The Marked Power Spectrum as a Practical Bispectrum Measure for Galaxy Redshift Surveys

Abstract

Modern datasets have the precision necessary to uncover new information by including higher-order, non-Gaussian information into cosmological inference. The marked power spectrum offers access to such information while preserving the structure of two-point correlators. This approach to higher-order statistics has the advantage that many modeling questions can directly benefit from progress already made in standard cosmological analyses using the power spectrum and correlation function, while increasing the data vector size negligibly and retaining much of the degeneracy-breaking power of the bispectrum. In this work, we first restructure the marked power spectrum to isolate its higher-order information and demonstrate its ability to break parameter degeneracies. We then investigate the effect of survey geometry on the marked power spectrum and find that a treatment similar to that of the power spectrum is sufficient. Additionally, we investigate the perturbative modeling and covariance structure of the marked power spectrum, shedding light on its degeneracy breaking power and cross-covariance with the power spectrum. Finally, we demonstrate that the cosmology dependence of the marked power spectrum is smooth, indicating that cosmological inference is possible by modeling the cosmology dependence through interpolation rather than analytical modeling.

Figures (8)

• Figure 1: The three diagrams contributing to higher-order information in the MPS for a cross-spectrum between marked field $\delta_M$ at $\mathbf{x}_1$ and unmarked field $\delta_g$ at $\mathbf{x}_2$. The red squares and circles represent the raw and smoothed overdensity operators $\delta_g$ and $\delta_{g,R}$, respectively. The operators on the left of each diagram belong to the marked field, $\delta_M$, and those on the right belong to the unmarked field, $\delta_g$. The left two diagrams contribute to $M_{13}$, whereas the rightmost diagram contributes to $M_{22}$. Notice that all diagrams include a $\delta^{(2)}\supset Z_2$, giving rise to a "leading-order" dependence on $b_2$ and $b_s$ that will be important later.
• Figure 2: Left: The change in $M_0$ (with $R=15h^{-1}\,\mathrm{Mpc}$ at $z=0.7$) from shifts in second-order bias $b_2$ (blue line; $\Delta b_2 = 2$) and shear bias $b_s$ (orange; $\Delta b_s =1$) of size equal to the $1\,\sigma$ uncertainty from current generation surveys Chudaykin25a. The $b_2$ dependence dominates despite both biases being quadratic in $\delta$. Overlaid on the $b_2$ curve is the error observed from $8\,h^{-3}\,\mathrm{Gpc}^3$ simulations (black error bars; §\ref{['sec:periodicbox']}) and the anticipated maximum effect of higher-loop contributions (blue shade; §\ref{['sec:nuisance']}) with $A_0 \lesssim0.11$, both of which are subdominant to the expected $b_2$ change. The dotted black line shows $1.7$ times the linear power spectrum $P_L$ demonstrating both that the shape of $\Delta M_0$ is degenerate with $P_L$ (Appendix \ref{['sec:Mshape']}) and that the higher-loop correction $A_0 P_L$ is subdominant by over an order of magnitude to the $b_2$ change. Right: The power spectrum and $R=15h^{-1}\,\mathrm{Mpc}$ marked power spectrum of two nuisance parameter sets $\{\theta_\mathrm{LRG}\}$ and $\{\theta_\mathrm{fit}\}$ at $z=0.7$. $\theta_\mathrm{LRG}$ is the LRG-like bias parameters that fit the DESI DR1 cutsky mock at $z=0.7$ (§\ref{['sec:cutsky']}) with $b_2\approx0.5$, while $\theta_\mathrm{fit}$ is another nuisance set with nearly identical power spectrum predictions but with $b_2=-1.5$. The similarity in $P_\ell(k)$ captures the $\Delta b_2\sim2$ fully expected in near-term redshift surveys, while $M_0(k)$ differs by a factor of 2, showing its potential to assist standard two-point analyses. The dotted vertical line indicates the $k_\mathrm{max}^M=0.12\,h^{-1}\,\mathrm{Mpc}$ used for this work.
• Figure 3: Some pieces of the Fourier-space DR1 cutsky window matrix $(\mathcal{W}_{\ell \ell'})_{ij}$ (Eqns. \ref{['eqn:windowP']} and \ref{['eqn:windowM']}). We choose $\ell=0$ and select $k_i$, for bins of width $\Delta k\sim0.01$. The real-space window matrix is shown in Fig. 6 of ref. Chaussidon25.
• Figure 4: $P_\ell$ and $M_0$ joint-fits to the average of 25 simulations in $8\,h^{-3}\,\mathrm{Gpc}^3$ periodic boxes, with each column corresponding to smoothing scales $R=10$, 15, $20\, h^{-1}\,\mathrm{Mpc}$, and each row corresponding to number densities $\bar{n}=10^{-3}$ and $3\times10^{-4} \, h^{3}\,\mathrm{Mpc}^{-3}$. In the top panels, the theory fits are in solid lines and simulation data points in circular markers. The bottom panels show the residuals of the fits using the standard deviation of simulations as errors and we add gray bands to demonstrate the $1\sigma$ range. For all panels, blue, orange, and red colors represent $P_0$, $P_2$, and $M_0$, respectively, and the fits adopt $k_\mathrm{max}^P=0.2\,h\,\mathrm{Mpc}^{-1}$ and $k_\mathrm{max}^M=0.12\,h\,\mathrm{Mpc}^{-1}$. The successful $<1\sigma$ fit for all cases demonstrate that the theoretical precision is enough to be adopted for near-term datasets. Furthermore the fits at different number densities indicate that potential stochastic effects are degenerate with other parameters and can be neglected (§\ref{['sec:stochasticity']}).
• Figure 5: The $P$-$M$ joint fits to DESI DR1 cutsky mocks in the LRG2 ($0.6<z<0.8$) bin with varying smoothing scales $R=10, 15,20\, h^{-1}\,\mathrm{Mpc}$. The top panels show the fits directly, with the window-convolved and unconvolved theory shown in solid and dashed lines against the mock measurements in circular markers. The bottom panels show the residual of the fits using simulation errors, with the $1\sigma$ range highlighted by the gray band. The fits are $<1\sigma$ for all cases, demonstrating that the survey geometry is propagated to the spectrum at precision satisfactory for any near-future data.