Phys. Rev. D 105, 123501 (2022) - Analysis method for 3D power spectrum of projected tensor fields with fast estimator and window convolution modeling: An application to intrinsic alignments

Analysis method for 3D power spectrum of projected tensor fields with fast estimator and window convolution modeling: An application to intrinsic alignments

Toshiki Kurita and Masahiro Takada

Phys. Rev. D 105, 123501 – Published 1 June 2022

Abstract

Rank-2 tensor fields of large-scale structure, e.g., a tensor field inferred from shapes of galaxies, open up a window to directly access 2-scalar, 2-vector, and 2-tensor modes, where the scalar fields can be measured independently from the standard density field that is traced by distribution of galaxies. Here we develop an estimator of the multipole moments of coordinate-independent power spectra for the three-dimensional tensor field, taking into account the projection of the tensor field onto plane perpendicular to the line-of-sight direction. To do this, we find that a convenient representation of the power spectrum multipoles can be obtained by the use of the associated Legendre polynomials in the form which allows for the fast Fourier transform estimations under the local plane-parallel (LPP) approximation. The formulation also allows us to obtain the Hankel transforms to connect the two-point statistics in Fourier and configuration space, which are needed to derive theoretical templates of the power spectrum including convolution of a survey window. To validate our estimators, we use the simulation data of the projected tidal field assuming a survey window that mimics the BOSS-like survey footprint. We show that the LPP estimators fairly well recover the multipole moments that are inferred from the global plane-parallel approximation. We find that the survey window causes a more significant change in the multipole moments of projected tensor power spectrum at $k ≲ 0.1 h {Mpc}^{- 1}$ from the input power spectrum, than in the density power spectrum. Nevertheless, our method to compute the theory template including the survey window effects successfully reproduces the window-convolved multipole moments measured from the simulations. The analysis method presented here paves the way for a cosmological analysis using three-dimensional tensor-type tracers of large-scale structure for current and future surveys.

Received 28 February 2022
Accepted 12 May 2022

DOI:https://doi.org/10.1103/PhysRevD.105.123501

Physics Subject Headings (PhySH)

Large scale structure of the Universe

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Toshiki Kurita ^1,2,* and Masahiro Takada ¹

¹Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study (UTIAS), The University of Tokyo, Chiba 277-8583, Japan
²Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

^*toshiki.kurita@ipmu.jp

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Vol. 105, Iss. 12 — 15 June 2022

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Figure 1
A schematic illustration of setups used for validation of the LPP power spectrum estimators for the projected tensor field. Gray shaded region in each panel denotes a hypothetical survey window for which we take the BOSS Northern Galactic cap (NGC) footprint; its solid angle area $Ω_{sky} ≃ 8, 000 \deg^{2}$ and the radial width $Δ r = 500 h^{- 1} Mpc$ (see text for details). Each blue ellipse denotes a “shape” of the projected tidal field at its position; we define the field by projecting the 3D tidal field onto plane perpendicular to the LOS direction, denoted by the black arrow. Case (a): This setup is for validating the LPP estimators. For each pair taken in the power spectrum estimate, we use the local LOS direction, denoted as red arrow, to compute the phase factor and the associated Legendre polynomials, which are needed for estimation of the multipole moments (see text for details). Case (b): The definition of the projected tidal field is the same as Case (a), but we use the GPP estimator to measure the multipole moments of power spectra. The global LOS direction (red arrow) is set as the direction of the midpoint of the survey region, viewed by the observer position. Case (c): This is the case where we can employ the distant observer approximation. In this case we can use the global LOS direction to define the projected tidal field and to implement the GPP estimator. We use the same LOS direction as the red arrow in Case (b). To have a fair comparison, we employ the same survey window as Case (a) and (b). For all the three cases we use exactly the same simulation realizations of the underlying 3D tidal field.
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Figure 2
Left panel: the lowest-order multipole moment of the cross-power spectrum of the projected tidal field, $P_{γ δ}^{(2)}$ , measured using the LPP or GPP estimators in each of the three setups illustrated in Fig. 1. The blue circle symbols show the result for Case (a), i.e., from the LPP estimator developed in this paper, while the green-triangle symbols are the result for Case (c), the GPP estimator under the distant observer approximation. The orange inverted-triangle symbols are the result for Case (b), which uses the inconsistent LOS directions for the projected tensor field and for the power spectrum estimator. The dashed line denotes the input matter power spectrum from which the tidal field is simulated. Right panel: similar to the left panel, but the results for the quadrupole moment of the autopower spectrum of the projected tidal field, $P_{+}^{(2)} (k)$ . To make the comparison easier, we multiply all the results by normalization factor to match them with the input $P (k)$ amplitude (see text for details).
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Figure 3
Multipole moments of the autocorrelation function of survey window, $Q_{ℓ}$ [Eq. (53)], computed using the pair-counting method for the BOSS-like survey window in Fig. 1.
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Figure 4
The upper panels compare the multipole moments of power spectra, measured by the LPP estimator (points with error bars), with the theoretical predictions including the window effects (solid lines). The lower panels show the fractional differences. We plot the mean of 1000 realizations with error bar of the mean in each $k$ bin, and normalize all the moments as in Fig. 2. Left panel: the blue circles, orange inverted triangles and green triangles show the matter spectrum ${\tilde{P}}^{(0)}$ , the lowest-order $L = 2$ moment of cross spectrum ${\tilde{P}}_{γ δ}^{(L = 2)}$ and the lowest-order $L = 4$ moment of “minus”-auto spectrum ${\tilde{P}}_{-}^{(L = 4)}$ , respectively, which are the lowest order multipole moments for each spectrum. Right: the blue circles, orange inverted triangles, and green triangles show the $ℓ = 0$ , 2, 4-th moments of the “plus”-auto spectrum ${\tilde{P}}_{+}^{(ℓ)}$ , respectively. The dark and light gray-shaded regions in the lower panel correspond to 1% and 3% fractional differences.
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Figure 5
Similar to Fig. 4, but this plot shows the leakage moments that arises due to the anisotropy of BOSS-like survey window; in other words, the moments should be vanishing if there is no window effect or for an isotropic window. The blue circles and orange inverted triangles are the quadrupole moment of the matter spectrum ${\tilde{P}}^{(2)}$ and the $L = 4$ th moment of the cross spectrum ${\tilde{P}}_{γ δ}^{(L = 4)}$ , respectively. The blue open and blue dashed lines denote the negative values (we plot their absolute values). Note that the range of $y$ -axis is different from that in Fig. 4.
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Figure 6
Comparison of the BOSS-like survey window effect on the cross power spectrum $P_{γ δ}^{(L = 2)}$ with the modification in $P_{γ δ}^{(L = 2)}$ due to the spin-2 local-type primordial non-Gaussian (PNG) initial condition. Here we show the quadrupole moment of $P_{γ δ}$ because it is the lowest-order moment in our formulation and carries most of the PNG information. The blue curves show the power spectra in the Gaussian condition and the red curves are those in the presence of the spin-2 PNG. The solid (dashed) lines correspond to the results with (without) the window convolution. We chose $f_{NL}^{s = 2} = - 180$ for demonstration, because the effect without the window convolution (dashed-red curve) is at the comparable level with the Gaussian result with the window convolution (solid-blue curve).
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Physical Review D

covering particles, fields, gravitation, and cosmology

Analysis method for 3D power spectrum of projected tensor fields with fast estimator and window convolution modeling: An application to intrinsic alignments

Toshiki Kurita and Masahiro Takada

Phys. Rev. D 105, 123501 – Published 1 June 2022

Abstract

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Physical Review D

covering particles, fields, gravitation, and cosmology

Analysis method for 3D power spectrum of projected tensor fields with fast estimator and window convolution modeling: An application to intrinsic alignments

Toshiki Kurita and Masahiro Takada

Phys. Rev. D 105, 123501 – Published 1 June 2022

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