1.8 Forecasts for Euclid
1.8.5 Testing the non-linear corrections for weak lensing forecasts
In order to fully exploit the scientific potential of the next generation of weak lensing surveys, accurate predictions of the matter power spectrum are required. The signal-to-noise ratio of the cosmic shear signal is highest on angular scales of 5−10 arcminutes, which correspond to physical scales of∼1 Mpc. Restricting the analysis to larger scales does not necessarily solve the problem, because the observed two-point ellipticity correlation functions are still sensitive to small scale structures projected along the line-of-sight. This may be avoided using a full 3D shear analysis [see 294, 675, for details], but using only the larger scales increases the statistical uncertainties due to cosmic variance.
Currently only N-body simulations allow us to capture the non-linear structure formation, but for a survey such asEuclidan accuracy of∼1% is needed [622, 626]. This accuracy goes beyond the claimed ±3% uncertainty of the popular halofit code [1106]. However, the accuracy can be improved provided the simulations are started with adequate initial conditions, with a large volume, sufficient time stepping and high mass resolution. For instance [580] obtained an accuracy of∼1% out tok∼1hMpc−1 for a gravity-only simulation.
It is important to distinguish between gravity-only simulations, which are used to make the forecasts, and hydrodynamical simulations that attempt to capture the modifications to the matter power spectrum due to baryon physics. Although most of the matter in the Universe is indeed believed to be in the form of collissionless cold dark matter, baryons represent a non-negligible fraction of the total matter content. The distribution of baryons traces that of the underlying dark matter density field and thus gravity-only simulations should capture most of the structure formation. Nonetheless, differences in the spatial distribution of baryons with respect to the dark matter is expected to lead to changes that exceed the required accuracy of 1 per cent.
Various processes, which include radiative cooling, star formation and energy injection from supernovae and active galactic nuclei, affect the distribution of baryons. Implementing these processes correctly is difficult, and as a consequence the accuracy of hydrodynamic simulations is under discussion. That baryon physics cannot be ignored was perhaps most clearly shown in [1193] who looked at the changes in the matter power spectra when different processes are included. This was used by [1072] to examine the impact on cosmic shear studies. The results suggest that AGN feedback may lead to a suppression of the power by as much as 10% atk∼1hMpc−1.
[1072] showed that ignoring the baryonic physics leads to biases in the cosmological parameter estimates that are much larger than the precision of Euclid. In the case of the AGN model, the bias in w is as much as 40%. Unfortunately our knowledge of the various feedback processes is still incomplete and we cannot use the simulations to interpret cosmic shear signal. Furthermore, hydrodynamic simulations are too expensive to simulate large volumes for a range of cosmological parameters. To circumvent this problem several approaches have been suggested. For instance, [168] proposed to describe the changes in the power spectrum by Legendre polynomials, and to marginalise over the nuisance parameters [also see 676, for a similar approach]. Although this leads to unbiased estimates for cosmological parameters, the precision decreases significantly, by as much as 30% [1272].
Instead [1072] and [1070] examined whether it is possible to model the effects of baryon physics using a halo model approach, in which the baryons and stars are treated separately from the dark matter distribution. The model parameters, rather than being mere nuisance parameters, correspond to physical quantities that can be constrained observationally. These works showed that even with this still rather simple approach it is possible to reduce the biases in the cosmological parameters to acceptable levels, without a large loss in precision.
The forecasts do not include the uncertainty due to baryon physics, hence the results implicitly assume that this can be understood sufficiently well that no loss in precision occurs. This may be somewhat optimistic, as more work is needed in the coming years to accurately quantify the impact of baryon physics on the modelling of the matter power spectrum, but we note that the initial results are very encouraging. In particular, [1070] found that requiring consistency between the two- and three-point statistics can be used to self-calibrate feedback models.
Another complication for the forecasts is the performance of the prescriptions for the non-linear power spectrum for non-ΛCDM models. For instance, [860] showed that, usinghalofitfor non- ΛCDM models, requires suitable corrections. In spite of that, halofit has been often used to calculate the spectra of models with non-constant DE state parameterw(z). This procedure was dictated by the lack of appropriate extensions of halofitto non-ΛCDM cosmologies.
In this paragraph we quantify the effects of using thehalofitcode instead ofN-body outputs for nonlinear corrections for DE spectra, when the nature of DE is investigated through weak lensing surveys. Using a Fisher-matrix approach, we evaluate the discrepancies in error forecasts forw0,wa and Ωmand compare the related confidence ellipses. See [290] for further details.
The weak lensing survey is as specified in Section 1.8.2. Tests are performed assuming three different fiducial cosmologies: ΛCDM model (w0=−1,wa = 0) and two dynamical DE models,
still consistent with the WMAP+BAO+SN combination [704] at 95% C.L. They will be dubbed M1 (w0 = −0.67, wa = 2.28) and M3 (w0 = −1.18, wa = 0.89). In this way we explore the
dependence of our results on the assumed fiducial model. For the other parameters we adopt the fiducial cosmology of Secton 1.8.2.
The derivatives needed to calculate the Fisher matrix are evaluated by extracting the power spectra from theN-body simulations of models close to the fiducial ones, obtained by considering parameter increments±5%. For the ΛCDM case, two different initial seeds were also considered, to test the dependence on initial conditions, finding that Fisher matrix results are almost insensitive to it. For the other fiducial models, only one seed is used.
N-body simulations are performed by using a modified version of pkdgrav [1123] able to handle any DE state equationw(a), withN3= 2563particles in a box with sideL= 256h−1Mpc.
Transfer functions generated using the camb package are employed to create initial conditions, with a modified version of the PM software by [680], also able to handle suitable parameterizations of DE.
Matter power spectra are obtained by performing a FFT (Fast Fourier Transform) of the matter density fields, computed from the particles distribution through a Cloud-in-Cell algorithm, by using a regular grid withNg = 2048. This allows us to obtain nonlinear spectra in a largek-interval.
In particular, our resolution allows to work out spectra up to k ' 10hMpc−1. However, for k >2 – 3hMpc−1neglecting baryon physics is no longer accurate [642, 1023, 207, 1271, 576]. For this reason, we consider WL spectra only up to`max= 2000.
Particular attention has to be paid to matter power spectra normalizations. In fact, we found that, normalizing all models to the same linear σ8(z = 0), the shear derivatives with respect to
w0, wa or Ωm were largely dominated by the normalization shift at z = 0, σ8 and σ8,nl values
being quite different and the shift itself depending onw0, wa and Ωm. This would confuse thez
dependence of the growth factor, through the observationalz-range. This normalization problem was not previously met in analogous tests with the Fisher matrix, as halofitdoes not directly depend on the DE state equation.
Figure 27: Likelihood contours, for 65% and 95% C.L., calculated including signals up to`'2000 for the ΛCDM fiducial. Here simulations andhalofityield significantly different outputs.
Figure 28: On the left (right) panel, 1- and 2-σcontours for the M1 (M3) model. The two fiducial models exhibit quite different behaviors.
As a matter of fact, one should keep in mind that, observing the galaxy distribution with future surveys, one can effectively measureσ8,nl, and not its linear counterpart. For these reasons, we
choose to normalize matter power spectra toσ8,nl, assuming to know it with high precision.
In Figures 27 we show the confidence ellipses, when the fiducial model is ΛCDM, in the cases of 3 or 5 bins and with `max = 2000. Since the discrepancy between different seeds are small,
discrepancies betweenhalofitand simulations are truly indicating an underestimate of errors in thehalofitcase.
As expected, the error on Ωm estimate is not affected by the passage from simulations to halofit, since we are dealing with ΛCDM models only. On the contrary, usinghalofitleads to underestimates of the errors onw0 andwa, by a substantial 30 – 40% (see [290] for further details).
This confirms that, when considering models different from ΛCDM, nonlinear correction ob- tained throughhalofitmay be misleading. This is true even when the fiducial model is ΛCDM itself and we just consider mild deviations ofwfrom−1.
Figure 28 then show the results in the w0-wa plane, when the fiducial models are M1 or M3.
It is evident that the two cases are quite different. In the M1 case, we see just quite a mild shift, even if they areO(10%) on error predictions. In the M3 case, errors estimated through halofit
exceed simulation errors by a substantial factor. Altogether, this is a case when estimates based onhalofitare not trustworthy.
The effect of baryon physics is another nonlinear correction to be considered. We note that the details of a study on the impact of baryon physics on the power spectrum and the parameter estimation can be found in [1071]