Obtaining the Best Cosmological Information from CMB, CMB Lensing and Galaxy Information

Obtaining the best cosmological information from CMB, CMB lensing and galaxy information.

Larissa Santos

P.H.D thesis in Astronomy Supervisors: Amedeo Balbi and Paolo Cabella

Rome 2012 Contents

1 Prologue 1

2 Introduction: The standard cosmology 3 2.1 The ΛCDM model: an overview ...... 3 2.2 The expanding universe ...... 4 2.2.1 The Friedmann Robertson-Walker metric ...... 4 2.2.2 Cosmological distances ...... 5 2.3 The discovery of CMB ...... 6 2.4 Temperature anisotropies ...... 7 2.4.1 Primary temperature anisotropies ...... 8 2.4.2 Secondary temperature anisotropies ...... 9 2.5 CMB temperature angular power spectrum ...... 10 2.6 Cosmological parameters ...... 12 2.7 Polarization ...... 13 2.8 The matter power spectrum ...... 16 2.9 The ΛCDM model + primordial isocurvature ﬂuctuations . . . . 17 2.10 The CDM model + massive neutrinos + time evolving dark energy equation of state ...... 19 2.11 The script of the present thesis ...... 20

I Asymmetries in the angular distribution of the CMB 22

3 The WMAP satellite and used dataset 23 3.1 Internal linear combination maps ...... 24 3.2 The WMAP masks ...... 25

4 The quadrant asymmetry 28 4.1 Mehod: The two-point angular correlation function ...... 29 4.2 Results ...... 30 4.2.1 The cold spot ...... 38

i 4.3 Discussion ...... 38 4.4 Conclusions ...... 41

II Constraining cosmological parameters 43

5 The Planck satellite 44

6 The SDSS galaxy survey 46

7 Mixed models (adiabatic + isocurvature initial ﬂuctuations) 48 7.1 Isocurvature notation ...... 48 7.2 Methodology ...... 49 7.2.1 Information from CMB ...... 49 7.2.2 Information from galaxy survey ...... 52 7.3 Results ...... 53 7.4 Discussion and conclusions ...... 57

8 Euclid survey 66

9 SCM with massive neutrinos and a time evolving dark energy equation of state. 68 9.1 Methodology ...... 68 9.1.1 Information from CMB ...... 68 9.1.2 Information from galaxy survey ...... 71 9.2 Fiducial models ...... 74 9.3 Preliminary results ...... 75 9.4 Discussion and conclusions ...... 80

ii List of Figures

2.1 The CMB blackbody spectrum...... 7 2.2 Angular power spectrum obtained by the Wilkinson microwave anisotropy probe (WMAP)...... 11 2.3 Sensitivity of the temperature angular power spectrum to four diﬀerent cosmological parameters...... 14 2.4 The 7-year temperature-polarization cross correlation power spectrum (TE) and the EE power spectrum...... 15 2.5 The matter power spectrum for diﬀerent experiments datasets in z =0...... 17 2.6 Comparison of the temperature angular power spectrum for a purely adiabatic and purely isocurvature models...... 18

3.1 The WMAP satellite leaving Earth/Moon orbit for L2 ...... 24 3.2 WMAP temperature maps for the 7 years data release in 2011 are seen in the left column. The right column displays the diﬀerence between the 7 year maps and the previous maps from the 5 years WMAP data release...... 26 3.3 ILC 7 year map produced by the WMAP team...... 27

4.1 TPCF curves computed for the WILC7 map using the WMAP KQ85 mask, smoothed for illustration purposes.The NWQ (top) and the NEQ (bottom) appear as solid red lines. The shadow part depicts the standard deviation intervals for 1000 simulated maps produced with the ΛCDM spectrum...... 31 4.2 Same as Figure 4.1, but now with the SWQ at the top and the SEQ at the bottom...... 32 4.3 Same as Figure 4.1, but now using the WMAP KQ85 mask + |b| < 10◦ Galactic cut in the temperature WMAP7 data. From top to bottom, the curves refer to the NWQ and the NEQ, respectively...... 33

iii 4.4 Same as Figure 4.3. From top to bottom, the curves refer to the SWQ and the SEQ, respectively...... 34 4.5 Same as Figures 4.1-4.4, but now using |b| < 10◦ Galactic cut, again in the temperature WILC7 map. From top to bottom the curves refer to the NWQ and the NEQ, respectively...... 35 4.6 Same as Figure 4.5 for the SWQ (top) and the SEQ (bottom). . 36 4.7 Top: Comparison between the SEQ quadrant for the TPCF using the KQ85 mask + these circular masks centered on the cold spot can be seen with a radius of 5 degrees (red line), 10 degrees (yellow line) and 15 degrees (green line). The black line refers to the function without masking the Cold Spot. Bottom: TPCF for the WMAP7 map using KQ85 + circular masks centered on the speciﬁed angles with a radius of 15 degrees each. The red dashed-dot curve refers to the Cold Spot region...... 39 4.8 Comparison between NWQ (top-lef), NEQ (top-right), SWQ (bottom- left) and SEQ (bottom-right) for the TPCF using the KQ85 mask (solid blue line) and the KQ85y7 (solid red line)...... 40

5.1 Planck satellite ...... 44

6.1 SDSS survey telescope ...... 46 6.2 Large scale structure in the Northern equatorial slice of SDSS main galaxy redshift sample...... 47

7.1 Fisher contours for ΛCDM model plus a contribution of initial isocurvature ﬂuctuation with ﬁducials amplitude of α = 0.06,

correlation phase of β = 0 and scalar spectral index of niso = 2.7 for the axion scenario. The red and blue contours represent 95.4% and 68% C.L., respectively, for the unlesed CMB + SDSS (dashed lines) and for the lensed CMB + SDSS (solid lines)(see Table 9.3). 61 7.2 Fisher contours for ΛCDM model plus a contribution of initial isocurvature ﬂuctuation with ﬁducials amplitude of α = 0.06,

correlation phase of β = 0 and scalar spectral index of niso = 0.982 for the axion scenario. The red and blue contours represent 95.4% and 68% C.L. respectively for the unlesed CMB + SDSS (dashed lines) and for the lensed CMB + SDSS (solid lines) (see Table 9.5)...... 62

iv 7.3 Fisher contours for ΛCDM model plus a contribution of initial isocurvature ﬂuctuation with ﬁducials amplitude of α = 0.003,

correlation phase of β = −1 and scalar spectral index of niso = 0.984 for the curvaton scenario. The red and blue contours represent 95.4% and 68% C.L. respectively for the unlesed CMB + SDSS (dashed lines) and for the lensed CMB + SDSS (solid lines). In this case we consider β ﬁxed (see Table 7.7)...... 63 7.4 CMB power spectra derivatives in respect to the isocurvature parameters α (purple) and β(red). The dashed darker lines are related to the unlensed power spectra’s derivative and the solid lighter ones are related to the lensed power spectra’s derivative for the axion scenario of α = 0.06, β = 0. On the left column the

scalar spectral index is niso = 2.7 and on the right column the

scalar spectral index is niso = 0.982 ...... 64

7.5 The same as Figure 7.4 but now the scalar spectral index is niso = 0.982...... 65

8.1 An Euclid satellite illustration ...... 66

9.1 Fisher contours for our model 1. The red and blue contours represent 95.4% and 68% C.L. respectively for the unlesed CMB (dashed lines) and for Euclid galaxy survey (solid lines)...... 77 9.2 Fisher contours for our model 2. The red and blue contours represent 95.4% and 68% C.L. respectively for the unlesed CMB (dashed lines) and for Euclid galaxy survey (solid lines)...... 78 9.3 Fisher contours for our model 3. The red and blue contours represent 95.4% and 68% C.L. respectively for the unlesed CMB (dashed lines) and for Euclid galaxy survey (solid lines)...... 79

v List of Tables

2.1 Six-parameters ΛCDM ﬁt (Larson et al. , 2011) ...... 13

4.1 Calculated probabilities of finding the same asymmetries as in WMAP data in the MC simulations using the WMAP KQ85 mask and considering the ΛCDM model...... 37 4.2 Calculated probabilities of finding the same asymmetries as in WMAP data in the MC simulations using the WMAP KQ85 mask + |b| < 10◦ Galactic cut and considering the ΛCDM model. 37 4.3 Calculated probabilities of finding the same asymmetries as in WMAP data in the MC simulations using the |b| < 10◦ Galactic cut and considering the ΛCDM model...... 37 4.4 Sigma ratios for the simulated CMB maps considering the ΛCDM model and the three Galactic cuts...... 38 4.5 Calculated probabilities of finding the same asymmetries as in WMAP data in the MC simulations using the WMAP KQ85 mask and considering the MΛCDM model...... 41

7.1 Planck specifications ...... 50 7.2 Marginalized errors for ΛCDM model plus a contribution of initial isocurvature fluctuation with fiducials amplitude of α = 0.06,

correlation phase of β = 0 and scalar spectral index of niso = 2.7. 54 7.3 The same as Table 9.3, but in this case β = 0 will be kept fixed. 55 7.4 Marginalized errors for ΛCDM model plus a contribution of initial isocurvature fluctuation with fiducials amplitude of α = 0.06,

correlation phase of β = 0 and scalar spectral index of niso =

nad = 0.982...... 56 7.5 The same as Table 9.5, but in this case β = 0 will be kept fixed. 57 7.6 Marginalized errors for ΛCDM model plus a contribution of initial isocurvature fluctuation with fiducials amplitude of α = 0.003,

correlation phase of β = −1 and scalar spectral index of niso = 0.984...... 58

vi 7.7 The same as Table 9.4, but in this case β = −1 will be kept ﬁxed. 59

9.1 Values of kmax, the galaxy bias and the galaxy density for each redshift bin...... 74 9.2 Fiducial models ...... 75 9.3 Marginalized errors for ΛCDM model with three massive neutri-

nos with identical mass (mν = 0.310 eV) ...... 76 9.4 Marginalized errors for ΛCDM model with two massive neutrinos with identical mass and one massless in a inverted hierarchy mass

splitting (mν = 0.125 eV) ...... 76 9.5 Marginalized errors for ΛCDM model with two massless neutrinos

and one massive in a normal hierarchy mass splitting (mν = 0.190 eV)...... 76

vii Acknowledgements

I would like to thank, first of all, Paolo Cabella for all the support and deep collaboration throughout these three years. He was not only a great supervisor but a friend. Without him the development of this thesis as it is would not be possible. I also would like to thank Amedeo Balbi for the discussions, advices and to come up with ideas in the early stages of this project. I am grateful to Pasquale Mazzotta and Nicola Vittorio for the opportunity to start my phd here in ”Tor Vergata”. I cannot forget Giancarlo de Gasperis that helped me many times with my computational issues, being always kind and patient. A special thank to all my colleagues in the university that helped me completing this thesis, either with suggestions or simple daily conversations. I would like to express my greatest gratitude to Thyrso Villela Neto. He is for me much more than a professor, he is a great friend. I would like to thank him for all the conversations and suggestions in these almost six years of collaboration and friendship. He has been always very supportive and motivational to me. He introduced me to the world of cosmology and research, enabling my climb until here. The first part of this thesis was done in collaboration with him and Carlos Alexandre Wuensche that I also have much to thank. Alex was always helpful to me while I was doing my master course in the National Institute for Space Research (INPE, from the portuguese acronym) in Brazil and also here in Italy. He was in Rome for three months, being of crucial importance in the development of part of this work, not to mention the nice moments that we shared in the eternal city! I wish to thank my parents for the undivided support and interest, without them I would not be able to pursue my goals as I did. They were always there for me, they are my inspiration. I also would like to thank my sister for all the funny conversations that we always had, for all the nice words in difficult times and for all the laughs in happy days. I am grateful to all my family: aunts, uncles, cousins and dogs (specially Meg, Pompom and now the new member of the family Mel) for their unconditional love that made my days much happier! At last but not least, I want to thank my friends. All of them. The new

viii ones that made Rome a sweet home. The ones that are away from my life now, but always appreciated me for my work and motivated me. The friends that I made in each place that I have been around the world. Finally, the best friends ever and forever: Beatriz, Sarah, Paulo and Celso. They are the family that I chose to follow me through life. I owe to all the great people that passed through my life the person that I am today and all the victories that I have achieved, including this small step in my academic carrier. Thank you all for everything!

ix This thesis is dedicated to the memory of my grandfather Jo˜aoBento de Oliveira Filho. Abstract

This thesis is divided in two main parts. In the first part, we will investi- gate asymmetries in the CMB temperature angular distribution considering the ΛCDM model in the three, five and seven year WMAP data. We aim to analyze the four quadrants of the internal linear combination (ILC) CMB maps using three different Galactic cuts. Our analysis showed asymmetries between the southeastern quadrant (SEQ) and the other quadrants (southwestern quadrant (SWQ), northeastern quadrant (NEQ) and northwestern quadrant (NWQ)). Over all WMAP ILC maps, the probability for the occurrence of the SEQ- NEQ, SEQ-SWQ and SEQ-NWQ asymmetries varies from 0.1% (SEQ-NEQ) to 8.5% (SEQ-SWQ) depending on the Galactic cut. Moreover, we tested possible relations between the cold spot and the low quadrupole amplitude with the SEQ excess of power and found no evidence for them. In the second part, we explore the possibility of obtaining better constraints in two different kind of models from future astronomical data by means of the Fisher information matrix formalism. In particular, we consider how cosmic microwave background (CMB) lensing information can improve our parameter error estimation. The first set of models regards a possible isocurvature contribution in addition to the well known adiabatic perturbation to the ΛCDM model. We consider the axion and curvaton inflationary scenarios and use Planck satellite experimental specifications together with the Sloan Digital Sky Survey. We found substantial improvements for all the considered cosmological parameters. In the case of isocurvature amplitude this improvement is strongly model- dependent, varying between less than 1% and above 20% around its fiducial value. In the second set of models, we consider massive neutrinos and a time-evolving dark energy equation of state in the ΛCDM framework. We use Planck satellite experimental specifications together with the future galaxy survey Euclid in our forecast. In this case, we found improvements in all studied parameters considering Planck alone when CMB lensing information is used. However, we could not find improvements in the same parameters constraints when Euclid information was added to CMB lensing information. This issue requires further investigation. Chapter 1

Prologue

The interest in Astronomy dates back to very ancient times when it was still very much correlated to religious practices. For the Babylonian, for instance, the movement of visible celestial bodies was considered mystical omens. The aim to know about the origin of life, and thus the origin of the Universe, in this pre science point of view, took its apex in the Ancient Greece with names as Plato, Aristotle, Ptolemy etc. The first big revolution in the understanding of the cosmos came a long time later between 15th and 18th centuries with Copernicus, Kepler, Galileo and Newton. Their transformational ideas leaded to the foundations of modern cosmology. The second revolution took place in the last century starting from the for- mulation of General Relativity (GR) by Einstein. Following his steps, in 1922, the russian mathematician Alexander Alexandrovich Friedman found the solu- tion of GR field equations in a large scale scenario assuming the Universe as homogeneous and isotropic (Friedman, 1922). Years later, in 1929, Edwin Hub- ble discovered the expansion of the Universe by analyzing the relation between distances and velocities of Extra-Galactic Nebulae (Hubble, 1929). Driven by the legacy left by them, the Belgian priest Georges Lemaitre, was the first person to suggest that the Universe developed from a high density primordial state(Lemaitre, 1931). Based in Lamaitre s model for the Universe, Alpher, Bethe and Gamow published the famous article αβγ in 1948. They claimed by the study of the abundance of chemical elements that the Universe was formed initially by a gas made of neutrons in a state of high temperature and density. This gas would lately decay in protons and electrons as a result of the expansion of the Universe (Alpher et al. , 1948). In this same year, Gamow concluded that due to the high temperature the primordial Universe should also be mainly constituted by a

1 black body radiation (Gamow, 1948b,a). The existence of this radiation, known currently as the Cosmic Microwave Background (CMB), was also predicted by Alpher & Herman (1948). They calculated that the temperature of the CMB today would be of approximately 5K . These new elucidations in the study of cosmology form altogether a testable theory of the Universe, known as the Standard Cosmological Model (SCM). The ﬁrst question that comes to mind is: what makes this model standard? As we will see later on, this model, also known as Big Bang model, attends three observational data:

• Universe expansion (via Hubble law);

• light elements abundance in the present Universe (nucleosynthesis);

• CMB (almost perfect blackbody nature).

Nevertheless, recent developments leaded to observations beyond the SCM. Among them there is the evidence of non barionic matter in the Universe, called Dark Matter (DM) 1, which existence was suggested for the first time by Zwicky (1933). However, the direct evidence for the existence of DM came lately when- Clowe et al. (2004) studied the interaction between two clusters, one of them well known as the bullet cluster. There are, however three different classifica- tions for the DM: Hot (HDM), Warm (WDM) and Cold (CDM). It is believed nowadays that the Universe is predominantly formed by CDM and many candidates have been exhaustively studied. The nature of the DM is still unclear. Another important observation dates from 1998 when the study of type I supernovae suggested an accelerated expansion of the Universe (Riess et al. , 1998; Perlmutter et al. , 1999). The most popular way to try to explain this acceleration is invoking what is called dark energy. In this work, I will start using as a candidate, the cosmological constant, Λ introduced the first time by Einstein, since it is the simplest approach that agrees with observations. So, to adjust the model to the observed data some changes needed to be made in the SCM, introducing on it the CDM and the cosmological constant. This modified SCM is called ΛCDM model. From now on, the reader should understand SCM as the ΛCDM model. In the next chapter, I will introduce briefly the SCM and some of its pre- dictions in an attempt to contextualize the reader for the main goal of this thesis. I will also discuss about our object of study, one of the ΛCDM model observational pillars, the CMB, and its most important features.

1There is, however a small amount of barionic DM in the Universe, for example the MA- CHOS (Massive Astrophysical Compact Halo Objects)

2 Chapter 2

Introduction: The standard cosmology

2.1 The ΛCDM model: an overview

According to the ΛCDM model the very early universe was dense and hot, providing the frequent interaction among its constituents. Particles and radiation were then coupled as the so called primordial plasma. In this scenario, the probability that neutral atoms or bound nuclei could be formed was almost none. If it happened they were instantaneously destroyed due to their interaction with the high energetic photons present in the environment. Since we have evidence for an expanding universe, as time evolves its temperature gets cooler (below the binding energy of typical nuclei) enabling the formation of light elements’ nuclei (D, 3He, 4He). This event is known as the Big Bang Nucleosynthesis (BBN). The temperature of the universe continues to drop until the energy of the photons gets smaller then the ionization energy of the Hydrogen atom (13.6 eV). The free electrons are then captured by the Hydrogen nuclei, in a process known as recombination, forming neutral atoms. As the number of free electrons in the medium decreases, the mean free path of the photons increases due to the smaller occurrence of the Thomson scattering. Begins thereby the decoupling of matter and radiation. By the end of this process, the photons from the CMB could freely propagate in a transparent universe while baryons coalesced, due to gravitational instabilities, to form the structures that we see today.

3 2.2 The expanding universe

One of the questions that comes to mind as observers of the cosmos is concerning our location within the universe as a whole. The natural answer is that we should not be in any privileged position. That is exactly what is known as the cosmological principle. We start building the cosmological model assuming that the universe is homogeneous (it looks the same no matter how deep you go) and isotropic (it looks the same in any direction you turn) on cosmological scales. Adding also the fact already mentioned of the universe expansion, we set up the basis of our model. Considering these theoretical frame we make use of the results of an important tool in the study of cosmology: the general relativity.

2.2.1 The Friedmann Robertson-Walker metric

Let’s begin deﬁning the line element.

2 X µ ν ds = gµν dx dx , (2.1) µν

where gµν is the four dimensional metric and the indices µ and ν run from 0 to 3. The universe’s expansion can be described by the scale factor a deﬁned 1 as function of the resdhift 1 + z = a , recalling that the redshift is related to the Doppler eﬀect. It results in a stretch of light’s wavelength from a receding object and it can be approximated as z ' v/c for low redshifts. The metric is then written for an expanding, homogeneous and isotropic universe in terms of the scale factor. In cartesian coordinates, the line element becomes

dx1 + dx2 + dx3 ds2 = dx0 − a2(t) , (2.2) (1 + Kr2/4)2 where

r2 = (dx1)2 + (dx2)2 + (dx3)2. (2.3)

That is the Friedman-Robertson-Walker (FRW) metric (Friedman, 1922; Lemaitre, 1931; Robertson, 1935; Walker, 1937) and K describes the universe curvature as follow:

• K = −1 for an open universe;

• K = 0 for a ﬂat universe;

• K = 1 in a closed universe.

Another important quantity related to the scale factor is deﬁned as the Hubble parameter that describes the expansion rate of the universe,

4 da/dt H(t) = . (2.4) a

Today, H(0) = H0 = 71 ± 2.5 km/s/Mpc (Larson et al. , 2011), and it is usually parametrized in terms of the dimensionless quantity h = H(t)/100. We assume, as evidenced by observations (de Bernardis et al. , 2000; Larson et al. , 2011), that we live in a spatially ﬂat universe with its energy density 2 today equal to the critical density today: ρ0 = ρc = 3H0 /8πG, being G the Newton’s constant. Based on this assumption, we can write the evolution of the scale factor more generally via the Friedman equation

8πG H2(t) = ρ(t). (2.5) 3 With this concepts in mind, we move to the next section where I will describe some cosmological distances that are of fundamental importance in the development of the present work.

2.2.2 Cosmological distances

Measuring distances in an expanding universe is not a straightforward job. First of all, we should deﬁne a special coordinate system where the observer remains at rest relative to the expansion. It is reasonable to think that we can recover the physical distance, that surely depends on time, by multiplying our rest frame distance by the scale factor,

∆~xphysical = a(t)∆~x. (2.6)

The same idea applies to the time coordinate, deﬁning an important comoving distance, called conformal time,

dt dη = , a(t) Z t dt0 η = 0 . (2.7) 0 a(t )

Equation 2.7 shows the maximum distance that could be traveled by a photon since the beginning of time. In other words, no information could have propagated further than η. Regions separated by distances greater than this are causally disconnected. For this reason the conformal time can be also un- derstood as the comoving horizon. Furthermore, it is possible to know the distance between us and a distant emitter by calculating the maximum distance that a photon can travel from its

5 emission out from a certain redshif, z, at a time t1 until our detection at time t0:

Z t0 dt0 χ(a) = 0 . (2.8) t1 a(t ) Another way of obtaining distances is by measuring the angle, ∆θ, subtended by an object. Being aware of the physical size of the object, l, and restricting ourselves to small angles, we deﬁne the angular diameter distance

l D = . (2.9) a ∆θ More generally we can say that

c Z z dz Da(z) = , (2.10) 1 + z 0 H(z) c being the speed of light. It is of paramount importance that future galaxy surveys measure the angular diameter distance, providing a tool to better constrain the dark energy. Moving to the observational aspect of cosmology, I start discussing about one of the SCM main pillars, the CMB.

2.3 The discovery of CMB

The CMB was discovered accidentally by Penzias & Wilson (1965). They were calibrating an antenna of the Bell Telephone Laboratories in the USA when they found out that the received noise was not coming from the instrument nor from the atmosphere. This signal was measured in one wave length in many different directions in the sky and its temperature was estimated in 3.5K. However, The interpretation of this measurement as the primordial blackbody radiation predicted by Gamow in the 40’s was done by Dicke et al. (1965). Both the discovery and the interpretation of CMB became a landmark in the history of cosmology. Thereafter, cosmological models could be tested and a new experimental science was born. The search for CMB anisotropies began with Conklin (1969). He concluded that since the Earth is moving in respect to the CMB rest frame, a Doppler effect should be measured. Another evidence of the existence of this anisotropy, known as dipole effect, came in 1971 in an experiment done by Henry (1971). Years later, in 1976, Corey and Wilkinson, taking advantage of a balloon-borne experiment measured in a frequency of 19GHz an anisotropy on large angular scales (Corey & Wilkinson, 1976). Finally, in 1977, the dipole anisotropy was confirmed as having an amplitude of 10−3K by Smoot et al. (1977) in an experiment on board of an airplane U-2. Over the years, preciser measurements

6 were done in the dipole effect, such as by Fixsen et al. (1983); Lubin et al. (1985), these last ones being responsible also for obtaining an upper limit for the CMB quadrupole amplitude. In 1989, it was launched the COsmic Background Explorer (COBE) satellite that in addition to improving the measurement of the amplitude of the dipole anisotropy, 3.365±0.027K, using data from the Differential Microwave Radiome- ter (DMR) (Kogut et al. , 1993), enabled the discovery of CMB cosmological anisotropies in large angular scales Smoot et al. (1992). Another instrument on board of the COBE satellite was the Far Infrared Ab- solute Spectrophotometer (FIRAS) that enabled the confirmation of the CMB blackbody spectrum with T = 2.735 ± 0.06 (see Figure 2.1) (Mather et al. , 1990). This measurement was later improved by Fixsen & Mather (2002), T = 2.725 ± 0.001.

Figure 2.1: The CMB blackbody spectrum.

Mather et al. (1990)

2.4 Temperature anisotropies

The CMB temperature anisotropies are classiﬁed according to their origins. Primary anisotropies were generated in the CMB before the decoupling epoch while the secondary anisotropies arise during the path between the last scatter-

7 ing surface (LSS) and us (see Bersanelli et al. (2002) for a review). Hereafter, I will brieﬂy summarize the main sources that can generate both primary and secondary anisotropies, giving emphasis to three of them that are of crucial importance for the development of the present work: Adiabatic and isocurvature ﬂuctuations (primary anisotropies) and gravitational lensing (secondary anisotropy).

2.4.1 Primary temperature anisotropies

Adiabatic or curvature fluctuations generate CMB temperature anisotropies in a 2 degree angular scale. They are related to density perturbations in the photon-baryon fluid. Inflation theory (for a review in this topic see Boyanovsky et al. (2009) and references therein) proposes that the universe expanded expo- nentially fast just after the Big Bang when it was only 10−35 seconds old. This fast expansion amplified the quantum fluctuations generated by the scalar field that drove inflation. This process enabled the formation of acoustic waves, that modified by the gravitational potential were responsible for the density oscillations in the primordial plasma. Adiabatic perturbations account for only one mode, that is to say,

1 δρ 1 δρ 1 δρ 1 δρ γ = ν = B = CDM , (2.11) 4 ργ 4 ρν 3 ρB 3 ρCDM for a universe that consists of photons, massless neutrinos, baryons and CDM at early times. The adiabatic perturbations contribute to the CMB temperature variation by (for details see Kolb & Turner (1994), chapter 9)

∆T δρ ∝ . (2.12) T ρ Isocurvature or entropy flutuations are associated with initial stress perturbations between radiation and matter. The perturbations were then caused by fluctuations in the number density between different particle species in the primordial fluid(see, Hu et al. (1997) for details). Differently from the adiabatic case, entropy perturbations consist of 4 modes that corresponds to a baryon isocurvature mode, a cold dark matter isocurvature mode, a neutrino density (ND) and a neutrino velocity (NV) mode. These modes are written in the form:

δρCDM 3 δργ SCDM = − , (2.13) ρCDM 4 ργ

δρbaryon 3 δργ Sbaryon = − , (2.14) ρbaryon 4 ργ

8 3 δρND 3 δργ SND = − , (2.15) 4 ρND 4 ργ

3 δρNV 3 δργ SNV = − . (2.16) 4 ρNV 4 ργ The contribution to CMB temperature variation for the isocurvatue ﬂuctu- ation is given by δni ∆T Si = − 3 . (2.17) ni T The Sachs-Wolf eﬀect is caused by scalar perturbations in the FRW metric that in a Newtonian context is related to perturbations in the gravitational potential Ψ. Fluctuations in this potential were caused by the irregular distribution of baryonic and dark matter in the LSS. After the decoupling the photons emerged from this gravitational potential, losing energy, and thus imprinting temperature variations that can be seen in the CMB,

δT 1 = Ψ. (2.18) T 3 This effect was originally described by Sachs & Wolfe (1967) and generate CMB temperature fluctuations in large angular scales. The Doppler effect is caused by peculiar velocities in the primordial fluid. The contribution in the CMB temperature variation for this effect, for photons with velocity v in respect to the primordial plasma, is given by

δT v ' . (2.19) T c

2.4.2 Secondary temperature anisotropies

The integrated Sachs-wolfe effect occurs because potential wells present in the way between the LSS and us vary in time changing therefore the energy of the photons during its path. The Sunyaev-Zel’dovich effect distorts the CMB black body spectrum by inverse Compton scattering of the photons by relativistic electrons in cluster of galaxies. The gravitational lensing effect do not modify the energy of the photons but their trajectory. This effect has a very weak signal, making hard its detection. However, as new experiments are being developed, the precision in CMB measurements makes small effects distinguishable by observations. CMB lensing is one of these effects and it has important quantitative contributions that should be taken into account. The CMB photons are deflected during their travel between the last scattering surface and the observer by gravitational po-

9 tentials Ψ(χ, η) dependent on the comoving distance χ and the conformal time η. For CMB temperature anisotropy, this is quantitatively written as

∆T(˜ ˆn) ∆T(ˆn’) ∆T(ˆn + d) = = , (2.20) T T T where the temperature T of the lensed CMB in a direction ˆn is equal to the unlensed CMB in a different direction ˆn’. Both these directions, ˆn and ˆn’, differ by the deflection angle d as it can be seen in the third equality above. To first order, the deflection angle is simply the lensing potential gradient, d = ∇ψ. To use the CMB lensing information we the have to measure the lensing potential that is defined as:

∗ Z χ χ∗ − χ ψ(ˆn) ≡ −2 dχ ∗ Ψ(χˆn; η0 − χ), (2.21) 0 χ χ ∗ χ being the comoving distance and η0 − χ is the conformal time at which the photon was at position χˆn. We can study CMB lensing properties through the lensing potential, and the temperature and polarization power spectra (explained in further in text), as well as their cross correlation (for a review of CMB lensing see Lewis & Challinor (2006)). The lensing signal was detected for the ﬁrst time by cross-correlating WMAP data to radio galaxy counts in the National Radio Astronomy Observatory Very Large Array sky survey (Smith et al. , 2007). In other words, Cψg 6= 0. The detection of the gravitational lensing using CMB temperature maps alone and the measurement of the power spectrum of the projected gravitational potential were already done using the Atacama Cosmology Telescope and the South Pole Telescope (Das et al. , 2011; van Engelen et al. , 2012). While still waiting for more precise data sets, much work is being done to CMB lensing reconstruction techniques (e.g. Hu (2001); Okamoto & Hu (2003); Smith et al. (2012); Bucher et al. (2010); Carvalho & Tereno (2011)).

2.5 CMB temperature angular power spectrum

The standard procedure to construct the CMB angular power spectrum is to decompose the radiation ﬁeld in spherical harmornics

∞ l ∆T X X (θ, φ) = a Y (θ, φ), (2.22) T lm lm l=2 m=−l

being θ and φ the polar and azimuthal angles respectively. The alm’s are the spherical harmonic coeﬃcients that represent the amplitude of each multipole l

10 (l and m are integers l ≥ 0 and −l ≥ m ≥ l). An useful relation relates each multipole to a corresponding angular scale in a nearly ﬂat universe: l ∼ 180/θ.

The angular power distribution, Cl is deﬁned for our homogeneous and isotropic universe as

l 1 X C ≡ |a |2. (2.23) l 2l + 1 lm m=−l

Figure 2.2: Angular power spectrum obtained by the Wilkinson microwave anisotropy probe (WMAP).

Larson et al. (2011)

Figure 2.3 shows the angular power spectrum predicted by the ΛCDM model in agreement with the data collected by WMAP. It is possible to extract a lot of cosmological information by analyzing the power spectrum, one example is the so-called anomalies in the CMB angular distribution of temperature fluctuations. Detailed studies showed these unexpected results considering the cosmological concordance model. These peculiar features in CMB results were found for the first time in the Cosmic Background Explorer (COBE) satellite data, attracting a lot of interest since then. A quadrupole amplitude smaller than that expected according to the ΛCDM model was reported by the COBE team (Smoot et al. , 1992) and was confirmed by all WMAP data releases (Bennett et al. , 2003b; Hinshaw et al. , 2007, 2009; Jarosik et al. , 2011). Other anomalies were found in the WMAP data that were not expected according to the ΛCDM model either, such as the alignment between the quadrupole and octopole (e.g. (Bielewicz et al. , 2004; Schwarz et al. , 2004; Copi et al. , 2004; de Oliveira-Costa et al. , 2004; Bielewicz et al. , 2005; Land & Magueijo, 2005; Copi. et al. , 2006; Abramo et al. , 2006; Frommert &

11 Enßlin, 2010; Gruppuso & Burigana, 2009)), the low quadrupole and octopole amplitudes (e.g., (Mukherjee & Wang, 2003; Ayaita et al. , 2010; Cayón,2010; Cruz et al. , 2011)), the north-south asymmetry (e.g., (Eriksen et al. , 2004; Hansen et al. , 2004a; K.Eriksen et al. , 2004; Hansen et al. , 2004b; Donoghue & Donoghue, 2005; Hoftuft et al. , 2009; Paci et al. , 2010; Pietrobon et al. , 2010; Vielva & Sanz, 2010)), the anomalous alignment of the CMB features toward the Ecliptic poles (e.g., (Wiaux et al. , 2006; Vielva et al. , 2007)), and the cold spot (e.g., (Vielva et al. , 2004; Cruz et al. , 2005, 2007; Vielva, 2010)). Recently, Aluri & Jain (2012) analyzed in detail the signature of parity asymmetry first found by Kim & Naselsky (2010) in the WMAP best- fit temperature power spectrum, confirming this asymmetry on a 3-σ level. Aluri & Jain (2012) also concluded that their result is not due to residual foregrounds or to foreground cleaning. On the other hand, Bennett et al. (2011) reviewed the anomalies reported in the CMB temperature fluctuations and claimed that they are not statistically significant and, for this reason, do not in disagree with the ΛCDM concordance model. Nevertheless, in the first part of this work I will report an asymmetry in the WMAP temperature anisotropy data appearing in the two-point angular correlation function (TPCF) at scales above 100 degrees. Another way to extract information of the power spectrum is by its shape, as for example the position of the first acoustic peak, enable us to estimate the value of some cosmological parameters. That is what I am going to explain next.

2.6 Cosmological parameters

The cosmological parameters are observables used to describe the global dy- namics of the universe as well as the parametrization of some functions. One of the interests is to constraint the content of the universe today, such as the neutrinos, baryon, dark matter and dark energy densities considering in our case a ﬂat geometry.

ρν Ων = , ρc ρb Ωb = , ρc ρCDM ΩCDM = , ρc ΩΛ = 1 − (Ωb + Ωc). (2.24)

12 Table 2.1: Six-parameters ΛCDM ﬁt (Larson et al. , 2011)

Parameters Descripstion 7-year WMAP ﬁt 2 2 2 +0.057 10 Ωbh 10 × Physical baryon density 2.258−0.056 2 ΩCDM h Physical cold dark matter density 0.1109 ± 0.0056 ΩΛ Dark energy density (w = −1) 0.734 ± 0.029 2 −1 −9 ∆R Amplitude of curvature perturbation, k0 = 0.002 Mpc (2.43 ± 0.11) × 10 −1 ns Spectral index of density perturbations, k0 = 0.002 Mpc 0.963 ± 0.014 τ Reionization optical depth 0.088 ± 0.015

In this case the neutrino density is given for massless neutrinos, mν = 0 We deﬁne the equation of state parameter for the evolution of the universe in terms of total pressure and energy density of each specie present in the universe at a given epoch

P w = . (2.25) ρ Since we are considering the ΛCDM model the dark energy will refer to the cosmological constant with a constant energy density at all times and w = −1. Matter and radiation corresponds to w = 0 and w = 1/3 respectively. Figure 2.3 shows how the variation of some cosmological parameters can change the shape of the temperature CMB power spectrum, as for example, the peaks position and heights. In this way, the value of some cosmological parameters can be constraint with CMB data. Table 2.1 shows the best ﬁt values for six parameters obtained by WMAP team using the satellite 7 year dataset released. 2 The parameters ∆R and ns will be explained better in Section 2.8. Now we move on to a brief explanation of another CMB observable that will be used throughout the text: the polarization.

2.7 Polarization

The CMB polarization was first measured by Degree Angular Scale Interferom- eter (DASI) (Kovac et al. , 2002) confirming its theoretical prediction. When photons and matter were coupled together in the primordial universe, the radiation field could be characterized only by its temperature (monopole) and by the Doppler shift due to peculiar velocities in the fluid (dipole). As consequence, the primordial plasma was unpolarized. When recombination started, the photons mean free path increased and electrons started to form local quadrupoles inside the plasma. Polarized light was then produced via Thomson scattering.

13 Figure 2.3: Sensitivity of the temperature angular power spectrum to four different cosmological parameters.

Hu & Dodelson (2002)

The CMB can be studied by its angular spectra that includes, the temperature power spectrum (Eq refcl), the polarization power spectra and their cross correlation. See Figure 2.4.

TT T ∗ T Cl = almalm , EE E∗ E Cl = alm alm , BB B∗ B Cl = alm alm , TE T ∗ E Cl = almalm , (2.26)

TB EB being for scalar CMB ﬂuctuations Cl = Cl = 0 by parity. The ﬁelds E

14 and B are coordinate independent and are related to the stokes parameters by a non-local transformation (for a review in CMB polarization theory, see Cabella & Kamionkowski (2004).

Figure 2.4: The 7-year temperature-polarization cross correlation power spectrum (TE) and the EE power spectrum.

Larson et al. (2011)

In this sense I brieﬂy introduce the Stokes parameter formalism. The electric ﬁeld for a monochromatic plane electromagnetic wave propagating in the z direction is described by:

15 Ex = ax(t) cos[ωot − θx(t)]

Ey = ay(t) cos[ωot − θy(t)] (2.27)

The Stokes parameters can then be deﬁned as:

2 2 I = ax + ay , 2 2 Q = ax − ay ,

U = h2axay cos(θx − θy)i ,

V = h2axay sin(θx − θy)i . (2.28)

In the case of CMB there is not circular polarization since it cannot be generated by Thomson scattering (V=0) Finally, the eﬀect of lensing in CMB polarization is written in terms of the Stokes parameters Q(ˆn) and U(ˆn)

[Q + iU](ˆn) = [Q + iU](ˆn + d). (2.29)

2.8 The matter power spectrum

The matter power spectrum is related to the density contrast of the universe δ(~x,t), in other words, it is related to the diﬀerence between a local density and the global density of the universe. We can extract useful information from galaxy surveys by the two point correlation function

ξ( ~x1, ~x2, t) = hδ( ~x1, t)δ( ~x2, t)i. (2.30)

Expanding δ(~x,t) in Fourier modes we get

3 Z d k ~˙ δ(~k) = δ(~x,t)eik~x. (2.31) 2π3 The primordial power spectrum is deﬁned as

~ ∗ ~0 3 3 ~ ~0 hδ(k)δ (k )i = (2π) δ (k − k )PR(k) (2.32)

We assume that the primordial ﬂuctuations do not have any privileged scale 2 n n by h|δk| i ∝ k so PR(k) ∝ k . PR(k) stands for the power spectrum of primordial curvature perturbations and n = ns − 1, being ns the spectral scalar index. Normalizing the power spectrum we get

16 3 ns 2 k PR(k) 2 k ∆R(k) = 2 = ∆R(k0) . (2.33) 2π k0 The evolution of the primordial power spectrum depends on which component of the universes dominates.

ns 2 k 2 P (k, z) = ∆R(k0) T (k, z), (2.34) k0 being z the redshift. The transfer function T (k, z) is obtained by solving the Boltzmann equations for the distribution function from the universe components: CDM, dark energy, neutrinos, photons and baryons. The transfer function is obtained numerically for its usage in cosmology today. Figure 2.5 shows the matter power spectrum obtained from diﬀerent datasets for z = 0.

Figure 2.5: The matter power spectrum for diﬀerent experiments datasets in z = 0.

Tegmark & Zaldarriaga (2002)

In the next two section I will introduce the main goals of this thesis.

2.9 The ΛCDM model + primordial isocurvature ﬂuctuations

Since the early measurement of the ﬁrst acoustic peak in the cosmic microwave background (CMB) angular power spectrum (de Bernardis et al. , 2000; Hanany

17 et al. , 2000), a pure isocurvature model of primordial fluctuations was ruled out (Enqvist et al. , 2000) (see also Figure 2.6 and compare with Figure 2.3). In addition, recent CMB data from the WMAP satellite found no evidence for nonadiabatic primordial fluctuations (Komatsu et al. , 2011). These results are consistent with a single scalar field inflationary model prediction of perfectly adiabatic density perturbations. However, small contributions from isocurvature primordial fluctuations (in mixed models) cannot be excluded by the current data.

Figure 2.6: Comparison of the temperature angular power spectrum for a purely adiabatic and purely isocurvature models.

The standard inflationary scenario driven by a single field cannot account for isocurvature fluctuations. If we want to take into account isocurvature fluctuations, a multiple-field inflation has to be considered (for a general formalism, see Gordon et al. (2001), for example). We consider the alternative scenario where perturbations to a light field different from the inflaton (the curvaton) are responsible for curvature perturbations and may also generate isocurvature fluctuations (Lyth & Wands, 2002; Moroi & Takahashi, 2002; Lyth et al. , 2003). In this case, the isocurvature component is completely correlated or anticorrelated

18 with the adiabatic component. One advantage of the curvaton model is that it can produce the power asymmetry related to the one that will be shown in the first part of this work without violating the homogeneity constraint (Erickcek et al. , 2008). Thereby I relate the first part of the thesis with the second one. In the other scenario that will be taken into account, quantum fluctuations in a light axion field generate isocurvature fluctuations. Unlike the first scenario, this isocurvature component is fully uncorrelated with the adiabatic one. It is important to point out that axion particles can be produced in this scenario, which can contribute to the present dark matter in the universe (see Beltrán et al. (2007); Bozza et al. (2002); Hertzberg et al. (2008) and references therein). There are already many studies in the literature constraining the isocurvature contribution using different data sets [CMB, large-scale structure (LSS), type Ia supernovae (SN), Lyman-α forest and baryon acoustic oscillations (BAO)] (see, for example, Li et al. (2011); Carbone et al. (2011a); Larson et al. (2011); Komatsu et al. (2011); Mangilli et al. (2010); Bean et al. (2006); Beltrán et al. (2005); Crotty et al. (2003)). In the second part of the present thesis, one of the goals is to use the well- known Fisher information matrix formalism (for a short guide see Coe (2009)) to estimate whether better constraints to the isocurvature contribution can be obtained in the near future using measurements of the CMB temperature and polarization power spectrum from the Planck satellite, as well as the large-scale matter distribution observed by the Sloan Digital Sky Survey (SDSS), using CMB lensing information. We will see how this new information could improve the error prediction for some cosmological parameters, especially those related to the isocurvature mode.

2.10 The ΛCDM model + massive neutrinos + time evolving dark energy equation of state

As mentioned before, the discovery of the accelerated expansion of the universe (Perlmutter et al. , 1999; Riess et al. , 1998) led us to introduce a cosmological constant in the standard cosmological model as an attempt to explain the observations. The cosmological constant Λ with equation of state w = −1 is the simplest dark energy candidate that together with the CDM constitute the ΛCDM model. Although the latest observations are consistent with this concordance model (Larson et al. , 2011; Komatsu et al. , 2011), different candidates of dark energy cannot be discarded. Moreover, the cosmological constant scenario has two difficulties known as the fine-tuning and cosmic coincidence problems (see

19 Zlatev et al. (1999)). To overcome these problems alternative candidates for the dark energy have been proposed, as for example the quintessence (Caldwell et al. , 1998) that allows the possibility of a time-dependent equation of state (Linder, 2003). In the scope of this thesis, I will assume a redshift dependent equation of sate for the dark energy (see Equation2.25)

Pde(z) wde(z) = . (2.35) ρde(z) Finally, I will adopt a well-known dark energy equation of state

wde(a) = w0 + (1 − a)wa. (2.36)

Another important task in cosmology is to constrain the neutrinos’ masses. It was shown by neutrino oscillation experiments that neutrinos have non-zero masses challenging the standard model of particle physics. However, these experiments can only constrain the neutrinos mass-square diﬀerences and not their individual values (for a review in neutrino masses see de Gouvea (2009)). On the other hand, cosmological probes are most sensitive to the total neutrino masses, P mν . Using CMB radiation data only, from WMAP 7-year, an upper limit P to mν of 1.3 eV at 95% C.L was found (Komatsu et al. , 2011). Recovering

Equation 2.24, the neutrino density equation is now replaced for mν 6= 0 by

m Ω = ν . (2.37) ν 94h2eV Since it was shown before by Hannestad (2005) that the dark energy equation of state and the neutrinos’ total mass parameters are degenerated, some work has already been done to constrain both parameters simultaneously in a few dark energy scenarios, such as for models with a constant and a time-varying equations of state (Ichikawa & Takahashi, 2008; Hamann et al. , 2010; Carbone et al. , 2011b; Joudaki & Kaplinghat, 2012). A second goal is to forecast the constraint in total mass of neutrinos in a time evolving dark energy model, using CMB temperature and polarization power spectrum from the Planck satellite experimental setup (including also CMB lensing information), as well as the large-scale matter distribution that can be observed by EUCLID survey.

2.11 The script of the present thesis

In the ﬁrst part of this thesis, I will describe an asymmetry found in one quadrant of the sky in the WMAP temperature anisotropy data. We used the two-point angular correlation function to obtain the results. In further analysis, we tested a possible correlation between the quadrant asymmetry and the cold spot region.

20 In Chapter 3 I will brieﬂy describe the WMAP satellite and the data we used. In Chapter 4, I will present the article itself divided in sections: method, results, the cold spot, discussion and conclusions. The reference for the published article version is

• Santos, L., Villela, T. & Wuensche, C. A. 2012. Asymmetries in the angular distribution of the cosmic microwave background. A&A, 544(aug), A121.

In the second part, the first goal was to forecast the value of some cosmological parameters for a cosmological model allowing both isocurvature and adiabatic primordial perturbations using Planck satellite information and SDSS galaxy survey. We made use of CMB gravitational lensing information to know if it could improve the previous results found in literature. With this in mind, we used the Planck predicted CMB temperature, polarization, deflection angle and the cross correlation between the deflection angle and temperature power spectra. In Chapter 5 and 6 I will briefly introduce the reader to the Planck experiment and SDSS galaxy survey. Chapter 7 I will present the article that was published from this work. It is divided into 4 sections: isocurvature notation, methodology, results and discussion and conclusions. The reference for the published article version is

• Santos, L., Cabella, P., Balbi, A. & Vittorio, N. 2012. Forecasting isocuva- ture models with CMB lensing information: Axion and curvaton scenarios. PRD, 86(jul), 023002.

The second goal is to forecast cosmological parameters for models with massive neutrinos and a time varying dark energy equation of state. In this case, we make use of Euclid survey as a galaxy survey and as a baryonic acoustic oscillation (BAO) experiment. We also use Planck information to predict the CMB temperature and polarization power spectra as well as the deflection angle and the cross correlation between the deflection angle and temperature power spectra. We include in this new scenario the cross correlation between the deflection angle and the E polarization power spectrum. In Chapter 8, the Euclid experiment will be introduced. Chapter 9 I will present part of the article that is in preparation.

21 Part I

Asymmetries in the angular distribution of the CMB

22 Chapter 3

The WMAP satellite and used dataset

The project of the WMAP satellite dates from 1995 however it was launched on June 30th 2001 from the base of EUA Air Force in Florida. The WMAP weights 836 kg and achieve a 3.8 m high. The lower disk has 6 solar panels in a diameter of 5.0 meters and it is responsible for the satellite energy supply. In addition, the instruments are all shielded as a protection against the solar radiation (Bennett et al. , 2003b). The WMAP orbits in the Sun-Earth L2 point, 1.5 × 106 km way from us, to avoid turbulences caused by our planet, such as by its magnetic fields. The instrument on board of the WMAP satellite consists of 2 Gregorian tele- scopes, being the primary mirror 1.4m × 1.4m. The secondary mirror is used as a protection against the Galactic signal (for details see Page et al. (2003)). The instrument also consists of 20 differential radiometers that use amplifiers based in high electron mobility transistors. These devices are cooled to a temperature of approximately 90K to diminish systematic errors (Jarosik et al. , 2003). The radiometers measure temperature variations in the sky enabling the generation of temperature maps in 5 different frequency bands centered in 22, 33, 41, 61 and 94 GHz named respectively K, Ka, Q, V and W bands. The WMAP angular resolution varies according to the frequency band, being 0.88◦, 0.66◦, 0.51◦, 0.35◦ and 0.22◦ for each of the cited frequency bands. Figure 3.2 shows the temperature maps in the 5 mentioned frequency bands for the WMAP 7 years data release. The WMAP was developed to get new and better measurements of the CMB comparing to previous experiments. The sensibility of WMAP is 45 times better then the one of COBE satellite, enabling a measurement of temperature

23 Figure 3.1: The WMAP satellite leaving Earth/Moon orbit for L2

http://wmap.gsfc.nasa.gov/media/990387/index.html

variation smaller than 20µK per pixel squared (Bennett et al. , 2003c). The temperature angular power spectrum obtained by WMAP showed also the CMB ﬁrst acoustic peaks as seen in Figure 2.3, an improvement in comparison to COBE spectrum that was obtained for low multipoles (plateau).

3.1 Internal linear combination maps

The data collected by the satellite is not only CMB signal, the Galactic foreground is the mainly contamination that we need to deal with. There are three main components of Galactic emission and their contribution depends in which frequency the data is collected. The synchrotron emission is related to the loss of energy by relativistic electrons due to the Galactic magnetic fields and dominates in low frequencies (for WMAP temperature maps it affects above all the K and Ka bands). The Bremsstrahlung emission is also associated to the energy loss by relativistic electrons but now due to their interaction with the electric fields generated by ions in the Galaxy. This emission also dominates in low frequencies. The thermal emission from the dust dominates in high frequencies (mostly W band) and is related to dust heating in the interstellar medium. The internal linear combination (ILC) method was developed to decrease the Galactic signal preserving however the CMB signal producing a full CMB

24 sky map (Bennett et al. , 2003a; Hinshaw et al. , 2007; Gold et al. , 2009). The produced map is an internal linear combination from the 5 previous generated frequency maps. The coeﬃcients are adjusted to cancel the Galactic emission(see Figure 3.3). It is however important to point out that even though the ILC method decreases the Galactic contribution, it does not produce maps completely free from the Galactic contamination. This residual contamination must be taken into account when analyzing the CMB maps. The best way of doing that is to literally remove the aﬀected zone from the map using masks.

3.2 The WMAP masks

The masks were developed to avoid any kind of foreground (not only Galactic) in the CMB maps. The masks exclude pixels in the map dominated by foregrounds and preserve the zones dominated by the CMB. The masks produced for the first WMAP year were based on the K band map because of its severe Galactic contamination. The masks were characterized according to their rigor in excluding pixels, being the Km2 the most severe followed by the Km1, Kp0, Kp1 until Kp12 (Bennett et al. , 2003a). Finally these masks were summed up with the radio emitters point sources masks. For the 3 years dataset, the point sources masks were updated adding new sources that were detected later. The masks were again modified for the WMAP 5 year data release. Regions containing the Gum and ρ Oph Nebulae were included in their production. These new masks were done based on the K and Q maps. From these maps are subtracted the ILC map, remaining just the Galactic emission signal. Both maps are then combined. The Kp2 mask was substituted by the combined masks K and Q 85% (it preserves nominally 85% of the sky), named KQ85. The Kp0 mask changes to the KQ75 mask. These new masks are also summed up with the point sources mask, including 32 new sources from a preliminary catalog version for the WMAP fifth year (Gold et al. , 2009). For the 7 years of WMAP data the masks were basically modified around the edge of the Galactic cut, particularry in the Gum and Ophiuchus regions. The updated masks named KQ85y7 and KQ75y7 leaves in total 78.3% and 70.6% of the sky respectively (Gold et al. , 2011).

25 Figure 3.2: WMAP temperature maps for the 7 years data release in 2011 are seen in the left column. The right column displays the diﬀerence between the 7 year maps and the previous maps from the 5 years WMAP data release.

Jarosik et al. (2011)

26 Figure 3.3: ILC 7 year map produced by the WMAP team.

27 Chapter 4

The quadrant asymmetry

The results presented here were derived from the analysis of three temperature ILC maps from the third, fifth and seventh year of WMAP data: WILC3 (Hinshaw et al. , 2007), WILC5 (Gold et al. , 2009) and WILC7 (Jarosik et al. , 2011). We computed the TPCF for the WMAP data and for the MC simulations to obtain, evaluate and finally compare the results with the ΛCDM model. The HEALPix (hierarchical equal area and isolatitude pixelization) package (synfast) (Górski et al. , 2005) was used to generate the MC simulations and analyze the maps. The simulated sky maps were generated with two different seed spectra. In the first run, we used the WMAP5 best-fit spectrum to the ΛCDM model, 1 available at LAMBDA , to generate 1000 simulations with Nside = 256 (pixel diameter ∼ 140). In the second run, we modified the ΛCDM model spectrum, substituting the amplitudes of the best-fit quadrupole and octopole by the values reported in the WMAP five-year data. This modified spectrum was used to generate 1000 simulations with the same resolution as the first run (the MΛCDM model). The data and the simulations were analyzed by means of the TPCF. Since it is hard to analyze the CMB angular distribution because of the foreground contamination, even using multifrequency maps, we chose to use three different Galactic cuts. When the first CMB maps covering the whole sky were released from the Relict satellite (Strukov & Skulachev, 1984), the safest way to deal with the foreground contamination was making parallel cuts above and below the Galactic plane. More recently, the masks developed by the WMAP team remove in a fair way the known point sources and the Galactic signal, but they are not perfect since the Galactic foregrounds are still not completely known. Smoot et al. (1992) also warned that the TPCF is highly affected when regions

1http://lambda.gsfc.nasa.gov

28 |b| < 10◦ are included in the analyses. Taking into account these results, we used the WMAP KQ85 mask, a Galactic cut |b| < 10◦ and the WMAP KQ85 mask + |b| < 10◦ Galactic cut in the present work. This last cut avoids the Galactic foreground and also the known point sources. All calculations were performed for degraded maps with Nside = 64. We analyzed the NWQ with 9906, 10200 and 9495 pixels for each Galactic cut. For the SWQ, 10423, 10200 and 9795 pixels were used, for the NEQ, 10250,10176 and 9417 pixels were used, and for the SEQ 9854, 10176, 9350 pixels were used, when the Galactic foreground was removed with the WMAP KQ85 mask, the Galactic cut |b| < 10◦, and the WMAP KQ85 mask + |b| < 10◦ Galactic cut, respectively. We also counted the pixels left in each quadrant when the new WMAP mask KQ85y7 was used to remove the Galactic foregorund, there were 9571 pixels in the NWQ, 10093 pixels in the SWQ, 9876 pixels in the NEQ, and 9217 pixels in the SEQ. The inﬂuence of the KQ85 mask asymmetry on the results was evaluated considering the three Galactic cuts. The same procedure was applied to the ΛCDM simulated sky maps. The MΛCDM realizations were analyzed using only the WMAP KQ85 mask. This modiﬁed spectrum enabled us to evaluate if the observed low quadrupole and octopole values can account for the results. We did not repeat the analysis using the KQ85y7 mask for the MC simulations because the computational time for computing the TPCF is prohibitive.

4.1 Mehod: The two-point angular correlation function

The TPCF function measures the angular correlation of temperature ﬂuctua- tions distributed in the sky and is deﬁned as

c(γ) ≡ hT (np)T (nq)i. (4.1)

T (np) and T (nq) are the temperature ﬂuctuations of the p and q pixels, respectively, and γ is the angular distance between the two pixels.

The pixels p and q are deﬁned by the coordinates (θp, φp) and (θq, φq), where 0◦ ≤ φ ≤ 360◦ and −90◦ ≤ θ ≤ 90◦. It is now possible to obtain the equation for γ:

cos γ = cos θp cos θq + sin θp sin θq cos(φp − φq). (4.2)

Finally, we deﬁne an rms-like quantity, σ, to compare the TPCF computed both for WMAP data and MC simulations (Bernui. et al. , 2006):

29 v u Nbins u 1 X σ = t f 2. (4.3) N i bins i=1

The fi corresponds to the TPCF for each bin i. We used the number of bins

Nbins = 90 to quantify our results.

4.2 Results

As mentioned before, we divided the CMB sky into quadrants and computed the TPCF for each quadrant, using the three diﬀerent Galactic cuts: the WMAP KQ85 mask, the Galactic cut |b| < 10◦ and the WMAP KQ85 mask + |b| < 10◦ Galactic cut. The results are shown in Figures 4.1 to 4.5. Asymmetries between the SEQ and the other quadrants can be easily noticed in the curves. On the other hand, the TPCF curves for the NWQ, the NEQ and the SWQ are mostly inside the gray-shadowed area, which corresponds to the MC 1σ interval. Tables 4.1, 4.2 and 4.3 show the probability for these asymmetries to occur for the ILC maps, and for the MC maps derived from the ΛCDM model. Both consider the above mentioned Galactic cuts. These tables also contain two sets of probabilities, P1 and P2. The probability P1 refers to the chance that exactly the same asymmetries found in the WMAP sky maps appear in the MC simulations:

• σSEQ/σNWQ(simul.) ≥ σSEQ/σNWQ(data)

• σSEQ/σSWQ(simul.) ≥ σSEQ/σSWQ(data)

• σSEQ/σNEQ(simul.) ≥ σSEQ/σNEQ(data).

The probability P2 extends the range of P1, comparing the chance that the ratio between the asymmetries found in the SEQ and in any of the three quadrants (from the MC simulations) to exceed that from the ratio between the SEQ and one given quadrant. The ** in the expressions below apply for any of the three quadrants (∗∗ = NW or SW or NE).

• σSEQ/σ∗∗Q(simul.) ≥ σSEQ/σNWQ(data)

• σSEQ/σ∗∗Q(simul.) ≥ σSEQ/σSWQ(data)

• σSEQ/σ∗∗Q(simul.) ≥ σSEQ/σNEQ(data).

30 Figure 4.1: TPCF curves computed for the WILC7 map using the WMAP KQ85 mask, smoothed for illustration purposes.The NWQ (top) and the NEQ (bottom) appear as solid red lines. The shadow part depicts the standard deviation intervals for 1000 simulated maps produced with the ΛCDM spectrum.

31 Figure 4.2: Same as Figure 4.1, but now with the SWQ at the top and the SEQ at the bottom.

32 Figure 4.3: Same as Figure 4.1, but now using the WMAP KQ85 mask + |b| < 10◦ Galactic cut in the temperature WMAP7 data. From top to bottom, the curves refer to the NWQ and the NEQ, respectively.

33 Figure 4.4: Same as Figure 4.3. From top to bottom, the curves refer to the SWQ and the SEQ, respectively.

34 Figure 4.5: Same as Figures 4.1-4.4, but now using |b| < 10◦ Galactic cut, again in the temperature WILC7 map. From top to bottom the curves refer to the NWQ and the NEQ, respectively.

35 Figure 4.6: Same as Figure 4.5 for the SWQ (top) and the SEQ (bottom).

36 Table 4.1: Calculated probabilities of ﬁnding the same asymmetries as in WMAP data in the MC simulations using the WMAP KQ85 mask and considering the ΛCDM model.

1 Map σSEQ/σNWQ P1 σSEQ/σSWQ P1 σSEQ/σNEQ P1 WILC7 4.6 0.5% 4.6 0.5% 6.7 0.1% WILC5 4.2 0.9% 4.1 0.9% 7.0 0.1% WILC3 3.9 1.3% 4.1 0.9% 6.8 0.1% 2 Map σSEQ/σNWQ P2 σSEQ/σSWQ P2 σSEQ/σNEQ P2 WILC7 4.6 1.7% 4.6 1.7% 6.7 0.2% WILC5 4.2 2.6% 4.1 2.8% 7.0 0.1% WILC3 3.9 3.5% 4.1 2.8% 6.8 0.1%

1Probability of ﬁnding, in this case, the asymmetry between the SEQ and the NWQ quadrants in the simulations. 2Probability of ﬁnding the asymmetry between the SEQ quadrant and any other quadrant in the simulations.

Table 4.2: Calculated probabilities of ﬁnding the same asymmetries as in WMAP data in the MC simulations using the WMAP KQ85 mask + |b| < 10◦ Galactic cut and considering the ΛCDM model.

Map σSEQ/σNWQ P1 σSEQ/σSWQ P1 σSEQ/σNEQ P1 WILC7 2.7 4.9% 2.3 7.4% 2.5 6.0% WILC5 2.3 7.4% 2.2 8.5% 3.2 2.7% WILC3 2.4 6.5% 2.4 6.5% 3.0 3.1%

Map σSEQ/σNWQ P2 σSEQ/σSWQ P2 σSEQ/σNEQ P2 WILC7 2.7 10.1% 2.3 16.0% 2.5 12.8% WILC5 2.3 16.0% 2.2 18.3% 3.2 5.4% WILC3 2.4 14.0% 2.4 14.0% 3.0 6.5%

Table 4.3: Calculated probabilities of ﬁnding the same asymmetries as in WMAP data in the MC simulations using the |b| < 10◦ Galactic cut and considering the ΛCDM model.

Map σSEQ/σNWQ P1 σSEQ/σSWQ P1 σSEQ/σNEQ P1 WILC7 3.7 1.6% 4.0 1.2% 2.8 4.8% WILC5 3.6 1.8% 3.6 1.8% 3.4 2.5% WILC3 3.4 2.5% 3.6 1.8% 3.1 3.2%

Map σSEQ/σNWQ P2 σSEQ/σSWQ P2 σSEQ/σNEQ P2 WILC7 3.7 3.2% 4.0 2.8% 2.8 8.8% WILC5 3.6 3.4% 3.6 3.4% 3.4 4.1% WILC3 3.4 4.1% 3.6 3.4% 3.1 5.8%

37 Table 4.4: Sigma ratios for the simulated CMB maps considering the ΛCDM model and the three Galactic cuts.

Galactic cut σSEQ/σNWQ σSEQ/σSWQ σSEQ/σNEQ +1.1 +1.2 +1.0 KQ85 1.0−0.5 1.1−0.6 0.9−0.5 ◦ +1.0 +1.0 +1.0 KQ85 + |b| < 10 1.0−0.5 1.0−0.5 1.0−0.5 ◦ +1.0 +1.0 +1.0 |b| < 10 1.0−0.5 1.0−0.5 1.0−0.5

4.2.1 The cold spot

Because the asymmetry was coincidentally found in the same quadrant as the so-called cold spot (see (Cruz et al. , 2007; Vielva & Sanz, 2010)), we tested if they are possibly correlated. We masked the cold spot region which is centered on φ = 209◦ and θ = 141◦, and calculated the TPCF for the SEQ using also the KQ85 mask. The cold spot mask radii were 5◦, 10◦ and 15◦. Finally, to look for a speciﬁc region in the SEQ that could account for the asymmetry, we scanned this quadrant, masking four diﬀerent regions in addition to the cold spot region. The masks were centered on coordinates chosen randomly on (φ = 315◦, θ = 157◦), (φ = 225◦, θ = 113◦), (φ = 270◦, θ = 135◦), (φ = 315◦, θ = 113◦), all of them with a radius of 15◦. The TPCF was computed using a mask in each position at a time. The results are shown in Figure 4.7 (bottom).

4.3 Discussion

Our analyses indicate that, even though the excess of power in the SEQ is independent of the chosen Galactic cut, the largest asymmetry between the SEQ and the other quadrants depends on the Galactic cut and also on which ILC map is used. Considering the analysis performed only with the KQ85 mask, the biggest asymmetry was found between the SEQ and the NEQ for all three ILC maps, the largest corresponding to the WILC5 map. Moreover, when only the KQ85 mask was used, the asymmetries between the SEQ and the other three quadrants for the other Galactic cuts are smaller. However, we found that the chance of having any asymmetry is quite low. Table 4.4 shows that most of the sigma ratios between the SEQ and the other three quadrants obtained from the TPCF computed from the data (see Eq. 4.3) are outside the standard deviation interval of the sigma ratios obtained from the TPCF computed for the MC simulations. Indeed, most of the sigma ratios between the SEQ and any other quadrant obtained from the TPCF computed for the simulated sky maps average one, as

38 Figure 4.7: Top: Comparison between the SEQ quadrant for the TPCF using the KQ85 mask + these circular masks centered on the cold spot can be seen with a radius of 5 degrees (red line), 10 degrees (yellow line) and 15 degrees (green line). The black line refers to the function without masking the Cold Spot. Bottom: TPCF for the WMAP7 map using KQ85 + circular masks centered on the speciﬁed angles with a radius of 15 degrees each. The red dashed-dot curve refers to the Cold Spot region.

39 Figure 4.8: Comparison between NWQ (top-lef), NEQ (top-right), SWQ (bottom-left) and SEQ (bottom-right) for the TPCF using the KQ85 mask (solid blue line) and the KQ85y7 (solid red line). expected, because we generated realizations with randomly distributed temperature fluctuations in the sky. Finally, it is possible to see that the excess of power in the SEQ is independent of the chosen Galactic cut. However, the chosen Galactic cut influences the size of this excess in comparison to the other quadrants. Furthermore, using the asymmetric mask does not change our results drastically. We also notice this by analyzing the effect of the new mask provided for the WMAP sseven-year data release (KQ85y7) in TPCF. There is no clear difference in the results of the TPCF for each quadrant using the WILC7 + KQ85 and WILC7 + KQ85y7 data sets, as can be seen from Figure 4.8. The values found for the sigma ratios between the SEQ and the other quadrants in this last case were

σSEQ/σNWQ = 4.6, σSEQ/σSWQ = 4.7 and σSEQ/σNEQ = 6.8 (see Table 4.1 for comparison). Furthermore, using the MΛCDM, which includes the WMAP best-ﬁt values for the quadrupole and octopole, instead of the ΛCDM to generate the MC simulations does not noticeably change neither P1 nor P2, as can be seen in Table 4.5. Based upon these results, we are conﬁdent to state that the low

40 Table 4.5: Calculated probabilities of ﬁnding the same asymmetries as in WMAP data in the MC simulations using the WMAP KQ85 mask and considering the MΛCDM model.

Map σSEQ/σNWQ P1 σSEQ/σSWQ P1 σSEQ/σNEQ P1 WILC7 4.6 0.7% 4.6 0.7% 6.7 0.2% WILC5 4.2 1.3% 4.1 1.3% 7.0 0.2% WILC3 3.9 1.5% 4.1 1.3% 6.8 0.2%

Map σSEQ/σNWQ P2 σSEQ/σSWQ P2 σSEQ/σNEQ P2 WILC7 4.6 1.7% 4.6 1.7% 6.7 0.4% WILC5 4.2 2.6% 4.1 2.9% 7.0 0.3% WILC3 3.9 3.9% 4.1 2.9% 6.8 0.4%

values for the quadrupole and octopole in the data are not responsible for this quadrant asymmetry. Finally, no evidence for a relationship between the SEQ asymmetry and the cold spot was found. The excess of power in the SEQ was most evident above the angular distance of 100 degrees. By masking the cold spot, we were able to identify small diﬀerences in the behavior of the TPCF, all of them remaining within the angular distance of 100 degrees. The same result was found for the other randomly chosen regions in the SEQ quadrant and no evidence was found that can relate the reported asymmetry to any of these chosen regions in the SEQ.

4.4 Conclusions

We found a significant asymmetry between the SEQ and the other quadrants by considering the temperature WMAP ILC maps. We calculated the probability of occurrence for this asymmetry using MC simulations, and 1 out of 1000 simulations for the ΛCDM model corresponded to the SEQ-NEQ asymmetry found in the WILC7 using the KQ85 mask. We also showed that different Galactic cuts do not influence the result in a significant way, leading us to believe that this effect is not caused by the asymmetric mask. Moreover, the use of KQ85y7 preserves the same asymmetries (SEQ-NEQ, SEQ-SWQ and SEQ-NWQ), as expected. Considering all Galactic cuts and maps used, the highest probability of having an asymmetry is 8.5%, for the WILC5 with the KQ85 mask + data clipping for |b| < 10◦. The possibility that the asymmetries described in this work were related to the reported lack of power in the quadrupole and octopole was tested. We constructed simulations based on a modified ΛCDM model, adjusting the amplitudes of the quadrupole and the octopole to their observational WMAP values.

41 No explicit relation between the quadrant asymmetries and the low amplitude of the first two non-zero multipoles was found. Furthermore, we found no evidence of a relationship between the cold spot region and the SEQ excess of power, as pointed out by Bernui (2009), who suggested that the Cold Spot is responsible for 60% of the Southern Hemisphere power. Masking this region and some other regions in the SEQ does not change the TPCF in a significant way, leading us to conclude that the asymmetries between the SEQ and the other quadrants are not related to any specific region in the SEQ. We conclude that the excess of power found in the SEQ is likely related to the north-south asymmetry, in which the South Hemisphere presents more power than the Northern one (see (Eriksen et al. , 2004; Hansen et al. , 2004a) and (Paci et al. , 2010; Pietrobon et al. , 2010; Vielva & Sanz, 2010) for recent discussions on the topic). Our results support the claims that there is indeed a north-south asymmetry and show that the excess of power occurs in the SEQ. Additional investigation is needed to find a better explanation for the north- south asymmetry. An explanation for these asymmetries is still missing. They could be primordial or caused by residual foregrounds or systematic effects. The upcoming CMB data from the Planck satellite when analyzed with more accurate foreground removal techniques will enable us to study in more detail the CMB anomalies reported in the literature. Finally, in addition to the SEQ excess of power, we can notice a lack of correlation in the SWQ, NWQ and NEQ in the TPCF in scales between 20 and 100 degrees for all Galactic cuts in the present work (see Figures 4.1-4.6). A lack of correlation in the TPCF was already reported by Copi et al. (2007) in scales above 60 degrees for a full sky analysis using the Kp0 mask in the first and third year of WMAP data.

42 Part II

Constraining cosmological parameters

43 Chapter 5

The Planck satellite

The Planck satellite is a project of the European Space Agency (ESA) and it was launched on May 14th of 2009 ( for an illustration see Figure 7.1). Planck is orbiting around the second Lagrange point of the Sun-Earth system located 1.5 million kilometers away from the Earth. The main goal of the Planck mission is to measure the CMB temperature anisotropies with unprecedented sensitivity, distinguishing temperature variation of about 1µK.

Figure 5.1: Planck satellite

Credit to ESA

The spacecraft weighted 1.9 tonnes in the time of its launch and it reaches

44 4.2 meters high. The satellite consists of a payload with 74 detectors that are sensitive to frequencies ranging from about 25GHz to 1000GHz. The angular resolution of the devices varies from nearly 30 arcminutes at the lowest frequencies to nearly 5 arcminutes at the highest one (Collaboration et al. , 2011). Planck’s Gregorian telescope was design with two mirrors, being the primary one of 1.9m ×1.5m and the secondary one of 1.1m × 1.0m. The secondary mirror is responsible for focusing the light in the low frequency instrument (LFI) and the high frequency instrument (HFI). The LFI consists of an array of diﬀerential radiometers cooled to 20 K covering 3 frequency bands centered at 30, 44 and 70GHz. This last frequency is the cleanest CMB window in the angular resolution of 13 arcminuts since it is near the minimum of the foreground emission. The other channels, besides measuring the CMB signal, will enable the studies of Galactic emission (Bersanelli et al. , 2010; Mennella et al. , 2011). The HFI instrument detectors (in this case bolometers) covers 6 frequency bands centered at 100, 143, 217, 353 and 545GHz. The ﬁrst 4 channels are design to measure CMB temperature and polarization signal. The last two are optimized to foreground understanding, enabling its separation from the CMB signal (Lamarre et al. , 2010).

45 Chapter 6

The SDSS galaxy survey

The SDSS is a photometric and spectroscopic survey that released its ﬁrst observational results in 2001. The 2.5m telescope located at the Apache Point in New Mexico can be seen in Figure 6.1. The imaging survey predominantly covers the Northern Galactic cap with nearly 10,000 square degrees. Based on the objects detected in the imaging survey the ones whose spectra will be taken can be chosen. Regarding the sample of galaxies, one of these targets is the Bright Red Galaxies (BRG) sample that contain about 105 galaxies. They can have their redshifts well determined by their SDSS spectra (for an example of see Figure 6.2) . For a technical introduction to the survey see York et al. (2000).

Figure 6.1: SDSS survey telescope

http://www.sdss.org

By the seventh data release of SDSS in 2009 there were over 1.6 million

46 Figure 6.2: Large scale structure in the Northern equatorial slice of SDSS main galaxy redshift sample.

http://www.sdss.org

spectra divided among galaxies (930,000), quasars(120,000) and stars (46,000) (Abazajian et al. , 2009). Currently, the SDSS team released the ninth data collection from SDSS including the extension of the original project York et al. (2000), named SDSS III. This extended project now includes surveys to study dark energy, the structure of the Milk Way and to search for exoplanets (Col- laboration et al. , 2012).

47 Chapter 7

Mixed models (adiabatic + isocurvature initial ﬂuctuations)

7.1 Isocurvature notation

In this work, we consider the standard isocurvature CDM mode generated during inﬂationary time that reads:

δρCDM 3 δργ SCDM = − . (7.1) ρCDM 4 ργ Following Bean et al. (2006), we write the isocurvature contribution to the purely adiabatic CMB power spectra in the form:

ad iso p cross Cl = (1 − α)Cl + αCl + 2β α(1 − α)Cl , (7.2) where the parameter α accounts for the isocurvature amplitude, while β stands for the isocurvature correlation phase, given by β = cos θ, −1 ≤ cos θ ≤ 1. In the same way, the matter power spectrum, P (k), can be decomposed into purely adiabatic, purely isocurvature and their cross correlation contribution:

P (k) = (1 − α)P (k)ad + αP (k)iso + 2βpα(1 − α)P (k)cross, (7.3) where

k ni−1 P i(k) = T i(k) , i = ad, iso or cross (7.4) k0 As stated by Bean et al. (2006) it is reasonable to assume a cross spectrum

48 1 independent of scale, ncross = 2 (nad + niso). We take the pivot value k0 = 0.002 Mpc−1 as used by the WMAP team.

7.2 Methodology

In order to search how an isocurvature contribution would aﬀect the measurements of cosmological parameters we apply the Fisher information matrix formalism to a Planck-like experiment, considering both temperature and polarization for the lensend and unlensed CMB spectrum, and to the SDSS. We consider both the axion and the curvaton scenarios in a ΛCDM model.

7.2.1 Information from CMB

The Fisher information matrix for the CMB temperature anisotropy and polarization is given by the approximation in (Zaldarriaga & Seljak, 1997):

X Y X X ∂Cl −1 ∂Cl Fij = (Covl )XY , (7.5) ∂pi ∂pj l XY

X where Cl is the power in the lth multipole, X stands for TT (temperature), EE (E-mode polarization), BB (B-mode polarization) and TE (temperature and E-mode polarization cross-correlation). We will not include primordial B- BB modes in the analysis since the measurement of the primordial Cl by Planck is expected to be noise-dominated. Therefore, our covariance matrix becomes therefore:

  ΞTTTT ΞTTEE ΞTTTE 2 Covl =  Ξ Ξ Ξ  . (7.6) (2l + 1)fsky  EETT EEEE EETE  ΞTETT ΞTEEE ΞTETE The elements of the covariance matrix in this unlensed case are:

TT TT 2 ΞTTTT = (Cl + Nl ) , (7.7)

EE PP 2 ΞEEEE = (Cl + Nl ) , (7.8)

BB PP 2 ΞBBBB = (Cl + Nl ) , (7.9)

TE 2 TT TT ΞTETE =(Cl ) + (Cl + Nl ) (7.10) EE PP × (Cl + Nl ),

49 TE 2 ΞTTEE = (Cl ) , (7.11)

TE TT TT ΞTTTE = Cl (Cl + Nl ), (7.12)

TE EE PP ΞEETE = Cl (Cl + Nl ), (7.13)

ΞTTBB = ΞEEBB = ΞTEBB = 0. (7.14)

TT PP In these equations, Nl and Nl are the Gaussian random detector noises for temperature and polarization, respectively, whose expression is written using 2 2 the window function, Bl = exp[−l(l + 1)θbeam/8 ln 2] and the inverse square of the detector noise level for temperature and polarization, wT and wP . The Full −2 Width Half Maximum , θbeam, is used in radians, and w = (θbeamσ) is the weight given to each considered Planck channel (Eisenstein et al. , 1999) . The experimental speciﬁcations can be checked in Table 7.1:

TT 2 2 2 2 −1 Nl = [(wT Bl )100 + (wT Bl )143 + (wT Bl )217 + (wT Bl )353] (7.15)

PP 2 2 2 2 −1 Nl = [(wP Bl )100 + (wP Bl )143 + (wP Bl )217 + (wP Bl )353] (7.16) Here we used four channels, 100, 143, 217 and 353 GHz of the Planck experiment as can be seen from the equations 7.15 and 7.16.

Table 7.1: Planck speciﬁcations

Frequency (GHz) θbeam σT (µK − arc) σP (µK − arc) 100 9.5’ 6.82 10.9120 143 7.1’ 6.0016 11.4576 217 5.0’ 13.0944 26.7644 353 5.0’ 40.1016 81.2944 http://www.rssd.esa.int/SA/PLANCK/docs/Bluebook-ESA- SCI(2005)1 V2.pdf

For the lensed case we have to perform a correction in the covariance matrix elements taking into consideration the power spectrum of the deﬂection angle T d and its cross correlation with temperature, Cl . We also change in this case the X ˜ X unlensed CMB power spectra, Cl , for the lensed ones, Cl . When we include these corrections, the covariance matrix becomes:

50 2 Cov = l (2l + 1)fsky   ξTTTT ξTTEE ξTTTE ξT T T d ξT T dd 0    ξEETT ξEEEE ξEETE 0 0 0    (7.17)  ξ ξ ξ 0 0 0   TETT TEEE TETE    .  ξT dT T 0 0 ξT dT d ξT ddd 0     ξ 0 0 ξ ξ 0   ddT T ddT d dddd  0 0 0 0 0 ξBBBB

Note that in this case we are taking into consideration the B-mode polarization generated by the CMB gravitational lensing from the E-mode polarization. In both cases, we used fsky = 0.65. The Fisher elements corrections for the lensed case are (Perotto et al. , 2006):

2 ˜ TT TT ξTTTT = Cl + Nl

2 2 ˜ TE ˜ T d 2 Cl Cl (7.18) − , EE 2 ˜ PP dd dd2 Cl + Nl Cl + Nl

2 ˜ EE PP ξEEEE = Cl + Nl , (7.19)

2 ˜ BB PP ξBBBB = Cl + Nl , (7.20)

1 TE 2 TT EE ξ = C˜ + C˜ + N TT C˜ + N PP TETE 2 l l l l l EE ˜ PP T d2 (7.21) Cl + Nl Cl − , dd dd 2 Cl + Nl

1 2 TT ξ = CT d + C˜ + N TT Cdd + N dd T dT d 2 l l l l l TE 2 dd dd ˜ (7.22) Cl + Nl Cl − , EE ˜ EE 2 Cl + Nl

dd dd2 ξdddd = Cl + Nl , (7.23)

2 ˜ TE ξTTEE = Cl , (7.24)

51 " T d2 # TE TT C ξ = C˜ C˜ + N TT − l , (7.25) TTTE l l l dd dd Cl + Nl

˜ TE ˜ EE PP ξTEEE = Cl Cl + Nl , (7.26)

T d2 ξT T dd = Cl , (7.27)

" ˜ TE 2 # T d ˜ TT TT (Cl ) ξT T T d = Cl (Cl + Nl ) − , (7.28) ˜ EE EE (Cl + Nl )

T d dd dd ξT ddd = Cl Cl + Nl , (7.29)

dd where Nl is the optimal quadratic estimator. Here we consider the TT quadratic estimator since it provides the best estimator for the Planck experiment (for a review, see Okamoto & Hu (2003); Hu & Okamoto (2002)) noise of the deﬂection ﬁeld and it can be written in the form (Hu, 2002):

−1 " TT TT 2 # dd X (Cl2 Fl1ll2 + Cl1 Fl2ll1) Nl = , (7.30) ˜ TT TT ˜ TT TT l1l2 2(Cl1 + Nl1 )(Cl2 + Nl2 )

r ! (2l1 + 1)(2l + 1)(2l2 + 1) l1 l l2 Fl1ll2 = 4π 0 0 0 1 × [l(l + 1) + l (l + 1) − l (l + 1)] . 2 2 2 1 1

Note that the Fisher matrix analysis approximates the likelihood as Gaus- sian function; however, the likelihood function could in general be non-Gaussian. Nonetheless, as stated in (Perotto et al. , 2006), CMB lensing information gives a more Gaussian-like function, breaking some parameters degeneracies and con- sequently providing a better error estimation.

7.2.2 Information from galaxy survey

The Fisher information matrix for the matter power spectrum obtained from galaxy surveys is given by (Tegmark, 1997):

Z kmax ∂ ln P (k) ∂ ln P (k) k2dk Fij = Veﬀ 2 , (7.31) kmin ∂pi ∂pj (2π)

Z 2 n¯(r)Pg(k) 3 Veﬀ(k) = d r. (7.32) 1 +n ¯(r)Pg(k)

52 2 We know that Pg(k) = b P (k) and using the speciﬁcations of the SDSS experiment for the BRG sample, called Luminous Red Galaxies (LRG) in more recent papers, we assume a linear and scalar independent bias b = 2 (Tegmark, 1997; Hutsi,¨ 2006). It is assumed that the expected number density of galaxies, 5 n¯(r), is independent of r, n = 10 /Vs, in a volume-limited sample to a depth of 103Mpc. The survey has an angle of π steradians; therefore the survey volume 9 becomes, Vs = 10 π/3 (Tegmark, 1997; Eisenstein et al. , 1999, 2001).

7.3 Results

For the purpose of our analysis, we consider as free cosmological parameters the adimensional value of the Hubble constant, h, the density of baryons and 2 2 cold dark matter, Ωbh and Ωch , the spectral index of scalar adiabatic perturbations, nad, the aforementioned isocurvature parameters, α and β, and the equation of state of dark energy w, assumed as a constant. We ﬁrst performed the forecast for Planck alone, with and without considering CMB lensing. Then, we introduced the forecast for SDSS, combining the results as (assuming that SDSS and CMB results can be well approximated as independent ones)

Total Planck SDSS Fij = Fij + Fij . (7.33)

We considered 3 diﬀerent scenarios to constrain the cosmological parameters.

First we use an axion scenario considering that a high niso = 1.9 ± 1 is favored by Lyman-α data (Beltrán et al. , 2005). Kasuya & Kawasaki (2009) proposed an axion model capable of generating isocurvature fluctuations with an extremely blue spectrum with 1 < niso ≤ 4. Taking into consideration that Planck could measure the existence of an isocurvature contribution with a high spectral index motivates our parameter forecast of such a model. We choose a fixed niso = 2.7 considering that Beltrán et al. (2005) tested the robustness of their result finding an extreme model with niso = 2.7. Bean et al. (2006) found this same value for the isocurvature spectral index for their best fit model considering an adiabatic-plus-isocuvarture CDM contribution, however, for a generally correlated isocurvature component with respect to the adiabatic one.

It was shown, however, that the chosen pivot scale affects the niso likelihood (Kurki-Suonio et al. , 2005). The previous mentioned articles (Beltrán et al. −1 , 2005; Bean et al. , 2006) used a pivot scale k0 = 0.05 Mpc that favors artificially large niso according to Kurki-Suonio et al. (2005). They chose instead −1 k0 = 0.01 Mpc and found that the likelihood for niso peaks at approximately −1 3. It was also shown that for k0 < 0.01 Mpc the results does not change drastically, concluding that our choice for niso = 2.7 is valid.

53 Table 7.2: Marginalized errors for ΛCDM model plus a contribution of initial isocurvature ﬂuctuation with ﬁducials amplitude of α = 0.06, correlation phase of β = 0 and scalar spectral index of niso = 2.7.

Parameter CMB alone CMB alone P(k) alone CMB + P(k) CMB + P(k) Planck Planck SDSS Planck + SDSS Planck +SDSS T + P T+ P+ lens T + P T + P + lens h 0.031 0.021 0.43 0.0073 0.0068 2 h Ωb 0.00013 0.00012 0.038 0.00012 0.00012 2 h Ωc 0.0011 0.00089 0.14 0.00093 0.00079 ns 0.0081 0.0045 0.70 0.0021 0.0016 α 0.0011 0.00089 0.28 0.00060 0.00056 β 0.00077 0.00040 0.17 0.00048 0.00035 w 0.071 0.048 2.21 0.012 0.012 Percentage of the parameters’ ﬁducial values for each error above Parameter CMB alone CMB alone P(k) alone CMB + P(k) CMB + P(k) Planck Planck SDSS Planck + SDSS Planck +SDSS T + P T+ P+ lens T + P T + P + lens h 4.21% 2.71% 58.42% 0.99% 0.92% 2 h Ωb 0.56% 0.52% 164.1% 0.52% 0.52% 2 h Ωc 1.03% 0.83% 130.96% 0.87% 0.74% ns 0.81% 0.45% 70% 0.21% 0.16% α 1.83% 1.48% not constrained 1.0% 0.93% β ----- w 7.1% 4.8% not constrained 1.2% 1.2%

2 In this case our fiducial model is given by h = 0.736, Ωbh = 0.02315, 2 Ωch = 0.1069, nad = 0.982 (niso = 2.7 fixed) α = 0.06, β = 0 and w = −1. Fisher contours and all the 1 sigma errors are shown in Fig. 7.1 and Table 9.3. Our first approach was to let β vary, obtaining in this way a lower and upper limit to the isocurvature correlation phase. Even if not predicted by the chosen inflationary scenario (axion type), we can still constrain a possibly nonzero measurement of β where this scenario is still valid. A second approach was to keep β fixed, as is showed in Table 7.3. It can be noticed that there is not a significant change in the constraints on the other cosmological parameters between Tables 9.3 and 7.3, especially for α. We found the best upper limits for the isocurvature contribution from this axion type of nonadiabatic fluctuation, considering a ΛCDM cosmological model, for the combined Planck (considering CMB lensing) + SDSS forecast with α < 0.061 (95% Confidence Level, CL) and −0.0007 < β < 0.0007 (95% CL). If β is not allowed to vary, the result for α’ s upper limit is not significantly changed. In this first scenario, we can also see in Fig. 7.1 that lensing information can break the degeneracy between α and β.

54 Table 7.3: The same as Table 9.3, but in this case β = 0 will be kept ﬁxed.

Parameter CMB alone CMB alone P(k) alone CMB + P(k) CMB + P(k) Planck Planck SDSS Planck + SDSS Planck +SDSS T + P T+ P+ lens T + P T + P + lens h 0.021 0.019 0.43 0.0072 0.0066 2 h Ωb 0.00012 0.00011 0.038 0.00011 0.00011 2 h Ωc 0.0011 0.00082 0.13 0.00082 0.00068 ns 0.0039 0.0036 0.38 0.0013 0.0013 α 0.00094 0.00084 0.12 0.000562 0.000558 w 0.045 0.042 0.91 0.012 0.012 Percentage of the parameters’ ﬁducial values for each error above Parameter CMB alone CMB alone P(k) alone CMB + P(k) CMB + P(k) Planck Planck SDSS Planck + SDSS Planck +SDSS T + P T+ P+ lens T + P T + P + lens h 2.85% 2.58% 58.42% 0.98% 0.90% 2 h Ωb 0.52% 0.47% 164.1% 0.47% 0.47% 2 h Ωc 1.03% 0.77% 121.61% 0.77% 0.64% ns 0.39% 0.36% 38% 0.135% 0.129% α 1.57% 1.40% not constrained 0.94% 0.90% w 4.5% 4.2% 91% 1.2% 1.2%

On the other hand, for the second axion scenario, where the cosmological parameters’ values are the same as the above ones, except for niso = 0.982 fixed (this assumption was made by Bean et al. (2006) and by the WMAP team following Dunkley et al. (2009)), the isocurvature amplitude is better constrained when β is kept fixed (see Tables 9.5, 7.5 and Fig. 7.2 for the Fisher constraints). The value for the limit of α for the Planck forecast only (without including CMB lensing and keeping β fixed) is comparable to the value found by the WMAP team: α < 0.11 (95% CL) for Planck, against α < 0.13 (95% CL) for WMAP 7-year data only (Larson et al. , 2011). However, if we consider the CMB lensing in the analysis we can improve this limit, obtaining α < 0.10 (95% CL) for Planck. Finally, combining Planck (including CMB lensing) + SDSS, we have α < 0.08 (95% CL) against α < 0.064 (95% CL) found earlier with WMAP + BAO + SN (Komatsu et al. , 2011). If, on the other hand, β is allowed to vary, we have that α < 0.12 (95% CL) as our best constraint (Planck + lensing information+ SDSS) and −0.11 < β < 0.11 (95% CL). Even though it is not better constrained, we found an upper limit to α when β is allowed to vary, giving us an extra bonus to also constrain the correlation phase.

Since the values chosen for niso in these first two scenarios are in the limit of the error bars found using Lyman-α data, niso = 1.9 ± 1, for the sake of completeness we tested another scenario for niso = 1.9 finding a significant

55 Table 7.4: Marginalized errors for ΛCDM model plus a contribution of initial isocurvature ﬂuctuation with ﬁducials amplitude of α = 0.06, correlation phase of β = 0 and scalar spectral index of niso = nad = 0.982.

Parameter CMB alone CMB alone P(k) alone CMB + P(k) CMB + P(k) Planck Planck SDSS Planck + SDSS Planck +SDSS T + P T+ P+ lens T + P T + P + lens h 0.051 0.036 0.31 0.0090 0.0081 2 h Ωb 0.00012 0.00011 0.028 0.00011 0.00011 2 h Ωc 0.0011 0.00089 0.095 0.00092 0.00077 ns 0.0033 0.0030 0.26 0.0031 0.0029 α 0.031 0.031 22.55 0.031 0.031 β 0.088 0.079 41.78 0.055 0.054 w 0.13 0.093 5.40 0.016 0.015 Percentage of the parameters’ ﬁducial values for each error above Parameter CMB alone CMB alone P(k) alone CMB + P(k) CMB + P(k) Planck Planck SDSS Planck + SDSS Planck +SDSS T + P T+ P+ lens T + P T + P + lens h 6.93% 4.89% 42.12% 1.22% 1.10% 2 h Ωb 0.52% 0.47% 120.95% 0.47% 0.47% 2 h Ωc 1.03% 0.83% 88.87% 0.86% 0.72% ns 0.33% 0.30% 26% 0.31% 0.29% α 52.48% 52.48% not constrained 52.48% 51.48% β ----- w 13% 9.3% not constrained 1.6% 1.5%

change only in the isocurvature amplitude α (β kept ﬁxed). In this case, we found that α < 0.066 (95% CL) against α < 0.062 (95% CL) for niso = 2.7 and

α < 0.1 (95% CL) for niso = 0.982, all of them considering Planck only with

CMB lensing information. As expected, α is better constrained for higher niso values. In the last scenario, the isocurvature primordial fluctuations are generated by the decay of the curvaton. Our fiducial parameters’ values are set to be, 2 2 h = 0.745, Ωb = 0.02293h ,Ωc = 0.1058h , nad = 0.984 (niso = 0.984 fixed, as generally predicted by curvaton scenarios (Bean et al. , 2006)), α = 0.003, β = −1 and w = −1. Unlike the first two cases, none of the isocurvature parameters can be constrained if β is allowed to vary, as it can be seen in Table 9.4. Nevertheless, the upper limit found for α (β fixed) is improved, with Planck only, α < 0.0068 (95% CL), compared to the one found for WMAP 7-year data only, α < 0.011 (95% CL). An even better constraint is reached considering the CMB lensing effect in our analysis (see Table 7.7 and Fig. 7.3), improving the Planck limit to α < 0.0054 (95% CL). For Planck (including CMB lensing) + SDSS, α < 0.0039 (95% CL) against α < 0.0037 (95% CL) for WMAP + BAO +

56 Table 7.5: The same as Table 9.5, but in this case β = 0 will be kept ﬁxed.

Parameter CMB alone CMB alone P(k) alone CMB + P(k) CMB + P(k) Planck Planck SDSS Planck + SDSS Planck +SDSS T + P T+ P+ lens T + P T + P + lens h 0.032 0.024 0.30 0.0090 0.0080 2 h Ωb 0.00012 0.00011 0.028 0.00011 0.00011 2 h Ωc 0.0010 0.00086 0.090 0.00092 0.00077 ns 0.0034 0.0030 0.26 0.0031 0.0029 α 0.025 0.020 2.62 0.010 0.0099 w 0.080 0.063 5.39 0.016 0.015 Percentage of the parameters’ ﬁducial values for each error above Parameter CMB alone CMB alone P(k) alone CMB + P(k) CMB + P(k) Planck Planck SDSS Planck + SDSS Planck +SDSS T + P T+ P+ lens T + P T + P + lens h 4.35% 3.26% 40.76% 1.22% 1.09% 2 h Ωb 0.52% 0.47% 120.95% 0.47% 0.47% 2 h Ωc 0.93% 0.80% 84.19% 0.86% 0.72% ns 0.34% 0.30% 26% 0.31% 0.29% α 41.67% 33.33% not constrained 17.5% 16.65% w 8.0% 6.3% not constrained 1.6% 1.5%

SN. Therefore, using other cosmological probes, such as BAO and SN to Planck with lensing information + SDSS, the error bars in the isocurvature amplitude can become even smaller, allowing for a very limited isocurvature contribution in the primordial ﬂuctuations when the it is completely anticorrelated with the adiabatic component.

7.4 Discussion and conclusions

In this second part we started studying a possible contribution of isocurvature initial perturbations in the pure adiabatic ﬂuctuations scenario from the well tested ΛCDM model. Using the Fisher formalism we obtained the best constraints possible for the isocurvature parameters using CMB and galaxy distribution information. One of the goals of this work has been to quantify how CMB lensing information can provide better constraints in the cosmological parameters, especially in the ones related to the isocurvature contribution. Moreover, we saw that CMB lensing information broke the parameter degeneracy between the isocurvature parameters α and β for one of the three studied scenarios. In all tested inﬂationary scenarios, the CMB lensing information improves the constraints of all chosen parameters, including the ones related to the isocur-

57 Table 7.6: Marginalized errors for ΛCDM model plus a contribution of initial isocurvature ﬂuctuation with ﬁducials amplitude of α = 0.003, correlation phase of β = −1 and scalar spectral index of niso = 0.984.

Parameter CMB alone CMB alone P(k) alone CMB + P(k) CMB + P(k) Planck Planck SDSS Planck + SDSS Planck +SDSS T + P T+ P+ lens T + P T + P + lens h 0.055 0.036 0.33 0.0094 0.0084 2 h Ωb 0.00012 0.00011 0.029 0.00011 0.00011 2 h Ωc 0.0011 0.00090 0.096 0.00093 0.00079 ns 0.0030 0.0028 0.26 0.0029 0.0027 α 0.029 0.030 17.63 0.028 0.029 β 4.98 5.19 3084.60 4.95 5.10 w 0.13 0.089 7.48 0.016 0.015 Percentage of the parameters’ ﬁducial values for each error above Parameter CMB alone CMB alone P(k) alone CMB + P(k) CMB + P(k) Planck Planck SDSS Planck + SDSS Planck +SDSS T + P T+ P+ lens T + P T + P + lens h 7.38% 4.83% 44.29% 1.26% 1.13% 2 h Ωb 0.52% 0.48% 126.47% 0.48% 0.48% 2 h Ωc 1.04% 0.85% 90.73% 0.88% 0.75% ns 0.30% 0.28% 26% 0.29% 0.27% α not constrained not constrained not constrained not constrained not constrained β not constrained not constrained not constrained 495% not constrained w 13% 8.9% not constrained 1.6% 1.5%

58 Table 7.7: The same as Table 9.4, but in this case β = −1 will be kept ﬁxed.

Parameter CMB alone CMB alone P(k) alone CMB + P(k) CMB + P(k) Planck Planck SDSS Planck + SDSS Planck +SDSS T + P T+ P+ lens T + P T + P + lens h 0.055 0.0345 0.31 0.0092 0.0083 2 h Ωb 0.00012 0.00011 0.027 0.00011 0.00011 2 h Ωc 0.0010 0.00088 0.092 0.00091 0.00077 ns 0.0027 0.0025 0.24 0.0026 0.0024 α 0.0019 0.0012 0.17 0.00047 0.00045 w 0.14 0.088 7.47 0.016 0.015 Percentage of the parameters’ ﬁducial values for each error above Parameter CMB alone CMB alone P(k) alone CMB + P(k) CMB + P(k) Planck Planck SDSS Planck + SDSS Planck +SDSS T + P T+ P+ lens T + P T + P + lens h 7.38% 4.63% 41.61% 1.23% 1.11% 2 h Ωb 0.52% 0.48% 117.75% 0.48% 0.48% 2 h Ωc 0.94% 0.83% 87.90% 0.86% 0.72% ns 0.27% 0.25% 24% 0.26% 0.24% α 63.33% 40% not constrained 15.67% 15% w 14% 8.8% not constrained 1.6% 1.5%

vature mode. If we consider Planck information alone (with β not allowed to vary) the smallest improvement obtained on α’s standard deviation is in the axion type inflation for nad 6= niso with a difference of 0.17% of its fiducial value between the lensed and unlensed analysis. This improvement gets bigger for the scenarios considered by WMAP reaching almost 9% (axion type with nad = niso) and 25% (curvaton type) (see the lower part where β is kept fixed in Tables 7.3, 7.5 and 7.7). Moreover, if CMB lensing can be measured, it would be possible to distin- guish between the axion models with niso = 0.982 and niso = 2.7 for instance.

The effect of CMB lensing is bigger for higher niso values as can be seen in the comparison of Figs. 7.1 and 7.2. We can better visualize this lensing effect on niso by analyzing the power spectra derivatives in respect to α and β in Figures 7.4 and der2. When the combined Planck + SDSS forecast is done, the improvement with the use of lensing information is not so significant for any of the scenarios. This is due to the poor ability of SDSS to constrain the parameters compared to Planck, especially when CMB lensing information is included. For a CMB experiment alone, or combined with any other precise experiments on galaxy distribution, lensing is important extra information in the attempt to know how well observations can constrain the presence of isocurvature contributions to

59 the primordial ﬂuctuations. An interesting forecast would include future galaxy surveys, such as Euclid, combined with planck CMB information including the lensing eﬀects.

60 Figure 7.1: Fisher contours for ΛCDM model plus a contribution of initial isocurvature ﬂuctuation with ﬁducials amplitude of α = 0.06, correlation phase of β = 0 and scalar spectral index of niso = 2.7 for the axion scenario. The red and blue contours represent 95.4% and 68% C.L., respectively, for the unlesed CMB + SDSS (dashed lines) and for the lensed CMB + SDSS (solid lines)(see Table 9.3).

61 Figure 7.2: Fisher contours for ΛCDM model plus a contribution of initial isocurvature ﬂuctuation with ﬁducials amplitude of α = 0.06, correlation phase of β = 0 and scalar spectral index of niso = 0.982 for the axion scenario. The red and blue contours represent 95.4% and 68% C.L. respectively for the unlesed CMB + SDSS (dashed lines) and for the lensed CMB + SDSS (solid lines) (see Table 9.5).

62 Figure 7.3: Fisher contours for ΛCDM model plus a contribution of initial isocurvature fluctuation with fiducials amplitude of α = 0.003, correlation phase of β = −1 and scalar spectral index of niso = 0.984 for the curvaton scenario. The red and blue contours represent 95.4% and 68% C.L. respectively for the unlesed CMB + SDSS (dashed lines) and for the lensed CMB + SDSS (solid lines). In this case we consider β fixed (see Table 7.7).

63 Figure 7.4: CMB power spectra derivatives in respect to the isocurvature parameters α (purple) and β(red). The dashed darker lines are related to the unlensed power spectra’s derivative and the solid lighter ones are related to the lensed power spectra’s derivative for the axion scenario of α = 0.06, β = 0. On the left column the scalar spectral index is niso = 2.7 and on the right column the scalar spectral index is niso = 0.982

64 Figure 7.5: The same as Figure 7.4 but now the scalar spectral index is niso = 0.982

65 Chapter 8

Euclid survey

The Euclid mission consists of a future spacecraft with 4.5 meters high and a launch mass of around 2100 kg (see Figure 8.1 for an illustration). The mission is planned to take oﬀ in 2020 with a main goal of studying the nature of dark energy and dark matter by accurate measurements of the universe’s accelerated expansion.

Figure 8.1: An Euclid satellite illustration

Credit to ESA

Euclid is optimized for measuring weak gravitational lensing (WL) and baryonic acoustic oscillations (BAO). The WL is a distortion observed in galaxies caused by light deﬂection due to mass inhomogeneities along the line of sight. This distortions are able to map dark matter by statistical measurements. More- over, the BAO refers to baryonic matter density ﬂuctuations imprinted in large

66 scale structure of galaxies, which provide a standard ruler for dark energy measurements. Euclid will orbit around the second Lagrange point of the Sun- Earth system providing imaging and spectroscopic measurements over at least half of the entire sky. Its spacial resolution is of 0.2 second and it will measure shapes and redshifts of galaxies and cluster of galaxies out to redshifts ∼ 2. Its lifetime is predicted for 7 years. For more information about Euclid mission see the red book (http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=48983).

67 Chapter 9

SCM with massive neutrinos and a time evolving dark energy equation of state.

9.1 Methodology

9.1.1 Information from CMB

In this section, we use the fisher matrix formalism as it was done in the Chapter 7. In the case of CMB temperature anisotropy and polarization, the fisher information is given by Equation 7.5. We repeat the same procedure for the CMB forecast without using lensing information (see Equations 7.6 to 7.16). For the lensed case however we performed a correction in the covariance matrix elements taking into consideration the power spectrum of the deflection T d angle and its cross correlation with temperature and E-polarization, Cl and Ed Cl . We used the same procedure introduced in Perotto et al. (2006) to Ed obtain the covariance matrix elements using the new information of Cl power spectrum. First of all, we make use of the effective χ2 defined in Equation (3.3) of Perotto et al. (2006).

X D |C| χ2 = (2l + 1) + ln − 3 (9.1) eff |C| ˆ l |C| where is our case D is deﬁned to be

68 TT EE dd D = Cˆ CEE CddCBB + CTT Cˆ CddCBB + CTT CEE Cˆ CBB BB TE Ed +CTT CEE CddCˆ + 2(Cˆ CEdCT dCBB + CTE Cˆ CT dCBB T d BB TT +CTE CEdCˆ CBB + CTE CEdCT dCˆ ) − CEd(Cˆ CBB CEd (9.2) BB Ed dd +CTT Cˆ CEd + 2CTT CBB Cˆ ) − CTE (Cˆ CBB CTE BB TE EE +CddCˆ CTE + 2CddCBB Cˆ ) − CT d(Cˆ CBB CT d BB T d +CEE Cˆ CT d + 2CEE CBB Cˆ ,

and |C| , |Cˆ| are the determinants of the theoretical and observed data covariance matrices

TT EE dd BB TE Ed T d BB |Cˆ| = Cˆ Cˆ Cˆ Cˆ + 2Cˆ Cˆ Cˆ Cˆ (9.3) TT BB Ed dd BB TE EE BB T d −Cˆ Cˆ (Cˆ )2 − Cˆ Cˆ (Cˆ )2 − Cˆ Cˆ (Cˆ )2,

|C| = CTT CEE CddCBB + 2CTE CEdCT dCBB (9.4) −CTT CBB (CEd)2 − CddCBB (CTE )2 − CEE CBB (CT d)2.

The theoretical covariance matrix M is given by   CTT CTE CT d 0  TE EE Ed   C C C 0  M =   . (9.5)  CT d CEd Cdd 0    0 0 0 CBB .

The ﬁsher matrix information is then derived from the second order derivative of the likelihood function, L, from an observing data set x given the real parameters p1, p2, p3, ..., pn:

∂2 ln L Fij = − , (9.6) ∂pi∂pj x 2 knowing that χeff ≡ −2 ln L, I derived the new elements for the covariance matrix

69 2 Cov = l (2l + 1)fsky   ξTTTT ξTTEE ξTTTE ξT T T d ξT T dd ξT T Ed 0    ξTTEE ξEEEE ξEETE ξEET d ξEEdd ξEEEd 0     ξTTTE ξEETE ξTETE ξT ET d ξT Edd ξT EEd 0  (9.7)      ξT T T d ξEET d ξT ET d ξT dT d ξT ddd ξT dEd 0  .    ξT T dd ξEEdd ξT Edd ξT ddd ξdddd ξddEd 0       ξT T Ed ξEEEd ξT EEd ξT dEd ξddEd ξEdEd 0  0 0 0 0 0 0 ξBBBB .

Note that here we are considering the B-mode polarization generated by the CMB gravitational lensing from the E-mode polarization since I am not including primordial B- modes in the analysis.

2 ˜ TT TT ξTTTT = Cl + Nl (9.8)

2 ˜ EE PP ξEEEE = Cl + Nl , (9.9)

dd dd2 ξdddd = Cl + Nl , (9.10)

2 ˜ BB PP ξBBBB = Cl + Nl , (9.11)

1 TE2 TT EE ξ = C˜ + C˜ + N TT C˜ + N PP , (9.12) TETE 2 l l l l l

1 h 2 TT i ξ = CT d + C˜ + N TT Cdd + N dd , (9.13) T dT d 2 l l l l l

1 h 2 EE i ξ = CEd + Cdd + N dd C˜ + N PP , (9.14) EdEd 2 l l l l l

2 ˜ TE ξTTEE = Cl , (9.15)

T d2 ξT T dd = Cl , (9.16)

Ed2 ξEEdd = Cl , (9.17)

Ed T d ξT Edd = Cl Cl , (9.18)

70 Ed TE ξEET d = Cl Cl , (9.19)

T d TE ξT T Ed = Cl Cl , (9.20)

˜ TE ˜ TT TT ξTTTE = Cl Cl + Nl , (9.21)

˜ TE ˜ EE PP ξEETE = Cl Cl + Nl , (9.22)

T d ˜ TT TT ξT T T d = Cl Cl + Nl (9.23)

T d dd dd ξT ddd = Cl Cl + Nl , (9.24)

Ed dd dd ξddEd = Cl Cl + Nl , (9.25)

Ed ˜ EE PP ξEEEd = Cl Cl + Nl , (9.26)

1 h TE TT i ξ = CT dC˜ + CEd C˜ + N TT , (9.27) T ET d 2 l l l l l

1 h EE TEi ξ = C˜ + N PP CT d + CEdC˜ , (9.28) T EEd 2 l l l l l

1 h TEi ξ = CEdCT d + Cdd + N dd C˜ , (9.29) T dEd 2 l l l l l We use fsky = 0.65 and the Planck speciﬁcation on Table 7.1. As free 2 2 2 parameters we use P = {h Ωb, h Ωc, h Ων , w0, wa}.

9.1.2 Information from galaxy survey

In this subsection, we show how the BAO information can be used to forecast errors in the dark energy parameters using the ﬁsher formalism. It was shown by Seo & Eisenstein (2003) that the Hubble parameter H(z) and the angular diameter distance Da(z) can be measured very precisely by using the BAO information present in the matter power spectrum. H(z) and Da(z) are expected to be determined as a function of redshift by future galaxy surveys. The goal is then to propagate the errors on H(z) and Da(z) to the constraints of dark energy parameters.

71 The observed matter power spectrum

We start deﬁning the observed galaxy power spectrum in a reference cosmology (in our case we use the ΛCDM model), distinguished by the subscript ”ref”, (diﬀerent from the true spectrum, referred as no subscript) that will be used to derive the cosmological parameters constraints using a galaxy survey that covers a wide range of redshifts. Following Seo & Eisenstein (2003),

2 Da(z)ref × H(z) Pobs(kref⊥, krefk) = 2 Pg(kref , kref ) + Pshot (9.30) Da(z) × H(z)ref

Where the Hubble parameter H(z) in a ﬂat Universe is related to the dark energy parameters through

q 3 3(1+w +w ) H(z) = H0 Ωm(1 + z) + Ωde(1 + z) 0 a exp (3wa(a − 1)), (9.31)

and the angular diameter distance was deﬁned in Equation 2.10. Pshot is the unknown Poisson shot noise.

The wavenumbers across and along the line of sight are denoted by k⊥andkk. It is important to point out that the wavenumbers in the reference cosmology are related to the ones in the true cosmology by

Da(z) k = k ref⊥ ⊥ Da(z) ref (9.32) H(z) k = k ref . refk k H(z)

Moreover, we deﬁne the galaxy power spectrum, Pg , including the redshift distortions:

2 2 2 2 G(z) −k2µ2σ2 P (k , k ) = b (z) 1 + βµ P (k)e r , (9.33) g ref⊥ refk G(0) matter,z=0

where µ = k · ˆr/k , being ˆr the unit vector along the line of sight and

Pmatter,z=0(k) was COBE normalized. The exponential damping factor is due to redshift uncertainties, where σr = cσz/H(z). G(z), β and b(z) are the growth function, the linear redshift space distortion parameter and the linear galaxy bias respectively. They will be deﬁned next.

72 Massive neutrinos cosmology: growth function and linear redshift space distortion

The growth rate of perturbations is deﬁned as

d ln G f ≡ , (9.34) d ln a where the growth function G(z) is related to the density of matter. In a 0.6 2 3 2 matter dominated Universe f ≈ Ωm(z) , with Ωm(z) = H0 Ωm(1 + z) /H (z). More generally, we use a growth factor dependent on the dark energy parameter (for a constant w) and massive neutrinos eﬀect computed by Kiakotou et al. α (2008), f = νΩm(z) with

α = α0 + α1[1 − Ωm(z)], 3 α = , 0 w (9.35) 5 − 1−w 3 (1 − w)(1 − 3w/2) α = . 1 125 (1 − 6w/5)3

ν is a numerical function dependent on Ωde and fν = Ων /Ωm (see Equation 17 and Table 5 of Kiakotou et al. (2008)). The linear redshift space distortion is also deﬁned as a function os the growth rate and the galaxy bias.

f β(z) ≡ . (9.36) b(z)

Fisher formalism

The Fisher matrix can be approximated by Equation 7.31

Z 1 Z kmax ∂ ln P (k, µ) ∂ ln P (k, µ) 2πk2dkdµ Fij = Veﬀ(k, µ) 3 . (9.37) −1 kmin ∂pi ∂pj 2(2π)

The eﬀective volume of the survey for a constant comoving number density is given by

2 n¯(r)Pg(k, µ) Veﬀ(k, µ) = V survey. (9.38) 1 +n ¯(r)Pg(k, µ) We use information of an Euclid like survey with area of 20000 deg2, redshift accuracy of σz/(1 + z) = 0.001 and a redshift range 0.5 ≤ z ≤ 2.1. Finally we dived our forecast into 15 redshift slices of ∆z = 0.1 centered in zi. We chose the

73 2 2 2 i initial set of parameters P = {h Ωb, h Ωc, h Ων ,H(zi), Da(zi),G(zi), β(zi),Pshot}. For each redshift bin we use the speciﬁcations on Table 9.1 (see Pavlov et al. (2012); di Porto et al. (2012) and references therein).

Table 9.1: Values of kmax, the galaxy bias and the galaxy density for each redshift bin.

−1 −3 3 zi Kmax (hMpc ) b(z) n(z) ×10 (h/Mpc) 0.55 0.144 1.0423 3.56 0.65 0.153 1.0668 3.56 0.75 0.163 1.1084 2.42 0.85 0.174 1.1145 2.42 0.95 0.185 1.1107 1.81 1.05 0.197 1.1652 1.81 1.15 0.2 1.2262 1.44 1.25 0.2 1.2769 1.44 1.35 0.2 1.2960 0.99 1.45 0.2 1.3159 0.99 1.55 0.2 1.4416 0.55 1.65 0.2 1.4915 0.55 1.75 0.2 1.4973 0.29 1.85 0.2 1.5332 0.29 1.95 0.2 1.5705 0.15

2 2 2 To obtain the constraints on our final set of parameters Q = {h Ωb, h Ωc, h Ων , w0, wa}, i first we marginalize our first fisher matrix over G(zi), β(zi),Pshot and use this sub matrix to change into the desired variables as

X ∂Pα sub ∂Pβ FDE,ij = Fαβ (9.39) ∂Qi ∂Qj α,β

9.2 Fiducial models

We assume 3 different cosmologies with all of them having as fiducial values w0 = −0.95 and wa = 0. The first model is assumed to have 3 massive neutrinos, being Ωm = Ωc + Ωb + Ων . As a second model we assume two massive neutrinos with identical mass and one massless in a inverted hierarchy mass splitting. In the third model, we will assume two massless and one massive neutrino in a normal hierarchy mass splitting. The free cosmological parameters for each model assume the values in Table 9.2.

74 Table 9.2: Fiducial models

Parameter Model 1 Model 2 Model 3 2 h Ωb 0.02219 0.02219 0.02258 2 h Ωc 0.01122 0.1122 0.01109 2 h Ων 0.01 0.0027 0.0021 Mν1 0.310 eV 0 0 Mν2 0.310 eV 0 .127eV 0 Mν3 0.310 eV 0.127 eV 0.198 eV

9.3 Preliminary results

We followed the same procedure shown in Chapter 7, performing the forecast for Planck alone, with and without considering CMB lensing. Moreover, we introduced the Euclid forecast, combining the results approximately as

Total Planck Euclid Fij = Fij + Fij . (9.40)

We considered the three previously mentioned models (Section 9.2). In Table 9.3 and Figure9.1, we show the 1 sigma errors and ﬁsher contours for model 1.

As the best upper limits for the neutrino density, w0 and wa we found, for the 2 combined Planck + Euclid, 0.00966 < h Ων < 0.01034, −0.976 < w0 < −0.924 and −0.094 < wa < 0.094 (all of them with 2 sigma conﬁdence level). We can roughly compare our results with Carbone et al. (2011b) since they are not using the ﬂat sky approximation. They found that the 1 sigma errors on w0 and wa are 0.0732 and 0.176 respectively (using Planck + Euclid + BOSS) against 0.013 and 0.048 (using Planck + Euclid) found by us. It is important to point out that our better result is also related to the fact that we use CMB information directly to constraint both w0 and wa. Carbone et al. (2011b) do not use CMB to constraint directly these parameters, but only to constraint other parameters of interest that are then summed up with the Euclid survey forecast that include w0 and wa. 2 For model 2, we have that 0.00244 < h Ων < 0.00296, −0.974 < w0 <

−0.926 and −0.088 < wa < 0.088 (95% C.L) as it can be inferred from Table 9.4 (see Figure 9.2 for the ﬁsher constraints). Finally in Table 9.5 and Figure 9.3, we show the 1 sigma errors and the ﬁsher ellipses for model 3, getting 2 with 95% C.L. 0.00186 < h Ων < 0.00234, −0.9666 < w0 < −0.9334 and

−0.058 < wa < 0.058.

75 Table 9.3: Marginalized errors for ΛCDM model with three massive neutrinos with identical mass (mν = 0.310 eV) .

Parameter Fiducial Planck Planck EUCLID Planck + EUCLID Planck + EUCLID value T + P T+ P+ lens T+P T + P + lens 2 h Ωb 0.002219 0.000109 0.000109 0.00064 7.71e-05 7.82e-05 2 h Ωc 0.01122 0.00092 0.00069 0.00025 9.9e-05 9.9e-05 2 h Ων 0.01 0.0031 0.0025 0.00054 0.00017 0.00017 w0 -0.95 0.028 0.029 0.040 0.013 0.013 wa 0 0.12 0.083 0.17 0.048 0.047

Table 9.4: Marginalized errors for ΛCDM model with two massive neutrinos with identical mass and one massless in a inverted hierarchy mass splitting (mν = 0.125 eV) .

Parameter Fiducial Planck Planck EUCLID Planck + EUCLID Planck + EUCLID value T + P T+ P+ lens T+P T + P + lens 2 h Ωb 0.02219 0.00012 0.00012 0.00048 7.9e-05 8.02e-05 2 h Ωc 0.01122 0.00080 0.00070 0.00015 8.6e-05 8.7e-05 2 h Ων 0.0027 0.0036 0.0011 0.00053 0.00013 0.00013 w0 -0.95 0.084 0.042 0.037 0.012 0.012 wa 0 0.084 0.057 0.15 0.044 0.044

Table 9.5: Marginalized errors for ΛCDM model with two massless neutrinos and one massive in a normal hierarchy mass splitting (mν = 0.190 eV) .

Parameter Fiducial Planck Planck EUCLID Planck + EUCLID Planck + EUCLID value T + P T+ P+ lens T+P T + P + lens 2 h Ωb 0.02258 0.00012 0.00012 0.00057 8.0e-05 8.12e-05 2 h Ωc 0.01109 0.00085 0.00066 0.00017 9.4e-05 9.4e-05 2 h Ων 0.0021 0.0027 0.0011 0.00061 0.00012 0.00013 w0 -0.95 0.044 0.029 0.034 0.0083 0.0083 wa 0 0.066 0.037 0.13 0.029 0.029

76 Figure 9.1: Fisher contours for our model 1. The red and blue contours represent 95.4% and 68% C.L. respectively for the unlesed CMB (dashed lines) and for Euclid galaxy survey (solid lines).

77 Figure 9.2: Fisher contours for our model 2. The red and blue contours represent 95.4% and 68% C.L. respectively for the unlesed CMB (dashed lines) and for Euclid galaxy survey (solid lines).

78 Figure 9.3: Fisher contours for our model 3. The red and blue contours represent 95.4% and 68% C.L. respectively for the unlesed CMB (dashed lines) and for Euclid galaxy survey (solid lines).

79 9.4 Discussion and conclusions

In this Chapter, we are considering massive neutrinos and a time evolving equation of state in the ΛCDM model. Using the Fisher formalism, we obtained the 2 best constraints possible for h Ων , w0 and wa considering Planck and Euclid survey. We, once more, tried to quantify the inﬂuence of CMB lensing information in the constraints of the parameters of interest. We saw from Tables 9.3 to 9.5 that in the case of CMB information alone we improve the constraints in basically all the studied cosmological parameters. When we sum up Euclid and CMB information however, we see no improvement in the using of lensing. This result makes us believe that the approximation

Total CMB+CMB lensing galaxy information Fij = Fij + Fij (9.41)

is a poor one when CMB lensing information is added, missing in this case a cross-correlation term between the two ﬁsher matrices. This is one of the prospectives of this work Finally, Euclid survey is a very precise future experiment and we also plan to test the CMB lensing information in more precise CMB polarization experiments, since the Planck noises related to polarization and the deﬂection angle power spectra are still not satisfactory.

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