The cosmological power of a joint analysis of Euclid and CMB surveys 

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The early Universe and the CMB

The most widely accepted theory is that soon after the Big Bang, at a time t . 10 35 s, the Universe went through an accelerated expansion phase called inflation. Inflation was proposed in the 1980s (Guth 1981; Hawking 1982) to solve at least two major observational problems: the extreme flatness of the Universe, and the homogeneity of the CMB at distances that are not causally connected.
This inflation is also responsible for expanding some tiny quantum fluctuations up to large scales (Mukhanov and Chibisov 1981; Linde 1982). These fluctuations in density and temperature, on top of the homogeneous background, will grow with time to form the anisotropies of the CMB, and later on the large scale structures like galaxies and clusters of galaxies as we know them.
After this inflationary phase and during the radiation dominated era, the Uni-verse continues to expand and to cool down, but at a decelerated rate. Between t 3 and 20 minutes, light atoms like Hydrogen and Helium were created in what is called the Big-Bang nucleosynthesis. At this time, the energy of photons was sufficiently high to prevent electrons and atoms to bind. The Universe was totally opaque, due to the permanent Compton scattering of photons on free electrons. This plasma of baryonic matter and photons was in thermal equilibrium and experienced acoustic oscillations under to the competing effects of gravity and radiative pressure. Meanwhile the dark matter, interacting only by gravity, was not subject to radiative pressure and was thermally decoupled. At a redshift of z 1000 (around t 380; 000 years after the Big Bang) and at a temperature of T 3000 K, photons were cooled enough to allow electrons to recombine with atoms. After this recombination, photons were able to travel freely in the Universe, and they still are today. We are now bathing in a sea of photons coming from this moment of the Universe. This relic radiation is what we call the cosmic microwave background (CMB). The energy spectrum of the photons of the CMB is a perfect black-body, with a present day temperature of TCMB = 2:726 K. Observations show small anisotropies in the temperature, of the order of T =T = 10 5 at a scale of 1 degree. Figure 1.4 presents the temperature anisotropies of the CMB as observed by the Planck satellite. These anisotropies are the precursor of the present day large scale structures. After recombination the matter is almost unaffected by photons, and under the influence of gravity, baryons will fall into the gravitational potential wells that are formed by the dark matter.

Late-time anisotropies

The observed anisotropies of the CMB have multiple origins. They are a combina-tion of primordial and late-time anisotropies. Primordial anisotropies were seeded by the inflation and evolved under the competing effect of gravity and radiative pressure until the recombination. Late-time anisotropies are created by the interac-tion between the photons of the CMB and the structures they crossed during their travel. Late-time anisotropies are a secondary effect but they contain a lot of cos-mological information. Their analysis shed light on the evolution of the universe Figure 1.4: The 2018 Planck map of the temperature anisotropies of the CMB. The grey outline shows the extent of the confidence mask. Credits: ESA and the Planck Collaboration.
from the time of recombination until today. We describe below some of the most important sources of late-time anisotropies, also known as secondary anisotropies (see Aghanim et al. 2008, for a review).
Reionisation When the first stars start to shine and to emit photons powerful enough to ionize their environment, the photons from the CMB will be Thom-son scattered on these free electrons. The main effects are the suppression of the CMB anisotropy on scales smaller than the horizon size at the epoch of reionisa-tion, corresponding5 to ‘ > 20 (Bond and Efstathiou 1984; Vittorio and Silk 1984), and the creation of a bump in the polarization power spectrum at large scales, for ‘ < 20 (Liu et al. 2001). The impact of reionisation on the CMB power spectrum is often described with the Thomson optical depth parameter . Reionisation process may happen inhomogeneously in the Universe, inside growing bubbles centred on the most massive and most star forming galaxies. This patchy reionisation creates secondary anisotropy on small scales (around ‘ 1000) due to the scattering of CMB photons onto free electrons of ionised gas in galaxies that have a non zero proper velocity compared to the CMB rest frame (Aghanim et al. 1996).

Growth of inhomogeneities

Even if the Universe on large scales is well described by the Friedmann equations derived in Section 1.1.4, we saw that the CMB is anisotropic, and observation of galaxies showed that they are not homogeneously distributed on intermediate scales (e.g. de Lapparent et al. 1986). We derive in this section the general formalism to describe the growth of structures from the small initial perturbation that are present after the inflation up to the large scale structures as we know them. We start from a simple case: the Newtonian and linearised case. The Newtonian regime simplifies the equations of general relativity we saw in Section 1.1.4, and is valid as long as we consider a region smaller than the Hubble horizon cH 1. For a more detailed description of the hypotheses and derivations see e.g. Peebles (1980).
We define the comoving coordinates x of a position in space such that the physical coordinate is r = xa. The proper velocity relative to the origin is u = ax + v(x; t), and v is the peculiar velocity. We define the overdensity field as the fluctuation around the mean density (x; t) = (t) (1 + (x; t)) (1.11). We consider matter as an ideal fluid with a potential . The general fluid equa-tions in comoving coordinates are given by @v 1 (v r)v + a 1 rp 1 r ; (1.12) + v = @t a a a a @ 1 r (1 + ) v = (1.13).

Halo mass function

A first step to model the non linear regime is to study the collapse of a spherically symmetric density distribution. In this configuration, an exact solution of the evolu-tion equations can be obtained, assuming an Einstein-de Sitter universe (Gunn and Gott 1972). Tracking the evolution of the density inside a collapsing sphere with a constant mass, one can show that the sphere collapses (the radius tends to zero) when the linear overdensity reaches c = 1:686. Even if not mathematically correct (it is an extrapolation of the linear equations), the c value is used as a criterion to describe collapsed objects: each time an overdensity reaches this threshold, it can be considered as collapsed, and the linear equations does not hold anymore.
In practice the sphere will not collapse to a point, but the particles will relax and reach the virial equilibrium when the total energy equals half of the potential energy. The ratio between the density inside a virialized sphere and the mean density of the universe (for an Einstein-de Sitter universe, i.e. m = 1) is equal to (White 2001) V (r < rvir) =18 2’178: (1.43).

Bias of haloes and galaxies

Both dark matter haloes and galaxies are sampling the matter density field, but they do not sample it uniformly, we observe numerous galaxies in dense environment, and almost no galaxies in less dense environments. We say that haloes, and the galaxies they host, are biased tracers of the underlying matter density field. This idea was first described in Kaiser (1984) and refined in Bardeen et al. (1986).
The threshold of linear density c = 1:686 seen in Section 1.6.2 explains why haloes are biased tracers of the density field. Due to the presence of both large and short wavelength modes in the density field, a large overdensity inside a void will not collapse as it cannot reach the c threshold, while a small overdensity in a very dense region will collapse. As such haloes can collapse more easily in dense regions. The Figure 1.10 illustrate this fact. Haloes are then populated with galaxies, making galaxies another biased tracer of the underlying matter density field.
The galaxy bias is often assumed to be scale independent, which is true at large scales (McDonald and Roy 2009; Desjacques et al. 2018) and modelled as function of redshift. We can then express the galaxy 3D density field as g(z; n^) = b(z) (z; n^) ; (1.45).
with g the galaxy density field and the matter density field. We then get this expression for the power spectrum of galaxies Pg(z; k) = b2(z)P (z; k) ; (1.46).
We see here that the bias behaves like a scaling factor which will affect the amplitude of the power spectrum. As such it is highly degenerate with the cosmo-logical parameter which governs the amplitude of the initial power spectrum As (see Equation 1.26).
Equation 1.46 shows a simplified version of the impact of bias on the observed galaxy power spectrum. Cosmological effects such as primordial non-gaussianities, massive neutrinos, dark energy and modified gravity can complicate this picture, and make the bias vary with the scale. Redshift space distortions (RSD), as introduced below, will also impact this relation (see Desjacques et al. 2018, for a review).

Galaxy formation efficiency

Galaxies form inside dark matter haloes. In the hierarchical clustering scenario, small haloes are accreted onto more massive ones and become “sub-haloes”. Galaxies are classified as either “satellite” if they are hosted by a sub-halo, or “central”. Central galaxies correspond to the most massive galaxy in the halo, located near its center of mass. For the sake of simplicity, we often refer to any halo or sub-halo hosting a galaxy as a “halo”.
If we make the simple hypothesis that the stellar content of a halo is directly proportional to its mass, the ratio of stellar mass to halo mass would be a con-stant. However, we do not observe a straightforward linear relation between the dark matter halo mass (Mh) and the stellar mass of a galaxy (M ).
This can be seen quantitatively if one compares the dark matter halo mass func-tion and the galaxy stellar mass function, see Figure 2.2. The stellar mass function (SMF) gives the number of galaxies per unit volume and per unit of mass. It has traditionally been modelled by a Schechter (1976) function, although for certain galaxy populations a combination of more than one such functions may provide a better fit to observations (Binggeli et al. 1988; Kelvin et al. 2014).
In Figure 2.2 where we compare the stellar mass function to the halo mass function, we see that there is a deficit of galaxies with a low stellar mass (M < 1011M ) and with a high stellar mass (M > 1011 M ), compared to the number density of dark matter haloes (see also e.g. Cole et al. 2001; Yang et al. 2003; Eke et al. 2006; Behroozi et al. 2010; Moster et al. 2010).
Although such comparisons between mass functions are phenomenological in nature (Mutch et al. 2013), they provide useful indications to models of galaxy formation in particular when the comparison spans a large redshift range. Two main physical processes have been proposed to explain the discrepancy at low and high mass. Massive galaxies often host Active Galactic Nuclei (AGN) which will heat the gas and expel the gas that would form the stars outside of the galaxy (for a review see Heckman and Best 2014). Small galaxies instead are not massive enough to retain the gas expelled by supernovae explosions (see e.g. Larson 1974; Powell et al. 2011) which decreases the available material to form stars. In both cases, these processes are lowering the efficiency of star formation, see Silk and Mamon 2012 for a detailed review on the feedback mechanisms.

Stellar to halo mass relation

Understanding how the stellar mass content of a galaxy relates to the mass of its dark matter halo is an alternative way of considering the process of galaxy formation.
Highly complete mass-selected galaxy surveys at high redshift (see, e.g., Ilbert et al. 2013) and accurate predictions for the halo mass function (Tinker et al. 2008; Watson et al. 2013; Despali et al. 2016) allows us to measure the stellar-to-halo mass relation (SHMR) of galaxies at different epochs.
There are many techniques to accomplish this. In the following, we will use the “sub-halo abundance matching » technique, where the number density of galaxies (from observations) and dark matter sub-haloes (from simulations) are matched to derive the SHMR at a given redshift (see, e.g., Marinoni and Hudson 2002; Behroozi et al. 2010, 2013, 2018; Moster et al. 2010, 2013, 2018; Reddick et al. 2013). This technique can also be implemented by assuming a non-parametric monotonic relation between the luminosity or stellar mass of the observed galaxies and sub-halo masses at the time of their infall onto central haloes (Conroy et al. 2006).

Table of contents :

1 From the Big Bang to galaxies 
1.1 Cosmological background
1.1.1 Content of the Universe
1.1.2 The metric of an expanding Universe
1.1.3 Redshift
1.1.4 Friedmann equations
1.1.5 The standard model of cosmology
1.2 The cosmic microwave background
1.2.1 The early Universe and the CMB
1.2.2 Late-time anisotropies
1.2.3 Summary of CMB observations
1.3 Summary of upcoming galaxy surveys
1.4 Growth of inhomogeneities
1.5 Statistics of the distribution of matter
1.5.1 Power spectrum
1.5.2 Angular power spectrum
1.5.3 Projected observables
1.6 Non linear scales
1.6.1 Modelling non linear evolution
1.6.2 Halo mass function
1.7 Astrophysical effects on galaxy power spectrum
1.7.1 Bias of haloes and galaxies
1.7.2 Redshift space distortions
2 Galaxy evolution within dark matter haloes 
2.1 Galaxies and haloes evolution
2.1.1 Gas cooling
2.1.2 Galaxy formation efficiency
2.1.3 Stellar to halo mass relation
2.2 Galaxies
2.2.1 The COSMOS field
2.2.2 Stellar mass function
2.3 Dark matter haloes
2.3.1 Halo mass function
2.3.2 Fit on dark matter simulation
2.4 Estimating the stellar-to-halo mass relation
2.4.1 Implementation
2.4.2 Fitting procedure
2.4.3 Main sources of SHAM uncertainties
2.5 Results: stellar-to-halo mass relation in COSMOS
2.5.1 Stellar-to-halo mass ratio
2.5.2 Variation of the peak halo mass with redshift
2.5.3 Impact of the halo mass function on our results
2.5.4 Stellar to halo mass ratio evolution at fixed halo mass
2.5.5 Interpretation of this evolution
2.6 SHMR from a hydrodynamical simulation
2.7 Cold molecular gas and dark matter haloes
2.7.1 Linking gas mass to stellar mass
2.7.2 Linking gas mass to halo mass
2.7.3 Results
2.8 What could explain a redshift evolution of quenching ?
2.9 Conclusions
3 The cosmological power of a joint analysis of Euclid and CMB surveys 
3.1 Interests of probe combination
3.2 Cosmological forecasts
3.2.1 The likelihood function
3.2.2 Fisher analysis
3.2.3 MCMC as an alternative to Fisher
3.2.4 Fisher matrix using angular power spectra
3.3 Modelling observables
3.3.1 Galaxy observables
3.3.2 CMB lensing
3.3.3 CMB temperature and polarisation
3.4 Euclid observables
3.4.1 Introducing the Euclid mission
3.4.2 Implementation of the Euclid observables
3.5 CMB experiments and noise models
3.5.1 Planck
3.5.2 Simons Observatory
3.5.3 CMB-Stage 4
3.6 Forecasting cosmological constraints
3.6.1 Extensions of CDM
3.6.2 Fisher matrix
3.6.3 Scenarios for the combined analysis
3.6.4 Limits of the numerical resolution
3.7 Results
3.8 Conclusion
4 An alternative probe for galaxy surveys 
4.1 Introduction
4.2 Surveys under consideration
4.2.1 The DESI experiment
4.2.2 The Euclid spectroscopic survey
4.2.3 Tomography
4.3 Observables
4.3.1 Galaxy angular density fluctuations
4.3.2 Angular redshift fluctuations
4.3.3 Angular power spectra
4.3.4 Numerical recipes
4.4 Signal to noise forecasts
4.5 Fisher forecasts
4.5.1 Results for the CDM model
4.5.2 Extension to CPL Dark Energy parametrization
4.5.3 Combining with CMB lensing
4.6 Discussion
4.7 Conclusion
5 Conclusion 
A CMB lensing 
A.1 Introduction
A.2 Statistical anisotropy
A.3 Quadratic estimator
B Impact of massive neutrinos 
C Related publications 

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