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Euclid preparation. Galaxy power spectrum and bispectrum modelling

Euclid Collaboration, K. Pardede, A. Eggemeier, D. Alkhanishvili, E. Sefusatti, A. Moradinezhad Dizgah, L. Christoph, A. Chudaykin, M. Kärcher, D. Linde, M. Marinucci, C. Porciani, A. Veropalumbo, M. Crocce, M. S. Cagliari, B. Camacho Quevedo, L. Castiblanco, E. Castorina, G. D'Amico, V. Desjacques, A. Farina, G. Gambardella, M. Guidi, J. Lesgourgues, C. Moretti, A. Pezzotta, A. Pugno, J. Salvalaggio, B. Altieri, S. Andreon, N. Auricchio, M. Baldi, S. Bardelli, P. Battaglia, A. Biviano, M. Brescia, S. Camera, G. Cañas-Herrera, V. Capobianco, C. Carbone, V. F. Cardone, J. Carretero, S. Casas, M. Castellano, G. Castignani, S. Cavuoti, K. C. Chambers, A. Cimatti, C. Colodro-Conde, G. Congedo, C. J. Conselice, L. Conversi, Y. Copin, F. Courbin, H. M. Courtois, A. Da Silva, H. Degaudenzi, S. de la Torre, G. De Lucia, H. Dole, F. Dubath, X. Dupac, S. Escoffier, M. Farina, R. Farinelli, F. Faustini, S. Ferriol, F. Finelli, P. Fosalba, S. Fotopoulou, N. Fourmanoit, M. Frailis, E. Franceschi, M. Fumana, S. Galeotta, K. George, B. Gillis, C. Giocoli, J. Gracia-Carpio, A. Grazian, F. Grupp, S. V. H. Haugan, W. Holmes, F. Hormuth, A. Hornstrup, K. Jahnke, M. Jhabvala, B. Joachimi, S. Kermiche, A. Kiessling, B. Kubik, M. Kunz, H. Kurki-Suonio, A. M. C. Le Brun, S. Ligori, P. B. Lilje, V. Lindholm, I. Lloro, G. Mainetti, D. Maino, E. Maiorano, O. Mansutti, S. Marcin, O. Marggraf, K. Markovic, M. Martinelli, N. Martinet, F. Marulli, R. J. Massey, E. Medinaceli, S. Mei, M. Melchior, Y. Mellier, M. Meneghetti, E. Merlin, G. Meylan, A. Mora, M. Moresco, L. Moscardini, C. Neissner, S. -M. Niemi, J. W. Nightingale, C. Padilla, S. Paltani, F. Pasian, K. Pedersen, W. J. Percival, V. Pettorino, S. Pires, G. Polenta, M. Poncet, L. A. Popa, F. Raison, A. Renzi, J. Rhodes, G. Riccio, E. Romelli, M. Roncarelli, R. Saglia, Z. Sakr, A. G. Sánchez, D. Sapone, B. Sartoris, P. Schneider, A. Secroun, G. Seidel, S. Serrano, E. Sihvola, P. Simon, C. Sirignano, G. Sirri, A. Spurio Mancini, L. Stanco, J. Steinwagner, P. Tallada-Crespí, I. Tereno, N. Tessore, S. Toft, R. Toledo-Moreo, F. Torradeflot, I. Tutusaus, J. Valiviita, T. Vassallo, G. Verdoes Kleijn, Y. Wang, J. Weller, G. Zamorani, F. M. Zerbi, E. Zucca, V. Allevato, M. Ballardini, A. Boucaud, E. Bozzo, C. Burigana, R. Cabanac, M. Calabrese, A. Cappi, T. Castro, J. A. Escartin Vigo, L. Gabarra, J. García-Bellido, V. Gautard, S. Hemmati, J. Macias-Perez, R. Maoli, J. Martín-Fleitas, N. Mauri, R. B. Metcalf, P. Monaco, M. Pöntinen, I. Risso, V. Scottez, M. Sereno, M. Tenti, M. Tucci, M. Viel, M. Wiesmann, Y. Akrami, I. T. Andika, G. Angora, S. Anselmi, M. Archidiacono, F. Atrio-Barandela, E. Aubourg, L. Bazzanini, J. Bel, D. Bertacca, M. Bethermin, F. Beutler, A. Blanchard, L. Blot, M. Bonici, S. Borgani, M. L. Brown, S. Bruton, A. Calabro, F. Caro, C. S. Carvalho, F. Cogato, S. Conseil, A. R. Cooray, S. Davini, G. Desprez, A. Díaz-Sánchez, S. Di Domizio, J. M. Diego, V. Duret, M. Y. Elkhashab, A. Enia, Y. Fang, A. Finoguenov, A. Fontana, F. Fontanot, A. Franco, K. Ganga, T. Gasparetto, E. Gaztanaga, F. Giacomini, F. Gianotti, G. Gozaliasl, A. Gruppuso, C. M. Gutierrez, A. Hall, C. Hernández-Monteagudo, H. Hildebrandt, J. Hjorth, J. J. E. Kajava, Y. Kang, V. Kansal, D. Karagiannis, K. Kiiveri, J. Kim, C. C. Kirkpatrick, S. Kruk, M. Lattanzi, L. Legrand, M. Lembo, F. Lepori, G. Leroy, G. F. Lesci, T. I. Liaudat, S. J. Liu, M. Magliocchetti, C. J. A. P. Martins, L. Maurin, M. Miluzio, G. Morgante, S. Nadathur, K. Naidoo, P. Natoli, A. Navarro-Alsina, S. Nesseris, L. Pagano, D. Paoletti, F. Passalacqua, K. Paterson, L. Patrizii, R. Paviot, A. Pisani, D. Potter, G. W. Pratt, S. Quai, M. Radovich, K. Rojas, W. Roster, S. Sacquegna, M. Sahlén, D. B. Sanders, E. Sarpa, A. Schneider, D. Sciotti, E. Sellentin, L. C. Smith, J. G. Sorce, K. Tanidis, C. Tao, F. Tarsitano, G. Testera, R. Teyssier, S. Tosi, A. Troja, D. Vergani, F. Vernizzi, G. Verza, P. Vielzeuf, S. Vinciguerra, N. A. Walton, A. H. Wright

Abstract

Higher-order correlation functions of the large-scale galaxy distribution offer access to information beyond that contained in standard 2-point statistics such as the power spectrum. In this work we assess this potential for the $\textit{Euclid}$ mission using synthetic catalogues of H$α$ galaxies based on the 54 $\, h^{-3} \, {\rm Gpc}^3$ Flagship I simulation, designed to reproduce the $\textit{Euclid}$ spectroscopic sample. We comprehensively validate the one-loop galaxy power spectrum and tree-level bispectrum predictions from perturbation theory in both real and redshift space. Assuming scale cuts consistent with our previous power spectrum study on the same catalogues, this modelling yields unbiased cosmological constraints for the bispectrum up to $k_{\rm max} = 0.15\,\, h \, {\rm Mpc}^{-1}$ in real space and $0.08 \, (0.1)\,\, h \, {\rm Mpc}^{-1}$ at the lowest (highest) redshift, corresponding to $z=0.9$ ($z=1.8$), for the monopole and quadrupole in redshift space using statistical uncertainties corresponding to the full simulation volume. With these scale cuts, adding bispectrum information to the power spectrum improves constraints on the amplitude of scalar perturbations and the matter density by up to 30 %, increasing the overall figure of merit for key cosmological parameters by a factor of about 2.5. Similar conclusions hold when statistical uncertainties are rescaled to a $\textit{Euclid}$-like volume, highlighting the importance of the bispectrum for fully exploiting the forthcoming $\textit{Euclid}$ data. Our analysis also provides the first detailed characterisation of the nonlinear bias model of H$α$ emitters, showing that bias relations calibrated on low-resolution \textit{N}-body simulations do not adequately describe the clustering of H$α$ galaxies at low redshift, whereas excursion-set and co-evolution relations for tidal biases remain accurate.

Euclid preparation. Galaxy power spectrum and bispectrum modelling

Abstract

Higher-order correlation functions of the large-scale galaxy distribution offer access to information beyond that contained in standard 2-point statistics such as the power spectrum. In this work we assess this potential for the mission using synthetic catalogues of H galaxies based on the 54 Flagship I simulation, designed to reproduce the spectroscopic sample. We comprehensively validate the one-loop galaxy power spectrum and tree-level bispectrum predictions from perturbation theory in both real and redshift space. Assuming scale cuts consistent with our previous power spectrum study on the same catalogues, this modelling yields unbiased cosmological constraints for the bispectrum up to in real space and at the lowest (highest) redshift, corresponding to (), for the monopole and quadrupole in redshift space using statistical uncertainties corresponding to the full simulation volume. With these scale cuts, adding bispectrum information to the power spectrum improves constraints on the amplitude of scalar perturbations and the matter density by up to 30 %, increasing the overall figure of merit for key cosmological parameters by a factor of about 2.5. Similar conclusions hold when statistical uncertainties are rescaled to a -like volume, highlighting the importance of the bispectrum for fully exploiting the forthcoming data. Our analysis also provides the first detailed characterisation of the nonlinear bias model of H emitters, showing that bias relations calibrated on low-resolution \textit{N}-body simulations do not adequately describe the clustering of H galaxies at low redshift, whereas excursion-set and co-evolution relations for tidal biases remain accurate.

Paper Structure

This paper contains 30 sections, 45 equations, 11 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Theoretical model
  3. Definitions and conventions
  4. Power spectrum
  5. Bispectrum
  6. Infrared resummation
  7. Redshift-space multipoles
  8. Alcock--Paczyński effect
  9. Data
  10. Flagship I simulation
  11. Power spectrum and bispectrum measurements
  12. Binning effect
  13. Power spectrum and bispectrum covariance
  14. Methodology and performance metrics
  15. Fitting procedure
  16. ...and 15 more sections

Figures (11)

  • Figure 1: The first row shows, for the four redshifts, the dependence on the triangle shape of the bispectrum measurements in real space in terms of the reduced bispectrum $Q_{\rm g}$ of Eq. (\ref{['eq:Q_real']}), averaged over the values of $k_1$ from $0.02$ to $0.16\, h \, {\rm Mpc}^{-1}$, as a function of the ratios $x_2\equiv k_2/k_1$ and $x_3\equiv k_3/k_1$. The same quantity in redshift space for the reduced bispectra defined in Eq. (\ref{['eq:Q_redshift']}) is displayed in the third and fourth rows for monopole and quadrupole respectively, with $k_1$ averaged here from $0.02$ to $0.11\, h \, {\rm Mpc}^{-1}$. The second and fifth row present the $\mathrm{S}/\mathrm{N}$ as function of $k_{\rm max}$ in the power spectrum and bispectrum respectively in real space -- Eq. (\ref{['eq:StoN_P']}) and Eq. (\ref{['eq:StoN_B']}) -- and redshift space -- Eq. (\ref{['eq:StoNPell']}) and Eq. (\ref{['eq:StoNBell']}).
  • Figure 2: Results of the power spectrum and bispectrum analysis in real space assuming the maximal model as a function of the bispectrum scale cut $k_{\rm max}^{B}$ and for different values of the power spectrum scale cut $k_{\rm max}^{P}$, denoted by the colour gradient. Each column corresponds to a different redshift. The top three rows show posterior means and 68% credible intervals for the three cosmological parameters (slightly displaced along the $x$-axis for readability), while the bottom three show respectively the goodness of fit in terms of the $p$-value, the FoB, and FoM. While not explicitly plotted, we have verified that the maximum a posteriori (MAP) values for the cosmological parameters fall within the 68% credible intervals. The dashed lines in the top three rows denote the fiducial values of the three cosmological parameters. The grey bands in the $p$-value and FoB panels represent the 68 and 95 percentiles of the associated distributions. The FoM values are normalised to the value computed at $k_{\rm max}^{P}=0.1\, h \, {\rm Mpc}^{-1}$ and $k_{\rm max}^B=0.08\, h \, {\rm Mpc}^{-1}$.
  • Figure 3: Constraints on nonlinear galaxy bias parameters for four different redshift bins as a function of the corresponding recovered value of the linear bias. These are compared to the predictions of the bias relations: Eq. \ref{['eq:b2_hod']} in the first panel; Eqs. (\ref{['eq:bGtwo_LL']}, \ref{['eq:bGtwo_fit']}) (based on co-evolution and excursion set, respectively) in the second panel; and Eq. \ref{['eq:bG3_relation']} (based on co-evolution) in the third panel. The grey vertical bands correspond to the $b_1$ estimates (and their 1$\sigma$ uncertainty) from the galaxy and matter power spectra described in PezzottaEtal2024. While not shown explicitly, we have verified that the MAP values for the bias parameters fall within the 68% credible intervals.
  • Figure 4: Performance of various 8-parameter models obtained either by imposing a relation among the bias parameters or by setting $\alpha_2=0$ in Eq. \ref{['eq:B_stochastic']}, compared to the 9-parameter maximal model. First row: difference in the DIC with statistically significant deviations lying outside the grey band. Second row: effective number of parameters $p_V$ constrained by the data. Third and fourth rows: marginalised constraints on the parameters $b_1$ and $\alpha_2$ (the grey band in the $b_1$ panels is identical to that in Fig. \ref{['fig:bias_relations_fixed_cosmology']}). All results are shown as a function of $k_{\rm max}^{B}$ at fixed $k_{\rm max}^{P}=0.15\, h \, {\rm Mpc}^{-1}$, with columns corresponding to the four redshift bins.
  • Figure 5: Same as Fig. \ref{['fig:cosmo_no_relation']}, but for the results of the joint power spectrum and bispectrum analysis in redshift space. While identical scale cuts are applied to the bispectrum multipoles, i.e., $k_{\rm max}^{B} \equiv k_{\rm max}^{B_0} = k_{\rm max}^{B_2}$, the power spectrum multipoles satisfy $k_{\rm max}^{P} \equiv k_{\rm max}^{P_0} = k_{\rm max}^{P_2} = k_{\rm max}^{P_4} + 0.05 ~ \, h \, {\rm Mpc}^{-1}$. In each redshift bin the FoM reference value is computed for the case $k_{\rm max}^{P}=0.1\, h \, {\rm Mpc}^{-1}$ and $k_{\rm max}^B=0.06\, h \, {\rm Mpc}^{-1}$. Similarly to Fig. \ref{['fig:cosmo_no_relation']}, we have checked that the MAP values for the cosmological parameters fall within the 68% credible intervals.
  • ...and 6 more figures