What will Large Scale Structure tell US about Dark Energy in the NEXT DECADE?

Santiago Casas

Postdoctoral Researcher

TTK, RWTH Aachen University

     @santiagocasas                                                                                        @sant87casas

https://www.pablocarlosbudassi.com/2021/02/the-infographic-and-artistic-work-named.html

https://www.esa.int/Science_Exploration/Space_Science/Euclid/Euclid_test_images_tease_of_riches_to_come

Do galaxies just randomly spread out across the sky?

https://www.esa.int/Science_Exploration/Space_Science/Euclid/Euclid_test_images_tease_of_riches_to_come

No they do not, there is actually a 2-point correlation (and higher orders) among them

Which expresses the excess probabilty of finding another galaxy as a function of scale

And tells us that some physical mechanism is at play

http://caastro.org/research-highlights/science-animations/

In this 2-point correlation function we can see geometric features that are directly related to the expansion history of the Universe

https://www.esa.int/Science_Exploration/Space_Science/Euclid/Euclid_test_images_tease_of_riches_to_come

It turns out that shape and orientation of galaxies is also correlated -- due to Weak Gravitational Lensing -- and this tells us about structures in the Universe

Euclid Space Satellite

Credits: www.esa.int/Science_Exploration/Space_Science/Euclid, www.euclid-ec.org, ESA/NASA/SpaceX, Euclid Consortium

Sun-Earth Lagrange point 2, 1.5 million km from Earth

Euclid consortium scientist visits Cannes. Credits: ThalesAlenia Space

The Euclid Consortium fingertip galaxy,  thanks to the contribution of many scientists within the EC, courtesy of Lisa Pettibone, Tom Kitching and ESA

Euclid Space Satellite

• Two instruments:
•  VIS (visible photometer): shape and orientation of  ~1 billion galaxies!
•  NISP (near infrared spectrograph): ~20 million galaxy spectra !
• 6 years nominal mission

• 15 000 square degrees in the sky
•  16 countries, ~1500 members
•  ~170 Petabyte of data!

VIS cosmic shear map

https://www.euclid-ec.org/blog/

Euclid preparation: I. The Euclid Wide Survey of ESA, R. Scaramella et al.

VIS cosmic shear map

Vera Rubin LSST

• Located in Chile, 8.4m telescope
• 3.2 Gigapixel camera!
• 20 billion galaxies
• Redshifts: 0 < z ~< 3  (photometric)
• 18,000 square degrees
• 11 years of observation
• 3x2pt, clusters, SNIa and more!

By LSST Project Office - http://www.lsst.org/gallery/telescope-rendering-2013, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=42054166

Vera Rubin LSST

DESI telescope

• 14 000 square degrees in the sky
• 20-40 million accurate spectra
• Redshifts: 0 < z < 2 (spectroscopic)
• Quasars up to z~3.5
• 5 years of observation
• Starting 2021
• Power spectra, Bispectra, Corr. Func.

Just 7 months of DESI data

5000 robotically controlled optical fibers

Tucson, Arizona, in the Schuk Toak District on the Tohono O’odham Nation

Credits: https://www.desi.lbl.gov

Just 7 months of DESI data

Vera Rubin LSST

Square kilometer array (SKAO)

• SKA Phase 1: SKA1-Low and SKA1-Mid
• SKA1-Low: 130,000 dipole antennas, 65km max. baseline (Australia)
• SKA1-Mid: ~200 dishes of ~15m diameter, max. baseline 150km (South Africa)
• Precursors: ASKAP, MEERKAT, HERA...
   
• €1.3 Billion, 16 countries, 710 Petabytes, 8 years construction

https://www.skao.int/

https://www.esa.int/Science_Exploration/Space_Science/Euclid/Euclid_test_images_tease_of_riches_to_come

With Stage-IV surveys we will have \(\approx 10^9\) galaxy shapes with photometric (approximate) and \(\approx 10^6\) (precise) redshifts and positions

Only 1/64th of the complete Euclid field of view is represented here, which in turn is equivalent to a mere quarter of the apparent size of the Moon. Consider the vast expanse of 15,000 square degrees, encompassing one-third of the entire sky!

How to measure:

• Positions of galaxies in the sky (angles)

• Redshifts of galaxies (photo or Spectro)

• Shapes (ellipticities of galaxies)

HUGE Scale StructureS

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

However, the Universe is not as nice and simple. It is a non-linear mess:

Dark Matter

Baryons

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

What happened in between, if in its infancy it was a fairly Gaussian, linearly perturbed, homogenous and isotropic Universe?

• \(\Lambda\)CDM is best fit to observations.
• Predictive model with few free parameters.
• Based on Einstein's GR:

https://www.cosmos.esa.int/web/planck/publications

• \(\Lambda\)CDM is still best fit to observations.
• Some questions remain:
• \(\Lambda\) and CDM.
• CDM hasn't been found directly
• Cosmological Constant Problem:

Tensions IN \(\Lambda\)CDM

\(H_0\) tension at 5\(\sigma\)

• Tensions between local and recombination-based measurements

Freedman et al

SH0ES, Riess et al

Planck 2018, VI

Tension with Planck in the
\(\sigma8\) - \(\Omega_m\) plane

Lange et al. arXiv: 2301.08692

• \( S_8 = \sigma_8 \sqrt{\Omega_{m,0}/0.3} \)
• So called "lensing is low" problem or S8 problem.
• At the moment just a discrepancy (no tension) at 2-3 \(\sigma\)
• Blind comparisons among surveys can rule out usual systematics below z<0.54 (A. Leauthaud et al.)
• Beyond \(\Lambda\)CDM modelling does not help with current nonlinear analysis

Planck 2018, VI

DES DRY3 arxiv:2207.05766

Tensions IN \(\Lambda\)CDM

• CMB LEnSING and ITS Cross-Correlation with LSS

• Cosmic VOIDS

• NON-Gaussian and HIGHER ORDER STatistics

• CLUSTERS of GALAXIES

• ....

We have many problems/Mysteries in cosmology

BUT let's focus on a classical one

why is the universe accelerating?

(assuming we don't like \(\Lambda\))

First Step: Modify Gravity

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

Modified Gravity and Dark Energy

• Add a scalar degree of freedom to the Einstein-Hilbert action
• (Non-)Minimal coupling, Kinetic terms

a SCALAR-tensor theory

• Horndeski theory: most general theory for an extra scalar degree of freedom, with second order equations of motion in 4D and no ghosts
• 5 Lagrangian terms, which have been largely ruled out by:
  • Late-ISW
  • GW170817
  • No self-acceleration -> uninteresting

Dynamical Dark Energy

• Dark Energy: Any dynamical cause for the accelerated expansion.

 Euclid was commissioned to test with observations the most common parametrization (CPL):

And possible deviations of the growth of structures

M. Chevallier and D. Polarski (2001), and E. Linder (2003),
Cosmology and Fundamental Physics with the Euclid Satellite (Amendola et al , Living Reviews in Relativity, 2018)

Dynamical Dark Energy

Euclid?

• Current bounds from CMB and LSS alone
•  \(\mathcal{O}(1)\) for \(w_0\) combining Planck+GC+WL+SNIa
• \(\mathcal{O}(0.01)\) with Euclid

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

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

Awardees of the Euclid STAR Prize Team 2019

Dynamical Dark Energy

Wolf, Ferreira 2310.07482

Problems with Dynamical DE CPL parametrization:

• It is just a first order Taylor expansion around \(a=1\)
• It is underdetermined, prior- and data-dependent
• parts of \( w_0 - w_a \) space do not match any meaningful model
• Many physical Quintessence models can recover same parameters even if we find \( w_0 \neq -1 ,  w_a \neq 0 \)

Raveri et al, 2107.12990

Parametrized modified gravity

Beyond \(\Lambda\)CDM the two linear gravitational potentials \(\Psi\) and \(\Phi\) are not equal to each other

We can describe general modifications of gravity (of the metric) at the linear perturbation level with 2 functions of scale (\(k\)) and time (\(a\))

Only two independent functions!

Another approach now also for perturbations:

Parametrized modified gravity

• Using Planck satellite data in 2015 and 2018, constraints were obtained on these two functions \(\mu\) and \(\eta\).
• Late-time parametrization: dependent on Dark Energy fraction

Parametrized modified gravity

Forecasts for Stage-IV : Euclid, DESI, SKA1, SKA2, only GC and WL

SC, Kunz, Martinelli, Pettorino, Phys.Dark Univ. 18 1703.01271

• Study more general case of free \(\mu\), \(\eta\) functions at each bin
• Non-linearities crucial for constraints
• Z-PCA decorrelation
• Constraints on \(\mu\), \(\eta\) of O(10%)

• For late-time and early-time scenario:
  Constraints on \(\mu\), \(\eta\) of O(3%)
• When combining GC+WL+Planck priors

Vera Rubin LSST

Radio x Optical Cosmology

SC, Carucci, Pettorino et al (2022) 2210.05705

• WL is better at measuring \(\Sigma\)
• GC is better at measuring \(\mu\)
• SKAO-all-probes constrains at 3% already

• DESI(GCsp)xSKAO(IM) helps in \(h, \sigma_8\) but not in MG parameters
• Combination of SKAO + one Stage-IV probe is as good as two Stage-IV
• Different noise and systematics -> break degeneracies

Classifying by screening

• Modified Gravity theories should recover GR at small scales, especially in dense regions, screening needed!

Brax, SC, Desmond, Elder 2201.10817  Universe 8

• Perturbations of \(\phi\) and matter coupled in Jordan frame -> Yukawa-like fifth forces:

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: \(Z(\phi_0) \gg 1\)

Classifying by screening

Brax, SC, Desmond, Elder 2201.10817  Universe 8

Screening mechanisms can be characterized by the inequality:

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

• \( f(R) \) Hu-Sawicki  (see later)
• \(k\)-essence + univ. coupling
• nDGP (3+1)d brane embedded in 5d

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

Text

SC et al, Euclid: Constraints on f(R) cosmologies from the spectroscopic and photometric primary probes, 2306.11053

f(R) Hu-Sawicki model

Modification of the Einstein-Hilbert action

Induces changes in the gravitational potentials *

Scale-dependent growth of matter perturbations

Small changes in lensing potential

Free parameter: \(f_{R0}\)

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

"Fifth-force" scale for cosmological densities

Hu, Sawicki (2007)

Text

SC et al, Euclid: Constraints on f(R) cosmologies from the spectroscopic and photometric primary probes, 2306.11053

f(R) Hu-Sawicki model

Codes used: for background and scale-dependent linear perturbations: MGCAMB and EFTCAMB

For non-linear power spectrum:

Winther et al (2019) fitting formula

Text

SC et al, Euclid: Constraints on f(R) cosmologies from the spectroscopic and photometric primary probes, 2306.11053

f(R) Hu-Sawicki model

 \(\sigma_{\log f_{R0}}=0.05\) (0.9%)

Full probe combination, optimistic Euclid constraints:

\(f_{R0}=(5.0^{+ 0.58}_{-0.52} \times 10^{-6})\)

Paper also contains impact of:

• Non-linear scales
• Cross-correlations
• Pessimistic settings
• LCDM-limit

Text

Scale-independent Modifications of GR

Forecasts for:

• Jordan-Brans-Dicke
• nDGP (5D brane model)
• k-Mouflage

Fruciante, Pace, Cardone, SC et al, Euclid: Constraining linearly scale-independent modifications of
gravity with the spectroscopic and photometric primary probes, 2306.12368

how do we do these forecasts again?

photometric primary probe

Directly constrains MG function \(\Sigma\) through Weyl potential

spectroscopic probe

BAO

Clustering

RSD

Euclid Collaboration, IST:Forecasts, arXiv: 1910.09273

Euclid will also measure the 2pt corr-func of spectroscopic galaxies in redshift space

spectroscopic probe

Rademacher, Linde, Lesgourgues, Moradinezhad, SC in prep.

The realistic GCspectro model:

CLASS 1-loop Effective Field Theory of LSS

Vera Rubin LSST

Square kilometer array (SKAO)

• Modelling is very analogous to GCsp, with brightness temperature on top and different biases
• GCsp-IM Cross-correlation in overlapping bins
• DESI : Two galaxy samples
• SKAO: HI Galaxies and 21cm-IM

\(P^{\rm IM}(z,k) = \bar{T}_{IM}(z)^2 \rm{AP}(z) K_{\rm rsd}^2(z, \mu; b_{\rm HI}) \)
\(FoG(z,k,\mu_\theta) \\ \times P_{\delta\delta,dw}(z,k)  \)

\( K_{\rm rsd}(z, \mu; b_{\rm HI}) = [b_{\rm HI}(z)^2+f(z)\mu^2] \)

\( b_{\rm HI}(z) = 0.3(1+z) + 0.6 \)

\( \bar{T}_{\mathrm{IM}}(z)= 189h \frac{(1+z)^2 H_0}{H(z)}\Omega_{HI}(z) \,\,{\rm mK} \)

\(\Omega_{HI}  = 4(1+z)^{0.6} \times 10^{-4} \)

Carucci et al (2020) 2006.05996

Jolicoeur et al (2020) 2009.06197

\(P^{{\rm IM} \times \rm{g}}(z,k) = \bar{T}_{\rm IM}(z) {\rm AP} (z) r_{\rm IM,opt}  K_{\rm rsd}(z, \mu; b_{\rm HI}) \)
\( \times K_{\rm rsd}(z, \mu; b_{\rm g}) FoG(z,k,\mu_\theta) P_{\delta\delta,dw}(z,k) \)

\( \times \exp[-\frac{1}{2} k^2 \mu^2 (\sigma_{\rm IM}(z)^2+\sigma_{\rm sp}(z)^2)]  \)

SC, Carucci, Pettorino et al (2022) 2210.05705

Brightness temperature of 21cm emission line

Fraction of neutral hydrogen in the Universe

Fisher Matrix Forecasts

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

Bayes Theorem:

Probability of the model parameters given the data

Fisher Information Matrix:

Curvature (Hessian) of the Likelihood

Gaussian Likelihood in data space:

How do we actually perform those forecasts?

J. Schaffmeister

Fisher Matrix Forecasts

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

Fisher Matrix for a Gaussian likelihood of angular power spectra:

Parameter covariance:

Defines an ellipse:

CosmicFish Code

Code: CosmicFish

S.Casas, M.Martinelli and M.Raveri

Soon to be released: New full pythonic version

https://github.com/CosmicFish/CosmicFish

Euclid likelihood NONLINEAR IST:NL

• Group of ~10 people
  In charge of implementing the non-linear prescriptions for IST:Likelihood
• GCspectro: EFT 1-loop RSD in multipoles
• 3x2pt photo:
  • Covariance, nonlinear Super-Sample
  • Bias expansion
  • Emulators for nonlinear, HMCode, Halofit, Bacco, EuclidEmu
  • Baryonic Feedback

• Different emulators with systematic offsets
• Euclid optimistic error bars can distinguish them
• Biases in parameter estimation to be resolved
• S.Casas, P. Carrilho, C. Moretti to lead KP papers

machine Learning and variational methods

• In era of Big Data (billions of raw data points) and hundreds of possible models and simulations:
  • Important to be as computationally efficient as possible
     
• Likelihood codes for Euclid 2-3s per evaluation, Einstein-Boltzmann codes 1-10 seconds (depending on model)
• Emulators for linear and nonlinear codes (CLASS, N-body, Lensing)
  • Bayesian Neural Networks give also uncertainty estimations
     
•  Automatic-Differentiable (AD) codes: Leverage variational methods, such as Hamilton MC, AD-Variational Inference, Exact Fisher and beyond

Text

Conclusions

• \(\Lambda\)CDM paradigm still best fit to observations, but tensions, unknowns and discrepancies start to crack the model.
• Several avenues of research important: Modified Gravity, cosmological Neutrinos, Early-Late time interaction.
• Stage-IV surveys, infrarred, optical and radio will improve error bars by one order of magnitude.
• Theoretical modelling (especially at non-linear scales) has to keep up with accuracy.
• Combination and synergies of surveys will break degeneracies.
• Big Data challenge needs to be resolved with ML and AI
• Unexpected unknowns awaiting for us in the data this decade!

BACKUP

the road ahead of horndeski

The "surviving Horndeski" Lagrangian:

In the EFT formalism, FLRW, linear and
(unitary gauge time \(\rightarrow \phi\) ) :

Parametrize free functions and check for stability in solutions

We have shown that certain classes of models will not be distinguishable from LCDM, even with future surveys,  at 1\(\sigma\), while others will be measured with 10%-60% precision in their parameters

Frusciante, Peirone, SC, Lima, 1810.10521, Phys.Rev.D 99

Credits: Tobias Liaudat, CosmoStat

Euclid Shape Pipeline

Credits: Rodlophe Cledassou, CNES

Euclid

doi: 10.1146/annurev.nucl.012809.104521

Age of the Universe

Temperature of the Universe

• Planck measurements, essentially cosmic-variance limited in TT
• Percent precision in cosmological parameters
• Degeneracies in \(\Omega_k\), \(\Omega_m\) and \(H_0\) broken with BAO and lensing
• \(H_0 = 67.37 \pm 0.54 \, \textrm{km/s/Mpc}\)

Planck 2018 results. VI. Cosmological parameters https://arxiv.org/abs/1807.06209

https://www.cosmos.esa.int/web/planck

Possible solutions and new directions

• \(\Lambda\) ?
• Inflation ?
   
• \(m_{\nu}\) ?
   
• LSS  ?
  
   
• Big data?

Outline of working fields

• Coupled Quintessence
• Higgs-Dilaton
• Modified Gravity
• Horndeski/EFToDE
   
• EFToLSS
• Emulators
   
• Machine Learning
• Variational inference
• Stage-IV surveys
  Radio and Optical

• In general: confidence on the distance ladder
• Concerns on calibration, anchors dust or
  metallicity, have basically
  faded out
• Riess et al 2022*:
  \(H_0 = 73.30 \pm 1.04 \, \textrm{km/s/Mpc}\)
  
  using Cepheids and
  high-z SNIa

* The Astrophysical Journal Letters, 934:L7 (52pp), 2022 July 20

Neutrinos Masivos en Cosmologia

Neutrinos masivos: Free streaming

CMB angular spectrum and matter power spectrum are both dependent on neutrino mass, N_eff and ordering

Linear Perturbations

Vlasov-Poisson system is a set of diff.eqn. in which all matter-radiation species are coupled

Slides by: Dennis Linde

f(R) Hu-Sawicki model

Text

Modification of the Einstein-Hilbert action

Induces changes in the gravitational potentials *

Scale-dependent growth of matter perturbations

Small changes in lensing potential

Free parameter: \(f_{R0}\)

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

f(R) Hu-Sawicki model

Text

Euclid: Casas et al (2022) in preparation

Codes used: for background and scale-dependent linear perturbations: MGCAMB and EFTCAMB

For non-linear power spectrum:

Winther et al (2019) fitting formula