Large 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

What happened between z=1000 and z=0?

Cosmologists looked up:

What happened between z=1000 and z=0?

(Image credit: DESI Collaboration/KPNO/NOIRLab/NSF/AURA/P. Horálek/R. Proctor)

• Positions of galaxies in the sky (angles)

• Redshifts of galaxies (photo or Spectro)

• Shapes (ellipticities of galaxies)

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

Expresses the excess probabilty of finding another galaxy as a function of scale

Strong hint that some physical mechanism is at play!!

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

Shape and orientation of galaxies is also correlated -- due to Weak Gravitational Lensing -- and this tells us about the content of (dark) mass-energy in the Universe

Some successful (STAGE-III) missions exploring these two aspects so far:

DES (Dark Energy Survey) and KiDS (Kilo-degree survey): photometric WL and Galaxy Clustering (angular)

SDSS, BOSS, eBOSS:

spectroscopic galaxy clustering

DES Y3 + KIDS-1000: CONSISTENT COSMOLOGY COMBINING COSMIC SHEAR SURVEYS (2305.17173v2)

Beyond - ΛCDM constraints from the full shape clustering measurements
from BOSS and eBOSS (2210.07304)

Everything roughly so far so good....

• $\Lambda$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 sound-horizon-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

Stage-IV surveys to the rescue?

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!

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.

DESI Data Release 1

5000 robotically controlled optical fibers

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

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

Vera Rubin LSST

DESI Baryon Acoustic Oscillations

In a nutshell: Measure a geometric scale that was imprinted on LSS at recombination

https://data.desi.lbl.gov/doc/papers/

Another tension we need to explain?

• 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

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

• 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 (flat)
• Non-flat FoM:
  500
• Optimistic flat:
  $\sigma_{w_0}=0.025$
  $\sigma_{w_a}=0.092$

Euclid preparation: VII. Forecast validation for Euclid cosmological probes, Blanchard et al. arXiv:1910.09273

Awardees of the Euclid STAR Prize Team 2019

Dynamical Dark Energy

• 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 (flat)
• Non-flat FoM:
  500
• Optimistic flat:
  $\sigma_{w_0}=0.025$
  $\sigma_{w_a}=0.092$

Awardees of the Euclid STAR Prize Team 2019

Euclid preparation: VII. Forecast validation for Euclid cosmological probes, Blanchard et al. arXiv:1910.09273

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

how do we do these forecasts again?

spectroscopic probe

BAO / AP-effect

Clustering

RSD

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

Euclid preparation: VII. Forecast validation for Euclid cosmological probes, Blanchard et al. arXiv:1910.09273

FoG

BAO-damping / IR-resummation

photometric primary probe

Euclid preparation: VII. Forecast validation for Euclid cosmological probes, Blanchard et al. arXiv:1910.09273

photometric primary probe

• Photometric galaxy density distribution.
• Bins convolved with photometric redshift PDF
• 10-15 bins in future Stage-IV surveys

• Window functions (Kernels projecting correlations into angular space) -> Limber
•  Cosmic Shear Kernels are "delocalized" in z -> k
• Localization techniques such as BNT to allow for clear cuts

Euclid: Forecasts for $k$-cut $3\times 2$ Point Statistics, P. Taylor, V. Cardone, ..., SC, et al.  2012.04672

Fisher Matrix Forecasts

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

Euclid preparation: VII. Forecast validation for Euclid cosmological probes, Blanchard et al. arXiv:1910.09273

Fisher Matrix Forecasts

Fisher Matrix for a Gaussian likelihood of angular power spectra:

Parameter covariance:

Defines an ellipse:

Euclid preparation: VII. Forecast validation for Euclid cosmological probes, Blanchard et al. arXiv:1910.09273

Fisher Matrix Forecasts

Euclid preparation: VII. Forecast validation for Euclid cosmological probes, Blanchard et al. arXiv:1910.09273

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

Can we trust Fisher constraints?

Euclid: Validation of the MontePython forecasting tools, 2303.09451 , SC, Lesgourgues, Schöneberg, et al.

• Validated IST:F Fisher forecasts against MCMCs with MontePython -> Gaussian posteriors for $w_0, w_a$
• It turns out Fisher approx. is good for a full 6yr data release of Euclid-like specs (optimistic and pessimistic).
• We also validated CAMB and CLASS to each other, the main Einstein-Boltzmann-Solvers on the market.
• We tested precise numerical accuracy of non-linear methods (Halofit and HMCode) and found differences in CAMB vs. CLASS.

The effect of Neutrinos

Simulations and images by my bachelor student: Yun Ling (RWTH), using CoNCEPT, developed by  J. Dakin (Aarhus, U. Zürich)

Can we trust Fisher constraints also in this case?

The effect of Neutrinos

Euclid Preparation: Sensitivity to Neutrino parameters. (Under internal review). Archidiacono, Lesgourgues, SC, Pamuk, et al.

• Neutrinos free stream below a scale of $k_{FS}$.
• They do not cluster below that scale.
• Suppression of the power spectrum.
• The non-relativistic transition imposes a minimum in $k_{FS}$

Dynamical Dark Energy + Neutrinos

Euclid Preparation: Sensitivity to Neutrino parameters. (Under internal review). Archidiacono, Lesgourgues, SC, Pamuk, et al.

• Use more "realistic" settings (HMcode, P_cb, no-fluid approx....) :
• $\sigma_{w_0} \approx 0.04$
  $\sigma_{w_a} \approx 0.13$
• $m_{\nu} \lt 200 \rm{ meV}$
• Euclid + Planck
• $\sigma_{w_0} \approx 0.04$
  $\sigma_{w_a} \approx 0.12$,
• $m_{\nu} \approx 60 \pm 40 \, \rm{ meV}$

Fisher is not so reliable anymore, with 7, 8 or 9 free cosmological parameters!!

Dynamical Dark Energy + Neutrinos

Euclid Preparation: Sensitivity to Neutrino parameters. (Under internal review). Archidiacono, Lesgourgues, SC, Pamuk, et al.

Let's go to even more realistic settings

Euclid: Likelihood and Nonlinear Challenges

• 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 error bars can distinguish among them
• Biases in parameter estimation to be resolved

Credit: P. Carrilho

Credit: SC, for IST:NL

Cosmological Likelihood for Observables in Euclid

Euclid: Likelihood and Nonlinear Challenges

Credit: SC, for IST:L

Cosmological Likelihood for Observables in Euclid

Weak Lensing

(Cosmic Shear)

+ Baryonic feedback

+ Intrinsic alignment

+ multiplicative bias

+ dnz uncertainties

 = 34 parameters

for LCDM flat

6x8 cores x 6 days

spectroscopic probe: Full Shape

Linde, Moradinezhad, Rademacher, SC, Lesgourgues (2402.09778)

The more-realistic GCspectro model, based on Senatore, Ivanov, Simonovic, Vlah, et al

CLASS 1-loop Code in development in Aachen, RWTH

Validated against CLASS-PT, Velocileptors

1-loop PT of density and velocity in redshift space

4 counterterms, 4 shot-noise, 4 higher-order biases

• Trade-off: larger error bars, more accuracy, less biasing
• Compared against IST:F model at different scales
• Many free parameters, good fit to ABACUS simulations

spectroscopic probe: Full Shape

• DESI full shape forecasts for w0, wa
• BGS and ELG
• As a function of k_max

Let's go Back to modified gravity

a SCALAR-tensor theory

Gregory Horndeski
https://www.horndeskicontemporary.com/works

And in the Lorentz Institute seminar room!

Costa Rica - Arenal Volcano

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

Gregory Horndeski
https://www.horndeskicontemporary.com/works

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!

Parametrized approach 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 $\Sigma$ (40% relative error)
• GC is better at measuring $\mu$ (20% relative error)
• SKAO-all-probes constrains at 3-5% relative error
• At Planck best fits

• 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 (2021), Review: Testing Screened Modified Gravity

• 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

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

Brax, SC, Desmond, Elder 2201.10817,  Universe 8 (2021), Review: Testing Screened Modified Gravity

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

f(R) Hu-Sawicki model

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

Non-linear matter power spectrum:

Winther et al (2019) fitting formula

Scale-dependent growth, change in forecasting pipeline

Current Euclid KP-JC6-SP paper in preparation (ledy by Kazuya), investigating biasing by Emulators/ReACT compared to simulations

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

Text

f(R) Hu-Sawicki model

 $\sigma_{\log f_{R0}}=0.05$ (1%)

Current cosmological limits ~approx:

$|f_{R0}|<10^{-6}$

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 (numerical)

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

Text

f(R) Hu-Sawicki model

Euclid preparation: Simulations and nonlinearities beyond ΛCDM
 Constraints on f (R) models from the photometric primary probes, Kazuya, SC, Pamuk, Bose, et al, (in prep.)

• What happens if we go to MCMC, can we trust our modelling?
• Not only fitting formulae, but many emulators available in the community
• Disagreements above the predicted Euclid error bars
• Baryonic modelling still unsufficient (and too degenerate) for full-analysis

Text

Scale-independent Modifications of GR

Forecasts for:

• Jordan-Brans-Dicke  (no-screening)
• nDGP (5D brane model)
• k-Mouflage
• We work mostly in QSA MGCAMB or hiclass and non-linear modelling: Halo model or fits to simulations

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

Current cosmo bounds $\omega_{BD} \gtrapprox 1000$, GR: $\omega_{BD} \rightarrow \infty$

$r_c = G_5 / 2G_N \, , \Omega_{rc} \equiv c^2 / (4 r_c^2 H_0^2 )$

Current cosmo bounds $\Omega_{rc} \lessapprox 0.27$, GR: $r_{c} \rightarrow \infty$

Vary just $\epsilon_{2,0}$ which in the limit $\epsilon_{2,0} \rightarrow 0$ turns the Kinetic term into a cosmological constant

Current cosmo bounds $−0.04 \lessapprox \epsilon_{2,0} \lessapprox 0$

Perform forecasts for limits close-to and far-from LCDM

Text

Scale-independent Modifications of GR

Example of result for JBD $\sigma(\log \omega_{BD})$:

(other results see paper)

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

 Full Euclid:

• JBD1: $\omega_{BD} = 800 ^{+200}_{-160}$
• JBD1: $\omega_{BD} = 2500 ^{+1070}_{-750}$

CosmicFish Code

Code: CosmicFish

S.Casas, M.Martinelli and M.Raveri

Soon to be released: New full pythonic version

https://github.com/santiagocasas/cosmicfishpie

Fisher Information Matrix:

Curvature (Hessian) of the Likelihood

Example: Fisher Matrix for a Gaussian likelihood of angular power spectra:

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 1-100 seconds (depending on model -> too slow!!
  
   
•  Automatic-Differentiable (AD) codes: Leverage variational methods, such as Hamilton MC, AD-Variational Inference, Exact Fisher and beyond

In collaboration with Johanna Schaffmeister and Sven Günther

jaxcosmo library https://github.com/DifferentiableUniverseInitiative

Campagne, Lanusse, Zuntz, SC, et al, 2302.05163

Text

Conclusions

• $\Lambda$CDM paradigm still best fit to observations, but tensions, unknowns and discrepancies start to show cracks in 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.
   
• Big Data challenge to be resolved with ML and AI
• Unknown unknowns awaiting for us in the data !

Merci!!

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?

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

photometric primary probe

Directly constrains MG function $\Sigma$ through Weyl potential

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

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

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

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