Quintessential scale dependence from separate universe simulations

By absorbing fluctuations into a local background, separate universe simulations provide a powerful technique to characterize the response of small-scale observables to the long-wavelength density fluctuations, for example those of the power spectrum and halo mass function which lead to the squeezed-limit $n$-point function and halo bias, respectively. Using quintessence dark energy as the paradigmatic example, we extend these simulation techniques to cases where non-gravitational forces in other sectors establish a Jeans scale across which the growth of density fluctuations becomes scale dependent. By characterizing the separate universes with matching background expansion histories, we show that the power spectrum and mass function responses depend on whether the long-wavelength mode is above or below the Jeans scale. Correspondingly, the squeezed bispectrum and halo bias also become scale dependent. Models of bias that are effectively local in the density field at a single epoch, initial or observed, cannot describe this effect which highlights the importance of temporal nonlocality in structure formation. Validated by these quintessence tests, our techniques are applicable to a wide range of models where the complex dynamics of additional fields affect the clustering of matter in the linear regime and it would otherwise be difficult to simulate their impact in the nonlinear regime.

Figure 1

(Top) Scale-dependent growth in ${\delta }_m$ as a function of the scale factor for super-Jeans (red solid) and sub-Jeans (blue dashed) long-wavelength modes. (Bottom) The ratio of ${\delta }_m$ in super-Jeans to sub-Jeans cases. When characterized as a separate universe, the former is closer to the global universe than the latter in the past for the same density fluctuation today.

Figure 2

(Top) Different components of the separate universe power spectrum responses at $z=0$: growth response (sub-Jeans, red solid; super-Jeans blue dashed), absolute value or negative of the dilation response (green dot-dashed), and the reference-density response (black dotted). (Bottom) Ratios of the mean of the total response to that of the sub-Jeans separate universe. Shaded bands reflect the error on the mean response of the simulations. The clear distinction between the sub-Jeans and super-Jeans power spectrum responses is the first important result of our separate universe simulations.

Figure 3

The response of the separate universe growth function as a function of the global scale factor $a$ in the global universe for super-Jeans (red solid) and sub-Jeans (blue dashed) cases. Above the Jeans scale, the response to the same ${\delta }_m$ at $a$ is smaller than below.

Figure 4

Linear perturbation theory predictions for the growth responses compared with the measurements from the 20 small-box separate universe simulations at $z=3$ (top) and 0 (bottom). The red solid and blue dashed lines with shaded areas show the sub-Jeans and super-Jeans separate universes measurements as in Fig. 2, whereas the red dot-dashed and blue long-dashed lines show the corresponding linear perturbation theory predictions, i.e. Eqs. (19)–(20). Note that the range of $y$ axes is smaller in the top than in the bottom panel. Also the cusp feature at $k\sim 0.078{\mathrm{Mpc}}^{-1}$ is a visual artifact due to binning, which we choose to be $\mathrm{\Delta }k=2\pi /L$.

Figure 5

Same as Fig. 4, but for the 1-loop predictions, i.e. Eq. (19) and Eq. (22).

Figure 6

Comparison of the squeezed-limit position-dependent power spectrum of big-box, global simulations (green dot-dashed), and the total power spectrum response of sub-Jeans (red solid) as well as super-Jeans (blue dashed) separate universe simulations at $z=0$. The shaded areas show the error on the mean. This figure summarizes our main power spectrum response results: the position-dependent power spectrum and the sub-Jeans power spectrum response agree significantly better than the difference in response across the Jeans scale confirming the scale dependence in this observable deep into the nonlinear regime.

Figure 7

Threshold mass shift as a response of varying ${\delta }_m$ at fixed cumulative abundance at $z=0$. The solid line and shaded region show the smoothed estimate and the bootstrap error.

Figure 8

(Top) The $z=0$ response biases measured from 20 sub-Jeans (red solid) and super-Jeans (blue dashed) separate universe simulations. The lines and shaded areas show the smoothed estimate and the bootstrap error. (Bottom) The ratios of the response biases to that of the sub-Jeans response bias. The difference between the sub-Jeans and super-Jeans response biases, which indicates that the linear halo bias is scale dependent in the presence of the scale-dependent growth, is the second central result of our separate universe simulations.

Figure 9

The ratios of the linear biases to that of the sub-Jeans response bias at $z=0$. The red solid and blue dashed lines show the sub-Jeans and super-Jeans response biases measured from 20 separate universe simulations, whereas the green dot-dashed line shows the clustering bias measured from 40 global simulations. The error of the clustering bias is measured from the scatter of the 40 simulations. This figure summarizes the main results on halo bias: the agreement between the clustering bias measured from global simulations and the sub-Jeans response bias verifies the observable difference in halo bias across the Jeans scale inferred from the separate universe simulations.

Figure 10

Fractional difference or “step” across the Jeans scale in the matter [red solid, Eq. (32)] versus the halo [blue dashed, Eq. (33)] power spectra. The step in the halo power spectrum is reduced by approximately a factor of 2 compared with the matter power spectrum at high mass where the Lagrangian bias contribution dominates. Bias prescriptions which assume locality at either the observed or initial redshift predict the same step or zero step respectively at high masses.

Figure 11

Ratio of the super-Jeans to sub-Jeans scale Lagrangian bias factors at $z=1$ (top) and $z=0$ (bottom) measured from separate universe simulations in comparison with the two models, PHS11 in Eq. (a1) and L14 in Eq. (a10), discussed in this Appendix. Oscillations are mainly due to spline smoothing. Increasing the number of knots gives flatter curves with larger errors but consistent averages. Large errors at $z=1$ for $M\gtrsim 10^{15}{M}_{\odot }$ are due to the lack of high redshift massive halos. The PHS11 model predictions rely on the standard assumption that Lagrangian bias is scale independent with respect to the initial density field, and our simulations rule this out.