New Probes of Large Scale Structure

We first need to account for galaxy bias Kravtsov and Klypin (1999)Desjacques et al. (2018). Galaxy bias is a term relating the galaxy number density contrast to the matter density contrast Gil-Marín et al. (2015)Gil-Marín et al. (2017). We use a model of second order bias, as done in recent treatments of galaxy surveys. Meanwhile, analytically the generalization to even higher order biases is straightforward. We adopt the most commonly used second-order galaxy bias model and assume all the bias parameters to be constants even while we are considering a large volume across a wide redshift range.

Observationally a galaxy catalog will be in the form of a light cone Carroll (1997) instead of a single redshift snapshot. The typical treatment is to cut a light cone into several thin redshift bins Chuang et al. (2017) and analyze the properties within each bin. Doing this, though, leads to loss of information on the long-wavelength modes along the line of sight. Thus in this paper we propose a method of considering all the galaxies in a light cone together, using the well-known Feldman-Kaiser-Peacock (FKP) estimator Feldman et al. (1994) to account for the evolution of the galaxy number density. Using an octant volume (the technique can be applied to a even more generalized shape), we test the quadratic estimator for long-wavelength modes using information from non-zero off-diagonal terms as in Li et al. (2020). It should be noticed that the FKP description corresponds to the monopole part of the estimator in redshift space (e.g. the Yamamoto estimator Yamamoto et al. (2006)Bianchi et al. (2015)). Because of this, our formalism will be able to reconstruct the large scale monopole power spectrum which is the main goal when studying the large scale matter distribution of the 3D universe.

We begin with a brief review of the formalism developed in Ref. Li et al. (2020), then present our treatment of the galaxy number density contrast in a light cone and build the quadratic estimator. Finally we apply the estimator to light-cone halo simulations and extract the large scale modes accounting for these effects.

When evaluating the two-point function of the full density contrast, cross-terms appear. For example, $\langle\delta_{\rm m}^{(1)}(\vec{k};z)\delta_{\rm m}^{(2)}(\vec{k}^{\prime};z)\rangle$ is proportional to $\delta^{(1)}_{\rm m}(\vec{k}+\vec{k}^{\prime};z)$ if both $\vec{k}$ and $\vec{k}^{\prime}$ correspond to short wavelengths but their sum is small (long wavelength). Explicitly, keeping terms up to second order,

Here $\vec{k}_{s}$ and $\vec{k}_{s}^{\prime}$ are two short-wavelength modes and $\vec{k}_{l}$ is a long-wavelength mode ($\vec{k}_{s},\vec{k}_{s}^{\prime}\gg\vec{k}_{l}$). They satisfy the squeezed-limit condition, $\vec{k}_{s}+\vec{k}_{s}^{\prime}=\vec{k}_{l}$, and $f$ is given by:

Here $P_{\rm m}^{(1)}$ is the linear matter power spectrum. Eq. (3) indicates that we can estimate the long-wavelength modes using small scale information with the following minimum variance quadratic estimator:

with $\vec{k}_{s}^{\prime}=\vec{k}_{l}-\vec{k}_{s}$. The normalization factor $A$ is defined by requiring that $\langle\hat{\delta}_{\rm m}^{(1)}(\vec{k}_{l};z)\rangle={\delta}_{\rm m}^{(1)}(\vec{k}_{l};z)$, and the weighting function $g$ is obtained by minimizing the noise. They can be expressed as:

where $P_{\rm m}$ is the nonlinear matter power spectrum including shot noise. With this choice of the weighting function $g$, the noise on the estimator $N(\vec{k}_{l};z)=A(\vec{k}_{l};z)$ if non-Gaussian terms in the four-point function are neglected. Therefore, the projected detectability of a power spectrum measurement using this quadratic estimator can be written as:

where $V$ is the volume of a survey and $\Delta k$ is the width of long-wavelength mode bins. We also take advantage of the fact that $A(\vec{k}_{l};z)$ does not depend on the direction of the long mode $\vec{k}_{l}$.

Galaxy bias describes the statistical relation between dark-matter and galaxy distributions. Similar to Eq. (1), we use the most commonly used Eulerian non-linear and non-local galaxy bias model np to second-order first proposed by McDonald and Roy (2009):

here “$\rm g$” denotes galaxy, and $b_{1}$ is the linear bias parameter relating galaxy and the matter density contrasts. The kernel $\mathpzc{F}_{2}$ is given by:

with $S_{2}$ given by:

Comparing the perturbative expansion of the galaxy density contrast Eq. (8) with that of the matter density contrast Eq. (1), we see that the difference with the first order term is an extra coefficient $b_{1}$. The second order term is also almost the same, with a simple replacement of the kernel function. This implies that we can easily generalize to the case of a galaxy catalog in a snapshot. In the case of a galaxy catalog in a light cone, the Feldman-Kaiser-Peacock (FKP) estimator is usually used to construct a weighted over-density that can be used to obtain the observed galaxy power spectrum Feldman et al. (1994):

with

Here $n_{g}$ is the observed galaxy number density and $n_{s}$ is the corresponding synthetic catalog (a random catalog with the same angular and radial selection function as the observations). The constant $\alpha$ is the ratio of the observed number density to the synthetic catalog’s number density. The FKP weight $w_{\rm FKP}(\vec{r})$ is usually defined as:

where $P_{0}$ is the typical amplitude of the observed power spectrum at the scale where the signal-to-noise of the power spectrum estimation is maximized, usually $k\sim 0.12\,h\rm Mpc^{-1}$. Note that in real surveys we will have other types of weights Gil-Marín et al. (2015)Gil-Marín et al. (2018), which can be easily included in the formalism of this section. The FKP estimator $F(\vec{r})$ is related to the observed galaxy power spectrum $P_{g,\rm obs}(\vec{k})$ by considering the following expectation value (diagonal elements) in Fourier space:

where $z_{\rm eff}$ is the effective redshift of the whole light cone. The window function $W(\vec{k})$, the shot noise spectrum $P_{\rm shot\;noise}$ and the windowed galaxy density contrast $\delta_{g,W}(\vec{k};z)$ are given respectively by¹¹1An interesting thing to notice here is that in the original paper introducing the FKP estimator Feldman et al. (1994) and in almost every succeeding work, $W(\vec{k})$ is defined as the complex conjugate of the quantity we use here. In these other cases, $W(\vec{k})$ only appears in the form of $|W(\vec{k})|^{2}$, and so this does not make a difference. However in our current work, we demonstrate using simulations that $e^{-i\vec{k}\cdot\vec{r}}$ should appear instead of $e^{+i\vec{k}\cdot\vec{r}}$ in the integrand, the same as for the definition of the Fourier transform.:

We want to calculate the off-diagonal term of the FKP estimator $F(\vec{k})$ given the fact that $F(\vec{k})$ is the observable from a galaxy light cone survey rather than $\delta_{g,W}$. Note that the two point functions of $n_{g}(\vec{r})-\alpha n_{s}(\vec{r})$ can be written as Feldman et al. (1994):

Assuming the squeezed limit $\vec{k}_{s}+\vec{k}^{\prime}_{s}=\vec{k}_{l}$ and using the expression above, we can write the off-diagonal term as:

with the“off-diagonal shot noise” $Q(\vec{k}_{l})$ given by:

The two point function $\langle\delta_{g,W}(\vec{k}_{s};z_{\rm eff})\delta_{g,W}(\vec{k}^{\prime}_{s};z_{\rm eff})\rangle$ up to second order can be simply expressed as:

by defining $\delta_{g,W}\equiv\delta_{g,W}(\vec{k}_{s};z_{\rm eff})$ and $\delta_{g,W}^{\prime}\equiv\delta_{g,W}(\vec{k}^{\prime}_{s};z_{\rm eff})$. One major difference here is that, for a non-cubical region, the leading order term would also be non-zero unlike the cubic volume in the last section II:

where $\delta_{\rm D}$ is the Dirac delta function. This term would vanish since in the case of a cube, $W(\vec{k})$ would be close to a Dirac delta function. For a non-cubic region, though, the term is no longer zero. Note that this leading order term can be fully determined numerically.

Using the expressions from Eq. (8) and Eq. (17), we can compute the second order two-point correlation of two short-wavelength modes $\delta_{g,W}(\vec{k}_{s};z_{\rm eff})$ and $\delta_{g,W}(\vec{k}^{\prime}_{s};z_{\rm eff})$. Using $\langle\delta^{(1)}_{g,W}\delta^{\prime(2)}_{g,W}\rangle$ as an example, we have:

Notice that we have computed the bracket $\langle\delta^{(1)}_{m}(\vec{k};z_{\rm eff})\delta^{(2)}_{g}(\vec{k}^{\prime};z_{\rm eff})\rangle$ in Ref. Li et al. (2020), with $F_{2}$ replaced by $\mathpzc{F}_{2}$. The result is:

Thus we can further express the bracket as:

Ideally, We would like to extract a term $\delta^{(1)}_{g,W}(\vec{k}_{l};z_{\rm eff})$ from this, where:

If one of the $W$ functions were a Dirac delta function, this would follow automatically. Here, $W$ is not a delta function, but given a large enough volume, $W(\vec{k})$ is peaked at $\vec{k}=0$ and also:

Thus we have the following approximations by first applying a redefinition of dummy variables:

With the calculation above, we can then recover the long-wavelength modes from the off-diagonal two-point functions of short-wavelength modes:

with

Notice that the $\mathpzc{f}$ is almost identical to the $f$ function in section II, simply with a replacement of the $F_{2}$ function and an extra coefficient $C$. The quadratic estimator can then be similarly formed, and is:

with $\vec{k}^{\prime}_{s}=\vec{k}_{l}-\vec{k}_{s}$ and $\mathpzc{g}$ being the weighting function. Notice here that the only difference is that we subtract off the non-zero leading order terms due to the non-cubical shape of the galaxy survey volume, and these two terms can be calculated numerically. By requiring that $\langle\hat{\delta}^{(1)}_{g,W}(\vec{k}_{l};z_{\rm eff})\rangle={\delta}^{(1)}_{g,W}(\vec{k}_{l};z_{\rm eff})$ we can similarly determine the normalization function $\mathpzc{A}$:

Similar to Ref. Li et al. (2020), by minimizing the noise we obtain the expression for the weighting function $\mathpzc{g}$:

Here $P_{g,\rm obs}$ is the full observed galaxy power spectrum including shot noise. With this choice of $\mathpzc{g}$ the noise term $\mathpzc{N}$ is identical to the normalization factor $\mathpzc{A}$. The projected detectability is defined as in Eq. (7):

Using the quadratic estimator Eq. (31) we can use the entirety of the small scale information from the non-cubical light cone to infer the large scale field of the windowed galaxy density contrast $\delta_{g,W}(\vec{r})$.

We consider the halo catalog in this light cone with halo masses between $2.2\times 10^{12}h^{-1}M_{\odot}<M<10^{14}h^{-1}M_{\odot}$, a wide mass bin. We obtained similar results using other mass bins as well. The effective redshift of this light cone is $z_{\rm eff}=0.76$. We assume the bias parameter $b_{1}$ and $b_{2}$ to be free parameters of the model, and use FAST-PT McEwen et al. (2016) to determine the bias parameters to be:

The remaining bias parameter $b_{s^{2}}$ can be constrained by assuming the bias model is local in Lagrangian space Baldauf et al. (2012):

We use the quadratic estimator, Eq. (31) to obtain the reconstructed Fourier space windowed galaxy density field, and transform it back into real space. Then, we compare this indirectly estimated result with the directly measured (using the FKP estimator) galaxy density in real space in Fig. 2. We use the information from small scale modes up to $k_{s,\rm max}=0.48\,h\,\rm Mpc^{-1}$. We also plot in Fig. 3 the directly measured large scale galaxy power spectrum with cosmic variance error bars (corresponding to Eq. (34) with $\mathpzc{A}=0$) versus the estimated large scale power spectrum using Eq. (31) with detectability given by Eq. (34). We see that our quadratic estimator gives a good estimation of the linear matter power spectrum and the uncertainty of the estimated result is only slightly larger than cosmic variance.

Note that in Fig. 2 because of the light cone, we cannot have a direct measurement of the $\delta_{g,W}$ field. Both the field computed from direct measurement of the Fourier modes (with FKP weighting) and the field derived from the quadratic estimator can be seen to encode almost the same large scale information from the light cone catalog. The large scales we are observing correspond to about $10^{-3}h\,\rm Mpc^{-1}$$<k_{l}<10^{-2}h\,\rm Mpc^{-1}$, where the magnitude of the observed power spectrum Fig. 3 is much greater than the shot noise term (in this case $P_{\rm shot\;noise}\simeq 1000\,(h^{-1}\rm Mpc)^{3}$). From Fig. 2 we see that as our quadratic estimator extracts large scale information, the cells with large over- and under-densities are especially well reconstructed. The difference with the true field becomes slightly larger when we go to higher redshift (corresponding to the panels on the right) and is worst for the very right panel.