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Register a new userMeasuring the galaxy power spectrum with multiresolution decomposition -- IV. redshift distortion
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In this paper, we develop a theory of redshift distortion of the galaxy power spectrum in the discrete wavelet transform (DWT) representation. Because the DWT power spectrum is dependent of both the scale and shape (configuration) of the decomposition modes, it is sensitive to distortion of shape of the field. On the other hand, the redshift distortion causes a shape distortion of distributions in real space with respect to redshift space. Therefore, the shape-dependent DWT power spectrum is useful to detect the effect of redshift distortion. We first established the mapping between the DWT power spectra in redshift and real space. The mapping depends on the redshift distortion effects of (1) bulk velocity, (2) selection function and (3) pairwise peculiar velocity. We then proposed $beta$-estimators using the DWT off-diagonal power spectra. These $beta$-estimators are model-free even when the non-linear redshift distortion effect is not negligible. Moreover, these estimators do not rely on the assumption of whether the pairwise velocity dispersion being scale-dependent. The tests with N-body simulation samples show that the proposed $beta$-estimators can yield reliable measurements of $beta$ with about 20% uncertainty for all popular dark matter models. We also develop an algorithm for reconstruction of the power spectrum in real space from the redshift distorted power spectrum. The numerical test also shows that the real power spectrum can be well recovered from the redshift distorted power spectrum.
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In this paper, we develop the method of analyzing the velocity field of cosmic matter with a multiresolution decomposition. This is necessary in calculating the redshift distortion of power spectrum in the discrete wavelet transform (DWT) representation. We show that, in the DWT analysis, the velocity field can be described by discrete variables, which are given by assignment of the number density and velocity into the DWT modes. These DWT variables are complete and not redundant. In this scheme, the peculiar velocity and pairwise velocity of galaxies or particles are given by field variables. As a consequence, the velocity dispersion (VD) and pairwise velocity dispersion (PVD) are no longer measured by number-counting or pair-counting statistic, but with the ensemble of the field variables, and therefore, they are free from the bias due to the number-counting and pair-counting. We analyzed the VD and PVD of the velocity fields given by the N-body simulation for models of the SCDM, $tau$CDM and $Lambda$CDM. The spectrum (scale-dependence) of the VD and PVD show that the length scale of the two-point correlation of the velocity field is as large as few tens h$^{-1}$ Mpc. Although the VD and PVD show similar behavior in some aspects, they are substantially different from each other. The VD-to-PVD ratio shows the difference between the scale-dependencies of the VD and PVD. More prominent difference between the VD and PVD is shown by probability distribution function. The one-point distribution of peculiar velocity is approximately exponential, while the pairwise velocitys is lognormal, i.e. of long tail. This difference indicates that the cosmic velocity field is typically intermittent.
The power spectrum estimator based on the discrete wavelet transform (DWT) for 3-dimensional samples has been studied. The DWT estimator for multi-dimensional samples provides two types of spectra with respect to diagonal and off-diagonal modes, which are very flexible to deal with configuration-related problems in the power spectrum detection. With simulation samples and mock catalogues of the Las Campanas redshift survey (LCRS), we show (1) the slice-like geometry of the LCRS doesnt affect the off-diagonal power spectrum with ``slice-like mode; (2) the Poisson sampling with the LCRS selection function doesnt cause more than 1-$sigma$ error in the DWT power spectrum; and (3) the powers of peculiar velocity fluctuations, which cause the redshift distortion, are approximately scale-independent. These results insure that the uncertainties of the power spectrum measurement are under control. The scatter of the DWT power spectra of the six strips of the LCRS survey is found to be rather small. It is less than 1-$sigma$ of the cosmic variance of mock samples in the wavenumber range $0.1 < k < 2$ h Mpc$^{-1}$. To fit the detected LCRS diagonal DWT power spectrum with CDM models, we find that the best-fitting redshift distortion parameter $beta$ is about the same as that obtained from the Fourier power spectrum. The velocity dispersions $sigma_v$ for SCDM and $Lambda$CDM models are also consistent with other $sigma_v$ detections with the LCRS. A systematic difference between the best-fitting parameters of diagonal and off-diagonal power spectra has been significantly measured. This indicates that the off-diagonal power spectra are capable of providing information about the power spectrum of galaxy velocity field.
The large-scale structure of the Universe should soon be measured at high redshift during the Epoch of Reionization (EoR) through line-intensity mapping. A number of ongoing and planned surveys are using the 21 cm line to trace neutral hydrogen fluctuations in the intergalactic medium (IGM) during the EoR. These may be fruitfully combined with separate efforts to measure large-scale emission fluctuations from galactic lines such as [CII], CO, H-$alpha$, and Ly-$alpha$ during the same epoch. The large scale power spectrum of each line encodes important information about reionization, with the 21 cm power spectrum providing a relatively direct tracer of the ionization history. Here we show that the large scale 21 cm power spectrum can be extracted using only cross-power spectra between the 21 cm fluctuations and each of two separate line-intensity mapping data cubes. This technique is more robust to residual foregrounds than the usual 21 cm auto-power spectrum measurements and so can help in verifying auto-spectrum detections. We characterize the accuracy of this method using numerical simulations and find that the large-scale 21 cm power spectrum can be inferred to an accuracy of within 5% for most of the EoR, reaching 0.6% accuracy on a scale of $ksim0.1,text{Mpc}^{-1}$ at $left< x_i right> = 0.36$ ($z = 8.34$ in our model). An extension from two to $N$ additional lines would provide $N(N-1)/2$ cross-checks on the large-scale 21 cm power spectrum. This work strongly motivates redundant line-intensity mapping surveys probing the same cosmological volumes.
Galaxy redshift surveys are one of the pillars of the current standard cosmological model and remain a key tool in the experimental effort to understand the origin of cosmic acceleration. To this end, the next generation of surveys aim at achieving sub-percent precision in the measurement of the equation of state of dark energy $w(z)$ and the growth rate of structure $f(z)$. This however requires comparable control over systematic errors, stressing the need for improved modelling methods. In this contribution we review at the introductory level some highlights of the work done in this direction by the {it Darklight} project. Supported by an ERC Advanced Grant, {it Darklight} developed novel techniques for clustering analysis, which were tested through numerical simulations before being finally applied to galaxy data as in particular those of the recently completed VIPERS redshift survey. We focus in particular on: (a) advances on estimating the growth rate of structure from redshift-space distortions; (b) parameter estimation through global Bayesian reconstruction of the density field from survey data; (c) impact of massive neutrinos on large-scale structure measurements. Overall, {it Darklight} has contributed to paving the way for forthcoming high-precision experiments, such as {it Euclid}, the next ESA cosmological mission.
The post-reionization ${rm HI}$ 21-cm signal is an excellent candidate for precision cosmology, this however requires accurate modelling of the expected signal. Sarkar et al. (2016) have simulated the real space ${rm HI}$ 21-cm signal, and have modelled the ${rm HI}$ power spectrum as $P_{{rm HI}}(k)=b^2 P(k)$ where $P(k)$ is the dark matter power spectrum and $b(k)$ is a (possibly complex) scale dependent bias for which fitting formulas have been provided. This paper extends these simulations to incorporate redshift space distortion and predict the expected redshift space ${rm HI}$ 21-cm power spectrum $P^s_{{rm HI}}(k_{perp},k_{parallel})$ using two different prescriptions for the ${rm HI}$ distributions and peculiar velocities. We model $P^s_{{rm HI}}(k_{perp},k_{parallel})$ assuming that it is the product of $P_{{rm HI}}(k)=b^2 P(k)$ with a Kaiser enhancement term and a Finger of God (FoG) damping which has $sigma_p$ the pair velocity dispersion as a free parameter. Considering several possibilities for the bias and the damping profile, we find that the models with a scale dependent bias and a Lorentzian damping profile best fit the simulated $P^s_{{rm HI}}(k_{perp},k_{parallel})$ over the entire range $1 le z le 6$. The best fit value of $sigma_p$ falls approximately as $(1+z)^{-m}$ with $m=2$ and $1.2$ respectively for the two different prescriptions. The model predictions are consistent with the simulations for $k < 0.3 , {rm Mpc}^{-1}$ over the entire $z$ range for the monopole $P^s_0(k)$, and at $z le 3$ for the quadrupole $P^s_2(k)$. At $z ge 4$ the models underpredict $P^s_2(k)$ at large $k$, and the fit is restricted to $k < 0.15 , {rm Mpc}^{-1}$.
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