pi ^2} \int d p P_{L}(p) \Bigr ] \Bigl [ P_{\mathrm{NL}}(k) \Bigr ] , \end{aligned}$$
(24)

where \(r = \frac{p}{k}\) and \(x = \frac{\mathbf {p}\cdot \mathbf {k}}{p k}\). The above equations are identical to Eqs. (36) of [2] when one replaces \(I_2\) and \(I_{3b}\) with those of the EdS case given by Eq. (15). Thus, the terms with \(I_{2}\) and \(I_{3b}\) represent the dark energy effect on the power spectrum. Also, both \(I_{2}\) and \(I_{3b}\) depend on the time and their values are changed depending on the measuring epoch. One interesting feature is that \(I_{3a}\) does not contribute the one-loop correction in the matter power spectrum. When we generalize the power spectrum in the SPT without using the EdS assumption, we obtain a similar matter power spectrum to [1]:

$$\begin{aligned} P^\mathrm{{SPT}}(k)&= P_{L}(k) + \frac{(2 \pi )^{-2} k^3}{2} \int _{0}^{\infty } \mathrm{d}r P_{L}(kr) \nonumber \\&\times \int _{-1}^{1} \mathrm{d}x P_{L} \Bigl ( k\sqrt{1+r^2-2rx} \Bigr )\nonumber \\&\times \Biggl [ \frac{(c_{21}+2c_{22}) r + (c_{21}-2c_{22}) x -2 c_{21} r x^2}{(1 + r^2 - 2rx)} \Biggr ]^2 \nonumber \\&+ (2\pi )^{-2} k^3 P_{L}(k) \int _{0}^{\infty } \mathrm{d}r P_{L}(kr) \nonumber \\&\times \Biggl [\!2c_{35} r^{-2}\! -\!\frac{1}{3} \Bigl (\! 4 c_{31} \!-\!8 c_{32} \!+\!3c_{33} \!+\!24c_{35} \!-\! 16 c_{36} \!\Bigr ) \nonumber \\&- \frac{1}{3}\Bigl (4 c_{31} -8c_{32} +12c_{33}-8c_{34}+6c_{35} \Bigr )r^2 \nonumber \\&+ c_{33} r^4 \!+\! \Bigl (\frac{r^2\!-\!1}{r} \Bigr )^3 \ln \Bigl |\frac{1+r}{1-r} \Bigr | \Bigl (c_{35} \!-\! \frac{1}{2}c_{33}r^2 \Bigr ) \!\Biggr ],\nonumber \\ \end{aligned}$$
(25)

where \(c_{21}\)–\(c_{36}\) are also given in the above reference. If we adopt the EdS assumption both for the LPT matter power spectrum and the SPT one, then \(P_{22}\) is the same for both approaches. However, the exact solutions will not be matched exactly for both cases. Also compared to SPT case where the magnitude of \(P_{13}\) is comparable to that to \(P_{13}\), the magnitude of \(P_{13}\) is much smaller than that of \(P_{22}\) in LPT. This shows that the EdS approximation in LPT is quite accurate.

Now we obtain the one-loop power spectrum for \(\Lambda \)CDM model. We run the camb to obtain the linear power spectrum [16] using \(\Omega _{b0} = 0.044\), \(\Omega _{m0} = 0.26\), \(h = 0.72\), \(n_{s} = 0.96\), and the numerical integration range for \(p\) in Eq. (24) is \(10^{-6} \le p \le 10^{2}\). In the left panel of Fig. 1, we show the linear power spectrum \(P_{L}\) (solid), the one-loop power spectrum \(P_{22}\) (dotted), \(|P_{13}|\) (dashed), and the nonlinear power spectrum \(P_{\mathrm{NL}} = P_{L} + P_{1\hbox {-}\mathrm {loop}}\) (dot-dashed). Absolute magnitude of \(P_{13}\) is smaller than that of SPT. Thus, the one-loop correction is larger than that of SPT. The one-loop correction is mainly contributed from the \(P_{22}\) and the coefficient \(I_2\) is not much deviant from that of EdS (\(-\frac{3}{7}\)) as shown in the appendix. That is why the EdS approximation is a good one for LPT. Also, there exists an additional exponential prefactor to get the total power spectrum. This is shown in the right panel of Fig. 1. The dot-dashed line represents \(P_{\mathrm{NL}}\) and the dotted line indicates \(P(k)\).

Fig. 1
figure 1

\(P_{\mathrm{NL}}(k)\) and \(P(k)\). a Solid, dotted, dashed, and dot-dashed line represent \(P_{L}\), \(P_{22}\), \(|P_{13}|\), and \(P_{ NL}\), respectively. b \(P_{\mathrm{NL}}\) and \(P(k)\) are indicated as dot-dahed and dotted lines, respectively

Full size image

Now we investigate the effect of dark energy on the one-loop power spectrum compared to the one with EdS assumption. The difference in \(P_{22} + P_{13}\) between them is shown in the left panel of Fig. 2. There exists only 0.2 % error in the \(k = 0.1 \,\mathrm {h}\,\mathrm{Mpc^{-1}}\) mode at the present epoch. When we investigate them at the different \(z\)s, then the error is about the same. The error can be about 2 % at large scales but the one-loop power spectrum is much smaller than the linear power spectrum at these scales. When we consider the total \(P_{\mathrm{NL}}\), the difference is even smaller. The error is less than 0.05 % for the same mode. This is shown in the right panel of Fig. 2 with the notation \(\Delta P_{\mathrm{NL}} = P_{\mathrm{NL}} - P_{\mathrm{NL}}^{(\mathrm {EdS})}\). This proves the goodness of the EdS assumption in LPT claimed in [17]. However, we need to pay attention to this EdS assumption when we consider more general models.

Fig. 2
figure 2

Errors in \(P_{1\hbox {-}\mathrm {loop}}\) and \(P_{\mathrm{NL}}\). a The percentage difference between the correct \(P_{1\hbox {-}\mathrm {loop}}\) and the one with EdS assumption. b The percentage difference between \(P_{\mathrm{NL}}\) and \(P_{\mathrm{NL}}^{(\mathrm {EdS})}\)

Full size image

We show that the EdS assumption is a good approximation to calculate the \(\Lambda \)CDM one-loop power spectrum in Lagrangian perturbation theory. However, when we consider general dark energy models we need to consider the fully consistent method by using the fact \(I_{n}\) depends on time. This also makes it possible to separate the temporal and spatial parts of the solutions. We might be able to extend this method to the early dark energy or the modified gravity theories. The upcoming redshift surveys will provide observational data of the large scale structure of the universe in a larger volume with a higher density. We obtain the accurate Lagrangian perturbation theory without using any assumption and this matches the requirement from future surveys. The obtained results are general for any background universe model including time varying dark energy models.