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Planck 2015 results. XIV. Dark energy and modified gravity

Planck Collaboration, P. A. R. Ade, N. Aghanim, M. Arnaud, M. Ashdown, J. Aumont, C. Baccigalupi, A. J. Banday, R. B. Barreiro, N. Bartolo, E. Battaner, R. Battye, K. Benabed, A. Benoît, A. Benoit-Lévy, J. -P. Bernard, M. Bersanelli, P. Bielewicz, A. Bonaldi, L. Bonavera, J. R. Bond, J. Borrill, F. R. Bouchet, M. Bucher, C. Burigana, R. C. Butler, E. Calabrese, J. -F. Cardoso, A. Catalano, A. Challinor, A. Chamballu, H. C. Chiang, P. R. Christensen, S. Church, D. L. Clements, S. Colombi, L. P. L. Colombo, C. Combet, F. Couchot, A. Coulais, B. P. Crill, A. Curto, F. Cuttaia, L. Danese, R. D. Davies, R. J. Davis, P. de Bernardis, A. de Rosa, G. de Zotti, J. Delabrouille, F. -X. Désert, J. M. Diego, H. Dole, S. Donzelli, O. Doré, M. Douspis, A. Ducout, X. Dupac, G. Efstathiou, F. Elsner, T. A. Enßlin, H. K. Eriksen, J. Fergusson, F. Finelli, O. Forni, M. Frailis, A. A. Fraisse, E. Franceschi, A. Frejsel, S. Galeotta, S. Galli, K. Ganga, M. Giard, Y. Giraud-Héraud, E. Gjerløw, J. González-Nuevo, K. M. Górski, S. Gratton, A. Gregorio, A. Gruppuso, J. E. Gudmundsson, F. K. Hansen, D. Hanson, D. L. Harrison, A. Heavens, G. Helou, S. Henrot-Versillé, C. Hernández-Monteagudo, D. Herranz, S. R. Hildebrandt, E. Hivon, M. Hobson, W. A. Holmes, A. Hornstrup, W. Hovest, Z. Huang, K. M. Huffenberger, G. Hurier, A. H. Jaffe, T. R. Jaffe, W. C. Jones, M. Juvela, E. Keihänen, R. Keskitalo, T. S. Kisner, J. Knoche, M. Kunz, H. Kurki-Suonio, G. Lagache, A. Lähteenmäki, J. -M. Lamarre, A. Lasenby, M. Lattanzi, C. R. Lawrence, R. Leonardi, J. Lesgourgues, F. Levrier, A. Lewis, M. Liguori, P. B. Lilje, M. Linden-Vørnle, M. López-Caniego, P. M. Lubin, Y. -Z. Ma, J. F. Macías-Pérez, G. Maggio, N. Mandolesi, A. Mangilli, A. Marchini, P. G. Martin, M. Martinelli, E. Martínez-González, S. Masi, S. Matarrese, P. Mazzotta, P. McGehee, P. R. Meinhold, A. Melchiorri, L. Mendes, A. Mennella, M. Migliaccio, S. Mitra, M. -A. Miville-Deschênes, A. Moneti, L. Montier, G. Morgante, D. Mortlock, A. Moss, D. Munshi, J. A. Murphy, A. Narimani, P. Naselsky, F. Nati, P. Natoli, C. B. Netterfield, H. U. Nørgaard-Nielsen, F. Noviello, D. Novikov, I. Novikov, C. A. Oxborrow, F. Paci, L. Pagano, F. Pajot, D. Paoletti, F. Pasian, G. Patanchon, T. J. Pearson, O. Perdereau, L. Perotto, F. Perrotta, V. Pettorino, F. Piacentini, M. Piat, E. Pierpaoli, D. Pietrobon, S. Plaszczynski, E. Pointecouteau, G. Polenta, L. Popa, G. W. Pratt, G. Prézeau, S. Prunet, J. -L. Puget, J. P. Rachen, W. T. Reach, R. Rebolo, M. Reinecke, M. Remazeilles, C. Renault, A. Renzi, I. Ristorcelli, G. Rocha, C. Rosset, M. Rossetti, G. Roudier, M. Rowan-Robinson, J. A. Rubiño-Martín, B. Rusholme, V. Salvatelli, M. Sandri, D. Santos, M. Savelainen, G. Savini, B. M. Schaefer, D. Scott, M. D. Seiffert, E. P. S. Shellard, L. D. Spencer, V. Stolyarov, R. Stompor, R. Sudiwala, R. Sunyaev, D. Sutton, A. -S. Suur-Uski, J. -F. Sygnet, J. A. Tauber, L. Terenzi, L. Toffolatti, M. Tomasi, M. Tristram, M. Tucci, J. Tuovinen, L. Valenziano, J. Valiviita, B. Van Tent, M. Viel, P. Vielva, F. Villa, L. A. Wade, B. D. Wandelt, I. K. Wehus, M. White, D. Yvon, A. Zacchei, A. Zonca

TL;DR

Planck 2015 data are used to systematically test dark energy and modified gravity theories beyond the cosmological constant. The authors employ two complementary modeling routes: background parameterizations that modify the expansion history and perturbation parameterizations (EFT/Horndeski and phenomenological mu/eta) that alter how perturbations evolve, complemented by concrete models like f(R) and coupled DE. Across data combinations, background-only models remain consistent with LCDM, while perturbation-focused tests reveal mild tensions that largely dissolve once CMB lensing is included. Overall, Planck strengthens LCDM constraints, with external probes helping to break degeneracies and further tighten the bounds on DE and MG scenarios; future data will be pivotal to probe more complex theories.

Abstract

We study the implications of Planck data for models of dark energy (DE) and modified gravity (MG), beyond the cosmological constant scenario. We start with cases where the DE only directly affects the background evolution, considering Taylor expansions of the equation of state, principal component analysis and parameterizations related to the potential of a minimally coupled DE scalar field. When estimating the density of DE at early times, we significantly improve present constraints. We then move to general parameterizations of the DE or MG perturbations that encompass both effective field theories and the phenomenology of gravitational potentials in MG models. Lastly, we test a range of specific models, such as k-essence, f(R) theories and coupled DE. In addition to the latest Planck data, for our main analyses we use baryonic acoustic oscillations, type-Ia supernovae and local measurements of the Hubble constant. We further show the impact of measurements of the cosmological perturbations, such as redshift-space distortions and weak gravitational lensing. These additional probes are important tools for testing MG models and for breaking degeneracies that are still present in the combination of Planck and background data sets. All results that include only background parameterizations are in agreement with LCDM. When testing models that also change perturbations (even when the background is fixed to LCDM), some tensions appear in a few scenarios: the maximum one found is \sim 2 sigma for Planck TT+lowP when parameterizing observables related to the gravitational potentials with a chosen time dependence; the tension increases to at most 3 sigma when external data sets are included. It however disappears when including CMB lensing.

Planck 2015 results. XIV. Dark energy and modified gravity

TL;DR

Planck 2015 data are used to systematically test dark energy and modified gravity theories beyond the cosmological constant. The authors employ two complementary modeling routes: background parameterizations that modify the expansion history and perturbation parameterizations (EFT/Horndeski and phenomenological mu/eta) that alter how perturbations evolve, complemented by concrete models like f(R) and coupled DE. Across data combinations, background-only models remain consistent with LCDM, while perturbation-focused tests reveal mild tensions that largely dissolve once CMB lensing is included. Overall, Planck strengthens LCDM constraints, with external probes helping to break degeneracies and further tighten the bounds on DE and MG scenarios; future data will be pivotal to probe more complex theories.

Abstract

We study the implications of Planck data for models of dark energy (DE) and modified gravity (MG), beyond the cosmological constant scenario. We start with cases where the DE only directly affects the background evolution, considering Taylor expansions of the equation of state, principal component analysis and parameterizations related to the potential of a minimally coupled DE scalar field. When estimating the density of DE at early times, we significantly improve present constraints. We then move to general parameterizations of the DE or MG perturbations that encompass both effective field theories and the phenomenology of gravitational potentials in MG models. Lastly, we test a range of specific models, such as k-essence, f(R) theories and coupled DE. In addition to the latest Planck data, for our main analyses we use baryonic acoustic oscillations, type-Ia supernovae and local measurements of the Hubble constant. We further show the impact of measurements of the cosmological perturbations, such as redshift-space distortions and weak gravitational lensing. These additional probes are important tools for testing MG models and for breaking degeneracies that are still present in the combination of Planck and background data sets. All results that include only background parameterizations are in agreement with LCDM. When testing models that also change perturbations (even when the background is fixed to LCDM), some tensions appear in a few scenarios: the maximum one found is \sim 2 sigma for Planck TT+lowP when parameterizing observables related to the gravitational potentials with a chosen time dependence; the tension increases to at most 3 sigma when external data sets are included. It however disappears when including CMB lensing.

Paper Structure

This paper contains 41 sections, 51 equations, 22 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Why is the CMB relevant for dark energy?
  3. Models and parameterizations
  4. Background parameterizations
  5. Perturbation parameterizations
  6. Modified gravity and effective field theory
  7. MG and phenomenological parameterizations
  8. Examples of particular models
  9. Data
  10. Planck data sets
  11. Planck low-$\ell$ data
  12. Planck high-$\ell$ data
  13. Planck CMB lensing
  14. Planck CMB polarization
  15. Background data combination
  16. ...and 26 more sections

Figures (22)

  • Figure 1: Typical effects of modified gravity on theoretical CMB temperature (top panel) and lensing potential (bottom panel) power spectra. An increase (or decrease) of $E_{22}$ with respect to zero introduces a gravitational slip, higher at present, when $\Omega_{\mathrm{de}}$ is higher (see Eq. (\ref{['eq:mudef']}) and Eq. (\ref{['eq:etadef']})); this in turns changes the Weyl potential and leads to a higher (or lower) lensing potential. On the other hand, whenever $E_{11}$ and $E_{22}$ are different from zero (quite independently of their sign) $\mu$ and $\eta$ change in time: as the dynamics in the gravitational potential is increased, this leads to an enhancement in the ISW effect. Note also that even when the temperature spectrum is very close to $\rm{\Lambda CDM}$ (as for $E_{11} = E_{22} = 0.5$) the lensing potential is still different with respect to $\rm{\Lambda CDM}$, shown in black.
  • Figure 2: $\Omega_{\rm m}$--$\sigma_8$ constraints for tomographic lensing from Heymans:2013fya, using a very conservative angular cut, as described in the text (see Sect. \ref{['sec:gallenssec']}). We show results using linear theory, nonlinear corrections from Halofit (HL) versions 1, 4, marginalization over baryonic AGN feedback (BF), and intrinsic alignment (IA) (the latter two using nonlinear corrections and Halofit 4). Coloured points indicate $H_0$ values from WL+HL4.
  • Figure 3: Parameterization $\{w_0, w_a\}$ (see Sect. \ref{['sec:w0wa']}). Marginalized posterior distributions for $w_0$, $w_a$, $H_0$ and $\sigma_8$ for various data combinations. The tightest constraints come from the Planck TT+ lowP+BSH combination, which indeed tests background observations, and is compatible with $\Lambda$CDM.
  • Figure 4: Marginalized posterior distributions of the ($w_0, w_a$) parameterization (see Sect. \ref{['sec:w0wa']}) for various data combinations. The best constraints come from the priority combination and are compatible with $\rm{\Lambda}$CDM. The dashed lines indicate the point in parameter space $(-1,0)$ corresponding to the $\Lambda$CDM model. CMB lensing and polarization do not significantly change the constraints. Here Planck indicates Planck TT+ lowP.
  • Figure 5: Reconstructed equation of state $w(z)$ as a function of redshift (see Sect. \ref{['sec:w0wa']}), when assuming a Taylor expansion of $w(z)$ to first-order ($N = 1$ in Eq. \ref{['eq:w_expansion']}), for different combinations of the data sets. The coloured areas show the regions which contain 95% of the models. The central blue line is the median line for Planck TT+ lowP+BSH. Here Planck indicates Planck TT+ lowP.
  • ...and 17 more figures