Third-Order Density Perturbation and One-Loop Power Spectrum in Dark-Energy-Dominated Universe

Third-Order Density Perturbation and One-Loop Power Spectrum in Dark-Energy-Dominated Universe

1Introduction

Revealing the nature of dark energy is fundamentally important not only for astrophysics but also for particle physics. Constraints on the dark energy from astronomical observations is very influential for them. Baryon acoustic oscillations (BAO) in the galaxy power spectrum provide a strong constraint on the dark energy using its acoustic scale as a standard ruler. Large galaxy surveys such as the Sloan Digital Sky Survey and two degree field already provide the constraint and future larger surveys are currently planned to detect the BAO more accurately[1]. Hence, an accurate theoretical model of the BAO is crucial, and many authors have been investigating the BAO using numerical simulation[5] and the perturbation theory (including the renormalized perturbation theory)[13].

Previously, several authors investigated the third-order density perturbation and derived the one-loop correction to the linear power spectrum in the EdS model[23]. Similarly for the cosmological constant model, Bernardeau (1994) presented the third-order perturbation solution (see also Refs. ). They found that the dependence of the cosmological model on the second- and third-order perturbations is very small, if the scale factor in the EdS model is replaced with the linear growth factor.1 However, since the theoretical model of the BAO should archive the subpercent accuracy to provide a strong constraint on the dark energy, it is useful to reinvestigate this topic to accurately check the above assumption. In this study, we calculate the third-order density perturbation, newly including the dark energy with the time-varying equation of state, and derive the one-loop power spectrum analytically for the first time. We compare our results with the approximate results based on the EdS model in detail, and discuss the effect of the dark energy on the power spectrum near the baryon acoustic scale.

Throughout this paper, we use as the density fluctuation, as the divergence of the peculiar velocity field, and as the conformal time. , and are the density parameter for the matter, the curvature and the dark energy at present. is the equation of state of the dark energy. The Hubble expansion rate is .

2Basics

The equation of motion determines the growth of the density field , and velocity field in the Fourier space is [31]

with

Equation (Equation 1) is the continuity equation, while equation ( ?) is the Euler equation with the Poisson equation. In the linear regime, one can neglect the mode-coupling terms on the right-hand sides of Eqs. (Equation 1) and ( ?). Then the linear solutions are

The linear growth factor is determined by

with the initial condition at . In the special case of the flat model () with the constant , the solution is given by the hypergeometric function[37].

The density field is formally expanded up to the third order as . We will show the second- and third-order solutions in the following sections.

3Second-order solution

Inserting the linear-order solutions of and into the right-hand sides of equations (Equation 1) and ( ?), one can obtain the second-order solution as

with

The second-order growth factors are determined by ordinary differential equations with the boundary condition at (see Appendix A). One usually approximately use , instead of , in equation (Equation 2). In order to demonstrate the validity of this approximation, we show the relative differences between and in Figure 1 for the constant in the flat model (). The results are shown by the contour lines in the plane for (top left panel) and (top right panel). As clearly seen in the figures, the relative errors are small, less than for and . The errors become larger for larger . This tenancy suggests for larger that the dark energy has been affecting the expansion rate since long time ago, and hence the large differences between and arise at present.

Figure 2 is the same as Figure 1, but for the time varying equation of state[39]

The results are shown in the plane with . As shown in the figure, for large , the relative differences become large. This is because the dark energy term in the hubble expansion , , becomes large for large in the past (). The relative errors are less than for and .

Figure 1:  Top panels: the contour lines show the relative differences between D_{2 A,B} and D_1^2 at present (a=1) in the \Omega_M-w plane. The flat cosmological model and the constant equation of state are assumed. The top left (right) panel is the result for D_{2 A} (D_{2 B}). Middle and bottom panels: same as top panels, but for the relative differences between D_{3} and D_1^3.
Figure 1: Top panels: the contour lines show the relative differences between and at present () in the plane. The flat cosmological model and the constant equation of state are assumed. The top left (right) panel is the result for (). Middle and bottom panels: same as top panels, but for the relative differences between and .
Figure 2:  Same as Fig. , but for the time-varying equation of state, w(a)=w_0+w_a a \left( 1-a \right). The results are shown in the w_0 - w_a plane in the flat cosmological model with \Omega_M=0.28.
Figure 2: Same as Fig. , but for the time-varying equation of state, . The results are shown in the plane in the flat cosmological model with .

4Third-order solution

Similarly, the third-order solution consists of six terms, as shown by

with

There are two additional conditions of

and hence only four terms in Eq. (Equation 3) are independent of each other. The growth factors are determined by the ordinary differential equations with the boundary conditions of in (see Appendix A). The middle and bottom panels in Figs. Figure 1 and Figure 2 are the same as the top panels, but for the relative differences between and . The results are shown for (middle left), (middle right), (bottom left), and (bottom right). The relative differences are less than for and and less than for and .

Our results of the second- and third-order solutions are consistent with the previous results of Bernardeau (1994) for the cosmological constant model (). Although we presented the results for only the density perturbations, one can easily obtain the velocity field perturbations by inserting Eqs. (Equation 2) and (Equation 3) to Eqs. (Equation 1) and ( ?).

5One-loop power spectrum

The one-loop power spectrum is the linear power spectrum with the leading correction arising from the second- and third-order density perturbations,

where , and . The first term is the linear power spectrum, and the second and third terms are the one-loop corrections. The explicit formulae for and are given in Appendix B.

One usually approximately apply the one-loop power spectrum in the EdS model to an arbitrary cosmological model by replacing the scale factor by the linear growth factor,

where the second and third terms are the corrections for the EdS model[27] (see also Appendix B). We compare the two power spectra in Eqs. (Equation 5) and (Equation 6) in order to quantitatively demonstrate the validity of the above approximation. We use CAMB (Code for Anisotropies in the Microwave Background)[41] to calculate the linear power spectrum with the cosmological parameters , , , and , consistent with the WMAP 5yr result[42].

Figure 3:  Relative differences of P_{22}(k) (top left), P_{13}(k) (top right), P_{22}(k)+P_{13}(k) (bottom left) and P(k) (bottom right) between the correct results and the approximate results denoted by [...]_{\rm EdS}. The cosmological model is consistent with the WMAP 5yr result.
Figure 3: Relative differences of (top left), (top right), (bottom left) and (bottom right) between the correct results and the approximate results denoted by . The cosmological model is consistent with the WMAP 5yr result.
Figure 4:  Relative differences of P_{22}(k) (top left), P_{13}(k) (top right), P_{22}(k)+P_{13}(k) (bottom left) and P(k) (bottom right) between the correct results and the approximate results denoted by [...]_{\rm EdS}. The cosmological model is consistent with the WMAP 5yr result.
Figure 4: Relative differences of (top left), (top right), (bottom left) and (bottom right) between the correct results and the approximate results denoted by . The cosmological model is consistent with the WMAP 5yr result.
Figure 5:  Relative differences of P_{22}(k) (top left), P_{13}(k) (top right), P_{22}(k)+P_{13}(k) (bottom left) and P(k) (bottom right) between the correct results and the approximate results denoted by [...]_{\rm EdS}. The cosmological model is consistent with the WMAP 5yr result.
Figure 5: Relative differences of (top left), (top right), (bottom left) and (bottom right) between the correct results and the approximate results denoted by . The cosmological model is consistent with the WMAP 5yr result.
Figure 6:  Relative differences of P_{22}(k) (top left), P_{13}(k) (top right), P_{22}(k)+P_{13}(k) (bottom left) and P(k) (bottom right) between the correct results and the approximate results denoted by [...]_{\rm EdS}. The cosmological model is consistent with the WMAP 5yr result.
Figure 6: Relative differences of (top left), (top right), (bottom left) and (bottom right) between the correct results and the approximate results denoted by . The cosmological model is consistent with the WMAP 5yr result.
Figure 7:  Same as top panels in Fig., but at various redshifts of z=0,1,3.
Figure 7: Same as top panels in Fig., but at various redshifts of .
Figure 8:  Same as top panels in Fig., but at various redshifts of z=0,1,3.
Figure 8: Same as top panels in Fig., but at various redshifts of .

Figure 6 shows the relative differences in , , and between Eqs. (Equation 5) and (Equation 6) at . The equation of states are , , , and . From the top panels, the error is for while for . On a small scale, these differences are small. In the bottom left panel, the error diverges at Mpc because the denominator of vanishes there. The approximate formula of predicts a slightly lower value than the correct result, because is almost the same as while is more negative than as shown in the top panels. However, as expected, the difference is very small at less than for . Figure 8 is the same as Figure 6, but at various redshifts of . Hence, from this figure, the EdS model approximation in Eq. (Equation 6) is sufficiently more accurate for higher redshifts .

Finally, we calculate the shift in the position of the first acoustic peak at /Mpc. Dividing by the no-wiggle model of Eisenstein & Hu (1999), we find that the position is shifted by only for .

In this chapter, we calculated the one-loop power spectrum, however it is not accurate in the strong nonlinear regime (Mpc). In fact, Jeong & Komatsu (2006) found that the one-loop power spectrum coincides with the nonlinear power spectrum from the numerical simulation within if is satisfied. This condition is rewritten as Mpc at . Hence, in order to extend our analysis to a smaller scale, further analysis of the cosmological dependence of the higher-order perturbation theory is necessary.

6Conclusion

We investigate the third-order density perturbation and the one-loop power spectrum in the dark-energy cosmological model. We present analytical solutions and a fitting formula with the general time-varying equation of state for the first time. It turns out that the cosmological dependence is very weak, for example, less than for Mpc for the power spectrum. However, our results may be useful in some cases when one needs a very highly accurate theoretical model of the BAO or in the study of the nonlinear evolution on a smaller scale (/Mpc).

Acknowledgements

We would like to thank Takahiko Matsubara and the anonymous referees for helpful comments and suggestions. This work is supported in part by a Grant-in-Aid for Scientific Research on Priority Areas No. 467 “Probing the Dark Energy through an Extremely Wide and Deep Survey with Subaru Telescope”.


ASecond- and Third-Order Growth Factors

The second-order growth factors are determined by the ordinary differential equations

with the initial conditions at :

For the flat model with the constant equation of state, the solutions are well approximated as

within a maximum error of for both and .

Similarly, the third-order growth factors are determined by

with the initial conditions at :

For with the constant , the solutions are well fitted by

within a maximum error of for both and . The other growth factors and can be obtained using Eq. (Equation 4).

BExplicit Expressions of and

Here, we present the explicit expressions of the one-loop correction terms and . From the results in §, we obtain

with

where is the cosine between and .

Similarly for , from the results in §, we obtain

with

By setting and in Eqs. (Equation 7) and (Equation 8), the correction terms reduce to the result in the EdS model, .

Footnotes

  1. Martel & Freudling (1991) and Scoccimarro et al. (1998) showed that this assumption is valid if . However, since , the approximation is not accurate.

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