Next-generation forecasts for screened and unscreened models of modified gravity

 

Santiago Casas, TTK, RWTH Aachen University

With collaboration from Euclid TWG WPs 1-6-7 and more...

 

 

Cosmic Microwave Background

Planck 2018 CMB Temperature map (Commander) .  wiki.cosmos.esa.int/planck-legacy-archive/index.php/CMB_maps

Large Scale Structure

Illustris Simulation: www.nature.com/articles/nature13316

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  • LCDM is still best fit to observations.
  • Concordance Cosmology:
  • Combination of different observables.
  • Lensing
  • CMB
  • Clustering
  • Supernovae
  • Clusters
G_{\mu \nu} + \Lambda g_{\mu \nu} = 8\pi G T_{\mu \nu}

The Standard \(\Lambda\)CDM model

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  • LCDM is still best fit to observations.
  • Some questions remain:
  • \(\Lambda\) and CDM.
  • Cosmological Constant Problem:

O(100) orders of magnitude wrong
(Zeldovich 1967, Weinberg 1989, Martin 2012).
 Composed of naturalness and coincidence
sub-problems, among others.

Quantum Gravity?

G_{\mu \nu} + \Lambda g_{\mu \nu} = 8\pi G T_{\mu \nu}

The Standard \(\Lambda\)CDM model

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The Standard \(\Lambda\)CDM model

  • \(\Lambda\)CDM is still best fit to observations.
  • Some questions remain:
  • H0 tension, now ~5\(\sigma\)
  • \(\sigma_8\) - \(\Omega_m\)  discrepancy at ~\(2\sigma\)

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Alternatives to \(\Lambda\)CDM

Ezquiaga, Zumalacárregui, Front. Astron. Space Sci., 2018

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Constraints on Theories

  • Background is well constrained to be around \(w=-1\)
  • Gravitational Wave speed = c
  • Galaxy morphology and solar system
  • Black holes
  • Coupling to baryons
  • Non-linear regime still pretty much unconstrained
  • Fifth forces
  • Neutrinos?

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Scalar field models

At lowest order in the perturbation of the scalar field \( \varphi \equiv \phi-\phi_0\)

 

  • Matter is coupled to the perturbed Jordan metric
  • In the case of a cosmological-stress energy tensor and a non-negligible scalar field mass :

 

\mathcal{L}_2=\frac{ Z(\phi_0)}{2} (\partial_\mu\varphi)^2 +\frac{ m^2_\phi(\phi_0)}{2} \varphi^2 - \delta g_{\mu\nu} \delta T^{\mu\nu}

Review: Brax, Casas, Desmond, Elder, arXiv:2201.10817

G_{\rm eff}=\left(1+ \frac{2\beta^2(\phi_0)}{Z(\phi_0)}e^{-m(\phi_0)r}\right) G_N

Yukawa term: Short range forces

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Screened Scalar fields

Different types of screening:

  • Chameleon: The mass \(m(\phi_0)\) increases sharply inside matter
  • Damour-Polyakov: The coupling \(\beta(\phi_0)\) vanishes inside matter
  • K-mouflage and Vainshtein: \(Z(\phi_0) \gg 1\)

 

G_{\rm eff}=\left(1+ \frac{2\beta^2(\phi_0)}{Z(\phi_0)}e^{-m(\phi_0)r}\right) G_N

Review: Brax, Casas, Desmond, Elder, arXiv:2201.10817

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Screened models

Z(\phi_0)= 1 + a(\phi_0) r_c^2 \frac{\Box\varphi}{m_{\rm Pl}} + b(\phi_0) \frac{(\partial \varphi)^2}{\Lambda^4}+ c(\phi)\frac{\Box^2 \varphi}{\Lambda^5}+\dots

As an effective field theory, the normalization factor can be expanded in a power series:

K-mouflage:  first derivative term \( \partial \varphi / \Lambda^2 \) dominates, which implies:

\vert \vec \nabla \Phi_N \vert \ge \frac{\Lambda^2}{2\beta(\phi_0) m_{\rm Pl}}

Screening where Newtonian acceleration \( a= - \vec \nabla \Phi_N \) large enough

Vainshtein:  second derivative term \( \Box \varphi \) dominates, which implies:

\nabla^2 \Phi_N \ge \frac{1}{2\beta(\phi_0) r_c^2}

Screening where spatial curvature is large

When the \( \Box^2 \varphi \) dominates → massive gravity

Review: Brax, Casas, Desmond, Elder, arXiv:2201.10817

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Screened models

\nabla^k \Phi_N \gtrsim C
  • Chameleon: \(k=0\) (surface N. potential is large)
  • K-mouflage: \(k=1\) (N. acceleration is large)
  • Vainshtein: \(k=2\) (curvature is large)

 

To summarize, screening mechanisms can be characterized by the inequality:

For DE applications and under some assumptions:

  • Chameleon screens everything above a certain potential threshold
  • K-mouflage does not screen galaxy clusters
  • Vainshtein screens all structures that turn non-linear

Review: Brax, Casas, Desmond, Elder, arXiv:2201.10817

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Examples of screened models

  • Chameleon: \( f(R) \) Hu-Sawicki
  • K-mouflage: \(k\)-essence + universal coupling
  • Vainshtein:  nDGP (3+1)d brane embedded in 5d

 

  • Solar system and other local constraints and instabilities forbid self-acceleration in these models
  • \(\Lambda\)CDM-like background
  • Just one free parameter each
  • Universal couplings

Review: Brax, Casas, Desmond, Elder, arXiv:2201.10817

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f(R) Hu-Sawicki model

Modification of the Einstein-Hilbert action

\newcommand{\sg}{\ensuremath{\sigma_{8}}} \newcommand{\de}{\mathrm{d}} S = \frac{c^4}{16\pi G} \int{\de^4 x \sqrt{-g} \left[R+f(R)\right]}

Induces changes in the gravitational potentials *

*for negligible matter anisotropic stress

-k^2\Psi =\frac{4\pi\,G}{c^4} \,a^2\mu\bar\rho\Delta\,
-k^2\left(\Phi+\Psi\right) = \frac{8\pi\,G}{c^4}\,a^2 \Sigma \bar\rho\Delta

Scale-dependent growth of matter perturbations

Small changes in lensing potential

\mu(a,k) = \frac{1}{1+f_R(a)}\frac {1+4k^2a^{-2}m_{f_R}^{-2}(a) }{1+3k^2a^{-2}m_{f_R}^{-2}(a)}
\Sigma(a)=\frac{1}{1+f_{R}(a)}\,

Free parameter: \(f_{R0}\)

f(R) = - 6 \Omega_{\rm DE} H_0^2 + |f_{R0}| \frac{\bar R_0^2}{R}\,

Hu, Sawicki (2007)

"Fifth-force" scale for cosmological densities

\(\lambda_C =32 \rm{Mpc}\sqrt{|f_{R0}|/10^{-4}}\)

Euclid: Casas et al (2022) in preparation

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f(R) as a scalar field theory

V_{\rm eff}(\phi)= V(\phi) +(A(\phi)-1) \rho
\tilde{g}_{\mu \nu} = A^2(\phi,X) g_{\mu \nu} + B^2(\phi,X) \partial_\mu \phi \partial_\mu \phi
A(\phi)= e^{\beta \phi/m_{\rm Pl}}
\frac{df}{dR}= e^{-2\beta \phi/_{\rm m_{Pl}}}
V(\phi)= \frac{m^2_{\rm Pl}}{2} \frac{ R \frac{df}{dR}-R}{(\frac{df}{dR})^2}

Universal coupling through a conformal transformation between Einstein and Jordan metrics

General Chameleon scalar models are given by specifying \(V\) and \(A\)

  • With a coupling function:
  • Map to a scalar field by:
  • Carefully chosen potential can realize chameleon mechanism:
     
  • Objects screened when:
\textcolor{green}{\Phi_N\gtrsim \frac{3}{2}\vert f_{R_0} \vert}

Review: Brax, Casas, Desmond, Elder, arXiv:2201.10817

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f(R) Hu-Sawicki predicitons

Codes used: for linear perturbations: MGCAMB and EFTCAMB

Scale-dependent growth

Fitting formula for non-linear power spectrum:

Winther, Casas, Baldi, Koyama, Li  (2019)

*Forge Emulator not available at time of first review

Euclid: Casas et al (2022) in preparation

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Scale-independent models

Euclid: WP1-WP6 et al (2022) in preparation

nDGP, K-mouflage and Jordan-Brans-Dicke have scale-independent growth

"Extreme cases" far away from LCDM and close to current upper bounds

Preliminary

Preliminary

nDGP: free parameter \(\Omega_{rc}\)  (related to the transition scale)

KM: free parameter \(\epsilon_2\)  (related to the conformal coupling amplitude)

JBD: free parameter \(\omega_{BD}\)  (related to the scalar coupling)

ReACT

HMCode

Halo+PT

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Scale-independent models

  • K-Mouflage presents a large enhancement of the lensing potential
  • Definitely detectable with next-generation WL observations

Preliminary

Euclid: WP1-WP6 et al (2022) in preparation

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Next-generation Galaxy Surveys

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DESI telescope

  • 14 000 square degrees in the sky
  • 30 million accurate galaxy spectra
  • Redshifts: 0 < z < 2
  • Quasars up to z~3.5
  • 5 years of observation

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Vera Rubin Observatory

  • Located in Chile, 8.4m telescope
  • 20 billion galaxies
  • Redshifts: 0 < z ~< 3
  • 18,000 square degrees
  • 11 years of observation

Santiago Casas, Oslo, 23.10.2020

Euclid Space Satellite

  • Two instruments:
  •  VIS (visible photometer): shape and orientation of 1.5 billion galaxies!
  •  NISP (near infrared spectrograph): 30 million galaxy spectra!
  • 15 000 square degrees in the sky
  •  16 countries, ~1500 members
  •  ~170 Petabyte of data!

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Photometric cross-correlations

Euclid preparation: VII. Forecast validation for Euclid cosmological probes.  arXiv:1910.09273

Also known as 3x2pt analysys

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Weak Lensing

C_{ij}^{\gamma\gamma}(\ell) = \frac{c}{H_0} \int{\frac{{\hat{W}}_i^\gamma(z) {\hat{W}}_{j}^\gamma(z)}{E(z) r^2(z)} P_{\Phi+\Psi}\left ( k_{\ell}, z \right ) dz}
P_{\Phi+\Psi} = \left[3\left(\frac{H_0}{c}\right)^2\Omega_\mathrm{M}^{0} (1 + z) \Sigma(k,z) \right ]^2 P_\mathrm{\delta \delta}

The cosmic shear angular power spectrum depends on the Weyl spectrum (of gravitational potentials \(\Phi+\Psi \))

Which is related to the matter power spectrum (of density contrast \(\delta \)) through

Information about background geometry, matter content and clustering

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Spectroscopic Galaxy Clustering

BAO

Clustering

RSD

Spec-z

Euclid preparation: VII. Forecast validation for Euclid cosmological probes.  arXiv:1910.09273

Slides by Dennis Linde, RWTH

Spectroscopic Galaxy Clustering

For mildly non-linear scales we need to use perturbation theory

  • SPT is not enough
  • EFT smoothing over terms ->  UV counterterms
    (see previous talk by Filippo)
  • Already being implemented into MCMC and Fisher pipelines like CosmicFish and CLOE

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The Matter Power Spectrum

Current data:

Image: https://www.cosmos.esa.int/web/planck/picture-gallery

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The Matter Power Spectrum

Euclid:

Scales from: ~ \(10^{-3}\) to \(10\) hMpc\(^{-1}\)

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Euclid: IST:Forecasts

  • Here: Flat \(w_0 w_a\mathrm{CDM}\)
  • GCsp+WL+GCph+XC
     
  • Figure of Merit:
    1257
  • Non-flat FoM:
    500
     
  • Optimistic:
    \(\sigma_{w_0}=0.025\)
    \(\sigma_{w_a}=0.092\)

Euclid preparation: VII. Forecast validation for Euclid cosmological probes.  arXiv:1910.09273

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Forecasts for f(R) from Euclid probes

Euclid: Casas et al (2022) in review

  • Combined constraints from GCsp and Photo probes

Preliminary

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Forecasts for f(R) from Euclid probes

Euclid: Casas et al (2022) in review

  • Combined constraints from GCsp and Photo probes

Preliminary

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Forecasts for f(R) from Euclid probes

Euclid: Casas et al (2022) in review

  • Transform into original space
  • Current LSS data: "just" upper bounds of the order of \(< 10^{-4}\)

Preliminary

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Forecasts for nDGP

Preliminary

  • GCsp does not constrain the free parameter very well
  • Most gain is at NL-scales for Photo probes

Euclid: WP1-WP6 et al (2022) in preparation

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Forecasts for K-Mouflage

Preliminary

  • For KM1:
    GCsp can constrain the free parameter at ~10%
  • Photo at ~1% (remember \(\Sigma_{WL}\) )
  • KM2 is basically \(\Lambda\)CDM, non detectable

Euclid: WP1-WP6 et al (2022) in preparation

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Bonus: Forecasts on parameterized MG

Casas, Pettorino, Camera, Martinelli, Carucci (in preparation)

Preliminary

  • DESI+Rubin have similar power than Euclid alone (under many assumptions)
  • Optical + Radio is also competitive and can remove systematics / degeneracies
  • Constraints on \(\mu, \, \Sigma \) of the order of ~5-10% under optimistic assumptions
  • PPN-approach screening assumed

Santiago Casas, ADE Marseille, May 2022

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Conclusions

  • Screening mechanisms can save scalar field models
  • Current constraints don't allow for self-acceleration
  • Screening mechanisms can be classified by the derivative order
  • Euclid and next-generation surveys will be powerful probes for Cosmology.
  • Primary LSS probes: Galaxy Clustering and Weak Lensing
  • Many challenges ahead in non-linear modelling
  • Next-generation surveys can constrain free parameters with percent precision accuracy

Thanks!
Merci!

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