Cosmic Reionisation and the Primordial Fluctuations in the Universe
Alexander van Engelen
Master of Science
Department of Physics
McGill University
Montreal, Quebec
August 31, 2007
A thesis submitted to McGill University in partial fulfilment of the requirements of the degree of Master of Science © Alexander van Engelen 2007 Library and Bibliothèque et 1+1 Archives Canada Archives Canada Published Heritage Direction du Bran ch Patrimoine de l'édition
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While these forms may be included Bien que ces formulaires in the document page count, aient inclus dans la pagination, their removal does not represent il n'y aura aucun contenu manquant. any loss of content from the thesis. ••• Canada ACKNOWLEDGEMENTS
1 thank my supervisor, Gil Holder, for inspiring this work and for helping me immensely during these first two years of graduate school. 1 look forward to more years of working with him as 1 continue my graduate studies. 1 also wish to thank
Paul Mercure for technical support on the computer cluster on which sorne of this work was performed; Gaelen Marsden for sharing sorne plotting and other scripts;
Sebastien Guillot for the translation of the abstract on page (iv); and my fellow group- and office-mates for much useful discussion. This work has made use of the
Legacy Archive for Microwave Background Data Analysis (LAMBDA), support for which is provided by the NASA Office of Space Science; and the publicly-available
CosmoMC package, including the CAMB program, by Antony Lewis and Sarah
Bridie. Finally 1 would like to thank my mother for her support all these years.
ii ABSTRACT
We investigate the effect of allowing freedom in the primordial power spec trum of curvature perturbations upon the measurement of other cosmological parameters, in particular the Thomson optical depth due to cosmic reionisation which is present in cosmic microwave background (CMB) observations. We find that the constraint on the optical depth from Wilkinson Microwave Anisotropy
Probe (WMAP) data broadens by approximately 10% upon allowing spectral freedom on large scales, and by a slightly larger factor when considering data from future experiments with lower noise in measurements of CMB polarisation.
We also present a reconstruction of the primordial power spectrum on the largest scales from WMAP, which is jointly obtained from this analysis.
iii RÉSUMÉ
Nous analysons les effets de l'ajout de degrés de liberté aux perturbations de la courbature originelle sur les mesures des autres paramètres cosmologiques.
Nous nous pencherons en particulier sur la profondeur optique de Thomson due a la ré-ionisation que l'on observe dans le fond diffus cosmologique (FDC). Nous re marquons que la contrainte pose sur la profondeur optique des donnes du satellite
Wilkinson Microwave Anisotropy Probe (WMAP) s'élargie de approximative ment 10%, tout en allouant, à grande échelle, de la liberté au spectre. De plus, ce pourcentage augmente légèrement si l'on considère des données d'expériences à venir avec moins de bruit dans les mesures de la polarisation du FDC. Enfin, nous présentons une reconstruction du spectre de puissance originel de WMAP à plus grande chelle, obtenue conjointement a cette analyse.
lV TABLE OF CONTENTS
ACKNOWLEDGEMENTS 11
ABSTRACT iii
RÉSUMÉ .. lV
LIST OF FIGURES vii
1 Introduction . 1
2 Background . 7 2.1 The Friedmann Universe and perturbations . 7 2.2 Boltzmann and fluid equations . 11 2.3 General solution ...... 13 2.4 Super-horizon modes ...... 15 2.5 Inflation and the origin of fluctuations 17 2.6 The Observed CMB ...... 19 2.7 Reionisation and the peak in the EE spectrum 20 2.8 Deviations from powerlaw spectra, and constraining them from data ...... 25
3 Application to current data 30 3.1 Choice of parameterisation 30 3.2 CMB power spectra . . . . 32 3.3 Using Monte Carlo Markov chains to constrain cosmology . 33 3.4 Results ...... 42
4 Forecasting for future data . 49
4.1 The Fisher information matrix for CMB experiments 49 4.2 Application 56 4.3 Results ...... 57
v 5 Conclusions ...... 63
vi LIST OF FIGURES Figure page
2-1 Sorne sample results of cosmological perturbation theory: the tem perature Cz's for a wide range of l's. The three regions indicated in the figure are SW, the super-horizon Sachs-Wolfe plateau; AP, the acoustic peaks; and SD, the region where there is noticeable Silk damping. The vertical lines are not meant to denote hard bound- aries between the regions, since there is sorne overlap...... 16
2-2 The effect of changing the Thompson optical depth to reionisation. The CMB EE power spectrum is plotted for an instant reionisa tion model with seven values of T between 0 and 0.32; this corre sponds to redshifts of reionisation between 0 (i.e. no reionisation) and 28. These plots are obtained using the cosmological Boltz 2 2 mann code CAMB with Ddmh = 0.12, Dbh = 0.022, Ho= 70km s -1 M pc-1 ...... 23
3-1 Primordial (top panel), CMB TT (centre panel), and CMB EE (bot- tom panel) power spectra for several 11-parameter "broken" spec- trum models, showing how the freedom allowed in the primordial spectrum propagates into the CMB. In addition to the seven pa- rameters describing the primordial power spectrum, the four other 2 2 parameters allowed to vary are {Dbh , Dch , h, Zre}, which for these models are {0.028, 0.092, 0.89, 13.3} (solid line), {0.050, 0.212, O. 72, 10.1} (dotted line), {0.039, 0.185, 0.75, 7.4} (dashed line) and {0.038, 0.165, 0.77, 4.9} (dot-dashed line). Note that for l < 20 the values of CfE increase with increasing Zre· The vertical grey bars are the binned uncer- tainty from the WMAP 3-year data from Hinshaw et al. (2007) (TT data) and from Page et al. (2007) (EE data). These particu- lar models are not good fits to the data but are shown for demon- strative purposes; they have values of -2ln L between 3586 and 3774...... 34
vii 3-2 The effect on the err and crE spectra of doubling or nulling each of the first 5 power spectrum parameters. Note that for the TE case the y-axis plots only one power of l...... 35
3-3 Samples from a Markov chain for the 11-parameter model, showing (top) the angle subtended by the sound horizon, a slow parame ter, and (bottom) the amplitude of the primordial power spectrum 3 1 at k4 = 1.5 x 10- Mpc- . Note that the fast parameter changes position much more often than the slow parameter...... 39
3-4 The constraints on the six power spectrum amplitude parameters, Bi = ln(1010 Ai), in the 11-parameter model. The 68% and 95% confidence limits for the amplitude in each bin, marginalised over all other cosmological parameters, shown in black and grey, re spectively. The crosses indicate the values of the parameters corre sponding to the maximum-likelihood point in the Markov chains. The dashed line is the best fit powerlaw to the WMAP data. The first 2a error bar reaches to low values and is not effectively con- strained away from zero...... 43
3-5 The constraints from the MCMC on the six power spectrum ampli tude parameters that are allowed to vary in the model. The thick and thin lines indicate 68% and 95% confidence limits, respectively; the crosses indicate the values of the parameters corresponding to the maximum-likelihood point in the chains. The constraint on 1 the seventh parameter, the spectral index above 0.05 Mpc- , is not shown. As in Fig. 3-4 the first power spectrum parameter is not constrained to be above zero for the 2a contour...... 45
3-6 The matrix indicating the degree of correlation between pairs of am plitude parameters (Bi, Bj), as defined in eq. 3.7. The rows and columns correspond to the amplitude parameters B 1 through B6 , and the pairings in the bottom triangle of this matrix correspond to the panels in Fig. 3-5...... 46
viii 3-7 The constraints on the four non-power spectrum parameters which are allowed to vary in the model, with (grey) the ACDM model and (black) the 11-parameter model described in the text. The plots along the diagonal show the 1-d marginalised (unnormalized) posteriors for the indicated parameters, and the off-diagonal plots show the contours of the joint 2-D posteriors...... 48
4-1 Constraints on tau for two cosmological models, the broken power spectrum (upper curve) and ACDM (lower curve), as a function of the inverse noise in the polarisation measurements, w. The only variables allowed to vary are the power spectrum parameters and T. The noise levels for WMAP (le ft) and Planck (right) are shown as vertical lines...... 59
4-2 The inverse Fisher matrix between parameters for the eight-parameter model normalized to have values of unity along the diagonal. The noise has been set to Planck levels. The 6 x 6 top-left corner can be compared to Fig. 3-6 which was obtained with the WMAP like- lihood...... 60
4-3 The effect of perturbing the EE power spectra by the 10" values of the parameters. The top left panel shows the effect of perturb-
ing the optical depth about the fiducial model by +Œ7 (top line)
and by -0"7 (bottom line). The top right panel shows the effect of perturbing the power spectral amplitude parameters by +ŒBi and -O"Bi, with each shade corresponding to a different i. These are for Planck noise levels. The bottom two panels show the same quantities but for an experiment with RMS sensitivity higher than Planck by a factor of 10. The grey envelope is the overall cosmic variance limit for a noise-free experiment, which sets the size of these bands...... 61
ix CHAPTER 1 Introduction
In the wake of rich datasets obtained mainly in the last two or three decades, the field of cosmology has converged upon a standard model describing both the expansion of the Universe and the growth of structure. This model contains approximately 10 free parameters, depending on the particular flavour chosen, and is a good fit to many kinds of datasets probing different physics, eras, and length scales. In roughly chronological order in cosmic time, these datasets include the measurements of the cosmic nuclear abundances, which were determined in the first sever al minutes after the big bang (Burles et al. 2001); observations of the spectrum and the anisotropies in the cosmic microwave background ( CMB), which offer a wealth of information about the evolution of the inhomogeneities in the
Universe during its plasma phase, and about the seeds for these inhomogeneities which were laid clown at very early times (Spergel et al. 2007); surveys of the galaxy distribution on large scales (Tegmark et al. 2004; Cole et al. 2005) and gravitationallensing (Refregier 2003) which demonstrate that the inhomogeneities in cold clark matter grew due to gravitational attraction and that galaxies, such as the one in which we live, formed in the high-density regions; and observations of high-redshift supernovae, which reveal that the Universe is not only expanding but is accelerating in this expansion (Riess et al. 2004).
1 In this thesis we focus on the CMB. The CMB was discovered by Penzias and Wilson (1965) as an isotropie background of photons at a temperature of rv 3 K, and was soon recognized as a relie of the hot big bang (Alpher et al. 1948), redshifted to this low temperature. Together with the light element abundances this observation led to the acceptance of the hot big bang model of the early
Uni verse.
Soon after this discovery it was realized that angular variations in the temperature of the CMB would trace the fluctuations in energy density present at early times (Sachs and Wolfe 1967) and could be connected, via the theory of gravitational structure formation, to the structures observed today. The detection of these angular variations on large angular scales by the Cosmic Background
Explorer (COBE) satellite (Smoot et al. 1992) confirmed these predictions, finding temperature variations on the order of 10-5 , consistent with structure formation models in which cold clark matter is the dominant component.
The large-angle variations detected by COBE are related to fluctuations on length scales which are so large that they were outside the cosmic horizon-the length scale over which causal physics can occur in standard big bang models-at the time the CMB photons last scattered from matter. The presence of such fluctuations is a prediction of theories of cosmic inflation (Guth 1981). These theories state that the seeds for fluctuations were generated at very early times, and were expanded beyond the cosmic horizon during an era of exponential expansion. As we will discuss in the next chapter, these theories generically predict a nearly scale-invariant power spectrum of Gaussian primordial perturbations.
2 The cosmic horizon, initially small, grew in size as time evolved. At the time that it outgrew the length scale of a given perturbation, the perturbation became active, evolving in amplitude in response to gravitational and pressure forces.
The photons in this era were tightly coupled through Compton scattering to the baryons (a term including both protons and electrons), and together they behaved as a single fluid. They also interacted indirectly with the perturbation in the clark matter density, through the effect of the clark matter on the gravitational potential. When the temperature, decreasing due to the expansion of the Universe, reached to the point where Compton scattering was no longer efficient, the protons and electrons were no longer kept from combining by the photons. At this time, known as recombination, the protons and electrons combined to form neutral atoms and the photons were free to propagate through the Universe with almost no further interactions. The angular variations in the CMB as observed on the sky today thus contain information about the fluctuations in several species, and for length scales that entered the cosmic horizon at many different epochs, up to and including recombination.
After a period following recombination known as the "clark ages", stars eventually formed in the Universe and reionised the atoms in the intergalactic medium by producing ionizing photons. The newly free electrons scattered with the relie CMB photons, producing two main observable effects: the damping of the angular variations on medium to small angular scales, and the extra polarisation of the CMB on large angular scales (Zaldarriaga 1997).
3 ~·. The process of reionisation is not completely understood. lt is thought that
spherical regions are ionised around stars, which expand until they percolate and
fill a larger space. The duration of this process is unknown but is thought to be
complete by redshift z ~ 6 (Fan et al. 2006). lt is also possible that reionisation
took place in two or more stages.
The polarisation of the CMB as seen today is complementary to the tem
perature observations because it is sourced only by Thomson scattering from
a local quadrupole anisotropy. The CMB that we observe obtains its polarisa
tion from only a few points in cosmic history: due to local quadruopoles during
recombination, and the more recent rescattering in the reionised Universe.
The quantity related to reionisation to which the observed CMB is most
sensitive is the optical depth due to Thomson scattering, which is a measure of
the free electron number density, integrated to recombination. This quantity
has been measured by the Wilkinson Microwave Anisotropy Probe (WMAP)
collaboration (Kogut et al. 2003; Page et al. 2007), as one of the base parameters
in the fits to their rich angular power spectra. Much work has since focussed on
the possible forms of the spatially averaged ionisation history as constrained by
this measurement (e.g. Kaplinghat et al. 2003; Haiman and Holder 2003) as well
as by other observations of reionisation including Gunn-Peterson absorption. It
is important to have an accurate measurement of the optical depth T in order to measure the amplitude of the primordial power spectrum accurately. The
amplitude of fluctuations in the nearby Universe, a 8 , is directly related to the
4 primordial amplitude; and a8 is degenerate with measurement of the equation of state for clark energy, which is of great interest for fundamental physics. The extra polarisation generated by reionisation is sensitive to the local
CMB quadrupole, and is generated at relatively recent times. The source of the anisotropy on such large angular scales, at the redshifts important for reionisation, is the primordial power spectrum on very large length scales-very little processing happens during the plasma phase on these super-horizon length scales as we will see in the next chapter. Most studies of reionisation have assumed a standard model for the power spectrum of primordial curvature perturbations, namely a nearly scale-invariant spectrum as generically predicted by theories of inflation.
However, as shown by Kinney (2001), the estimates of cosmological parameters can be significantly biased ( "fooled") if the primordial power spectrum is allowed to be non-standard, while still giving the same power spectrum of angular variations in the CMB to its inherit uncertainty, cosmic variance. Also, deviations from standard primordial power spectrum could provide elues about physics at the highest energy scales.
Tegmark and Zaldarriaga (2002) underscore the importance of keeping in mind the distinction between cosmological parameters that describe very early-universe physics from those which describe more recent phenomena. The contrast between these is particularly strong when comparing the primordial power spectrum with the reionisation signal: the former is thought to be set at a temperature of"" 1028 K, and the latter at a temperature of"" 30 K. Since such different physics takes place on these vastly different energy scales, in this thesis
5 we ask what are the observational constraints on the optical depth to reionisation when not assuming standard forms for the primordial power spectrum.
ln Chapter 2 we summarise aspects of cosmology which are relevant to this work, and present more quantitative descriptions of sorne of the topics described in this introduction. ln Chapter 3 we describe a reconstruction of the primordial power spectrum from WMAP data, and determine the loosening of constraints on other cosmological parameters when this freedom is allowed, including the optical depth to reionisation. ln Chapter 4 we project how this will change for future experiments with lower noise in the polarisation observations. We summarise in
Chapter 5.
6 CHAPTER 2 Background
In this chapter we summarise aspects of modern cosmology important for understanding the work clone in this thesis. These topics include the expansion of the Universe, the generation and growth of fluctuations, the physics of the cosmic microwave background including the generation of polarisation, and cosmic reionisation.
2.1 The Friedmann Universe and perturbations
Upon making the assumptions that the Universe is homogeneous and isotropie on the largest scales-an assumption which is verified by data (e.g. Hogg et al.
2005)-we can take the spacetime metric for the Universe as a whole to be given by the line element (2.1)
We have made the assumption that the spatial part of the metric is fiat. This assumption is supported by data (de Bernardis et al. 2000).
2 2 2 1 2 The comoving distance dx = (dx + dy + dz ) 1 between any two points in space remains constant by definition in a homogeneous universe; the physical distance is the comoving distance multiplied by the cosmological scale factor a. To find the function a(t) we can assume that the Universe is filled with a smooth, uniform fiuid with energy density p and pressure p. Upon placing these
7 assumptions into the Einstein field equations we obtain the Friedmann equations for the evolution of a(t),
81rG H2(a) = (~da) 2 - -p (2.2) a dt 3 1 d2a -47fG 3 (3p + p). (2.3) a dt2
The quantity His known as the Hubble parameter and is the expansion rate of the universe; this equation shows that the expansion rate depends upon the form of the constituent energy. Our Universe today contains several distinct components, each of which has a unique relationship between pressure and density, and each of which dilutes in its own way during the expansion. Since the energy densities are additive, we can rewrite the first Friedmann equation as
(2.4) where Dr, Dm and Dvac are the dimensionless energy densities, evaluated today, of radiation, matter, and vacuum energy in units of 87rG/3H'lJ. H0 is the Hubble
parameter evaluated today. If we had allowed for curvature in the spatial part of the metric, it could be viewed as an equivalent energy density and this equation
2 would contain an extra term proportional to a- .
As a result of dedicated observations, the values of the energy densities have been measured to high precision. The density of matter is Dm ~ 0.24, which is too
large to be accounted for by the visible baryons, which contribute Db ~ 0.05. The
density of the vacuum energy, a component with negative pressure that to a good
approximation does not dilute with the expansion, is Dvac ~ O. 76.
8 It is clear from eq. 2.4 that the Universe underwent several stages in its history. In particular there was an early time at which the densities of matter and radiation were approximately equal, aeq = flr/flm. Similarly, since flm c:::: flvac today to within an arder of magnitude, we are currently at the transition between matter domination and vacuum energy domination.
This thesis, and in fact much of the research in cosmology in the past decades, focuses on the large-scale fluctuations in the energy density, and hence in the metric. In this section and the several that follow we present, but do not derive, the exact equations of linear cosmological perturbation theory which are integrated to determine the CMB anisotropies. This is for several reasons: the solution on the large scales in which we are interested is easy to find as a special case once the full equations are shawn, and for completeness, since in Chapters 3 and 4 we use cosmological Boltzmann codes extensively.
For the early times we consider, namely the times prior to the "last scat tering", or recombination, of the CMB, the fluctuations are of sufficiently small magnitude (lo-5 on horizon scales) to be modelled as first-order perturbations to the background metric in eq. 2.1. The ten degrees of freedom in the symmetric perturbation metric can be characterized by the way they transform under spatial rotations. There are four scalar modes, two vector modes, and four second-rank tensor modes (Brandenberger 2004). The scalar modes couple to inhomogeneities in the matter and radiation distributions; the vector modes decay in an expanding universe so we do not consider them; and the tensors correspond to gravitational waves. We will focus on scalar perturbations, however, the tensor modes are of
9 interest, and a major effort is currently underway to observe their imprint on the
CMB (e.g. Oxley et al. 2004).
Not all of the possible perturbations to the metric are physical. Sorne are related to the choice of variables used in the perturbation metric, i.e. gauge artifacts. Only two of the four scalar modes are physical rather than gauge artifacts. We can choose a gauge in which the perturbed metric is diagonal, the conformai Newtonian gauge (Ma and Bertschinger 1995), for which the perturbed metric gives the line element (Dodelson 2003)
2 2 2 2 2 2 ds = (1 + 2w(x, t)) dt - a(t) (1 + 2(x, t)) (dx + dy + dz ), (2.5) where \]i « 1 is interpreted as the Newtonian gravitational potential and « 1 is the perturbation to the spatial curvature.
In the linear theory of perturbations the Fourier modes evolve independently; we will generally consider the Fourier transforms w(k, t) and (k, t), where k is the comoving wavenumber, the Fourier transform conjugate variable to the comoving distance introduced in eq. 2.1.
By inserting the perturbed metric into the Einstein field equations, we obtain two equations relating the two scalar metric variables to the energy and pressure,
k2 + 3~ ( + ~ \]i) (2.6)
k2 ( 10 the perturbation to the sum of pressures. Equation 2.6 reduces to the Poisson equation if à= O. The term "species" here refers to both radiation, which consists of photons and neutrinos, and matter, which consists of baryons and pressureless, collisionless dark matter. 2.2 Boltzmann and fluid equations The goal is to determine the evolution in phase space for the species present in the early Universe. We begin with the fiuid equations for the baryons and the dark matter. The cold dark matter (CDM) perturbation can be modeled as a fiuid with a density contrast and velocity everywhere in space, denoted by 6dm = 6pdm/ Pdm, and vdm respectively. These quantities obey the usual Euler and continuity equa- tions for a fiuid, which, modified to take into account the background expansion of the Universe, become -kvdm + 3<1?, and (2.8) V• dm + -Và dm - -ik'T' '±', (2.9) a where here also the overdots refer to derivatives with respect to conformai time and vdm is directed along k. The equation for the baryons is similar, but unlike CDM, baryons can interact with other species. During the era before recombination the baryons were coupled to the photons via Compton scattering, and they were also coupled to each other via Coulomb scattering. Due to the tight coupling, the overdensities in the protons occurred in the same locations in space as the overdensities in electrons to a good 11 approximation; we can define 6b to be the density contrast of either. The two forms of scattering lead to two extra terms proportional to the differentiai optical depth 7, Jb -kvb + 3<Ï>, and (2.10) Vb + ~Vb - -ik'I! + ~ (Vb + 3i81)' (2.11) where el is the local dipole moment of the photon distribution; T = -neO"Ta is the differentiai optical depth; O"T is the cross section for Thomson scattering, and R = Pb + Pb = ~ Pb (2.12) P-y + P-y 4 P-y is the ratio of energy densities in baryons and photons. 7 The quantity 7 is related to the visibility function, 9(17) = -T-e- . This 710 function, which is normalised so that f0 d17 9(17) = 1, has a simple interpretation which says that 9(17)d17 is the probability that a given photon last scattered from matter at a conformai time between 17 and 17 + d17. The peak of 9(17) occurs at the time when the most atoms were combining during recombination. In reionised models, 9 is nonzero for recent times as weiL In the case of the photons, rather than the Euler and continuity equations, we must consider the full Boltzmann equation. This is because in this case there are two relevant directions: the direction of propagation of the photon and the direction of the perturbation, k. We will define the eosine of the angle between them as p, = E;!. There will again be coupling between the photons and the 12 baryons. The Boltzmann equation for photons with temperature e is (2.13) II e2 + eP2 + epo (2.14) i ( -ep + ~(1- P2 (~t))II) , (2.15) where P2 is the second Legendre polynomial, e P refers to the strength of the polarisation, and e 2 is the local quadrupole moment. 2.3 General solution In arder to solve cosmological perturbation theory to linear arder-and to be able to predict the power spectrum of CMB fluctuations on the sky, as well as the matter power spectrum which is relevant for galaxy surveys-it is first necessary to expand the equations for e and ep, eqs. 2.13 and 2.15, into equations for the local multipole moments el and ePl· This results in a hierarchy of coupled differentia! equations in which èl depends on el-1, e[, and el+l, and similarly for the polarisation. The coupled nature of these equations is referred to as free streaming in cosmology, and physically represents the transfer of power to higher multipoles. For example if the radiation field is uniform, but with a hot spot in one location, every observer will see only a monopole including the observer at the hot spot. However, after the photons have had time to propagate, other observers will see the hot spot at sorne higher multipole. Equations 2.6, 2.8, 2.9, 2.10, 2.11, and these coupled equations for et and e Pl for 0 ::; l ::; lmax can be solved numerically for the set of variables 13 1 1 J: J: max max .c ,T,'J!, Udm, Vdm, ub, Vb, {8l }l=O , {8Pl }l=O . Th"IS must b de one 10r each value of k of interest. 1 The observed power on the CMB sky at multipole l is then an integral over comoving wavenumbers, (2.16) where the source term D..2 (k) is the power spectrum of the potential W at very early times, whose origin we discuss in the next section. This procedure for finding the solution is very laborious because in orcier to evaluate 0 1 for a given l, the functions 8 1, must be evaluated for each l' between 0 and lmax-typically thousands of terms-for just one value of k. A major breakthrough came with the new technique introduced by Seljak and Zaldarriaga (1996) in which sources are integrated along the photon past light corre. In this approach the multipole moments need to be evaluated only up to l c:::.:: 4 or 5, and the integral along the line of sight can make use of the spherical Bessel functions which only need to be evaluated once and then stored on disk. The result is an improvement in CPU time by 2 orders of magnitude for a comparable level of accuracy. 1 The functions W and - are equal if anisotropie stresses can be neglected. This approximation is frequently made and so we typically only need to solve for one of them, but otherwise we can add eq. 2. 7 to the system. 14 The new method to estimate the radiation multipoles is then (2.17) where j 1 is the spherical Bessel function of order l and the source function S is a sum of the perturbed quantities. Although this method, released in the software package CMBFast, makes it straightforward to integrate the equations numerically, sorne insight can be gained by taking an analytical approach. These equations, along with the appropriate initial conditions, contain all the information about the rich detail that is seen in the well-known CMB and galaxy power spectra. Analytic treatment of these equations such as in Hu and Sugiyama (1995) can motivate sorne of the features that are seen in the spectra, including the Sachs-Wolfe plateau, which is due to photons climbing out of potential wells by mostly unprocessed large-scale modes; the acoustic peaks at l ""' 100 - 500 which are due to driven oscillations in the photon-baryon fluid; and the exponential damping at l > 1000 which is due to diffusion damping (Fig. 2-1). 2.4 Super-horizon modes No general analytic solution to the coupled system described in the previous section is known that is applicable for all values of k. This thesis focuses on the largest length scales, which are super-horizon at all times up to (and past) trec- We will briefly discuss the analytic solution for (2003), section 7.1. First, all terms in the equations which are dependent onk can be dropped; this is equivalent to neglecting spatial derivatives over these 15 6000 ...... , "'~ -5 4000 1:: N ..,_::;..... u -::::- 2000 + SD 10 100 1000 10000 multipole 1 Figure 2-1: Sorne sample results of cosmological perturbation theory: the tempera ture C1's for a wide range of l's. The three regions indicated in the figure are SW, the super-horizon Sachs-Wolfe plateau; AP, the acoustic peaks; and SD, the region where there is noticeable Silk damping. The vertical lines are not meant to denote hard boundaries between the regions, since there is sorne overlap. large scales. We are left with equations for the local monopole, the local dark matter density contrast, and the gravitational potential . One of these can be eliminated, leaving two coupled first-order equations. The solution for (Kodama and Sasaki 1984) is (2.18) where a Pdm y=-=-, aeq Pr and we are neglecting the baryons. In the limit of large y this becomes - to (O). Thus as the large-wavelength mode passes through the epoch of matter radiation equality, the potential slightly decays. 16 ~·. We have not discussed the initial conditions for the system, for instance what sets the value of In this section we very briefly discuss the origin of the perturbations to Inflation is a proposai that is successful in explaining the flatness and homo geneity of the Universe in the standard big bang model. The idea is that early in the Universe's history the scale factor a(t) grew exponentially as a function of time, due to the effect of an (unknown) energy density component with negative pressure. In this scenario, small regions of the primordial universe get stretched 1 exponentially and quickly become larger than the Hubble length, H- . Quantum fluctuations in this energy density component become macroscopic. At the end of inflation, the Universe enters the standard big bang, power-law expansion, and as this proceeds the widely-separated regions eventually re-enter causal contact. Although the original proposai for inflation involved a scalar field tunneling through a potential barrier (Guth 1981), and later versions considered a field undergoing a second-arder phase transition (Linde 1982), the most common models of inflation considered today are slow roll models, in which a scalar field rolls slowly down a potential. The rolling must happen slowly because it is important that the potential energy dominates the kinetic energy. 17 A field with potential V (1.p) and kinetic energy ( ijJ )2 /2 is placed into the Friedmann equation, eq. 2.2. If ( ijJ )2 /2 < V (1.p) then the expansion rate is close to exponential, with the scale factor a(t) oc exp(Ht) where H 2 ~ s;av. The fluctuations in i.p satisfy (2.19) and the power spectrum of fluctuations in 1.p is (2.20) this is related to the Hawking temperature. The fluctuations are expected to be Gaussian. As the scalar field rolls down the hill, the Hubble constant slightly decreases. The amplitude of a given mode is set by the value of the Hubble constant at the time that mode exits the horizon, because once outside the horizon the amplitude is "frozen-in" and remains constant. By considering a conserved curvature perturbation it is possible to relate the fluctuations in 1.p to those in the scalar metric, 2 4 Pw(k) = g (aH)---:j; P8 10 (k)laH=k (2.21) where we evaluate at the time that the mode exits the horizon, k=aH. The net result of this is that because H was only slightly decreasing during inflation, the spectrum of the potential W is nearly scale invariant, ll2 (k)- ~;P(k) =As (:O) ns-l, (2.22) 18 where the scalar spectral index, ns, is given by quantities which measure how quickly the field rolls clown the potential, 2 2 - ( d -1 ) ( -1 d i.p 1dt ) n 8 - 1 = 4E - 2rt = 4 dt (H ) - 2 H d The parameters E and rt must both be small for slow-roll inflation to take place, and so ns - 1 should be close to O. For particular choices of the potential V (cp) the expected value of n 8 is close to O. 95; this matches the WMAP observation. The meaning of the power spectrum ll2 (k) is that for a mode at wavenumber k, the potential at very early times (such as (O) in eq. 2.18) will be a complex number, with lm( (O)) and Re( (O)) each drawn from a Gaussian distribution 2 3 with mean 0 and variance ll (k)/2k . 2 In this thesis we reconstruct ll ( k) directly from data. This approach is more model-independent, assuming less physics from these high energy scales, 1015 Ge V. 2.6 The Observed CMB On the largest angular scales, the only CMB anisotropy observed at a given position on the sky is the Sachs-Wolfe effect, (2.24) This is simply because a photon must climb out of the potential it finds itself in at arec to reach us today (White et al. 1994). For this reason we need to know the potential all the way to today, which we found in eq. 2.18. Because potentials are constant in a matter-dominated universe, the second term in 2.24 is zero for most of the photon's path. The exceptions to this are shortly following recombination, 19 which is not long after matter-radiation equality; and recent times, when the Universe becomes dominated by vacuum energy. A given map of t5T on the sky can be projected into the spherical harmonie basis, (2.25) Due to statistical isotropy, and Gaussianity, all the information is contained in the variance of the atm's: 1 +t Ct= (a"tmatm)· (2.26) 21 + 1 L m=-t Because for a given l the average is taken taken over only 2l + 1 values of min the sum, the resulting value of Ct from a single realisation has an uncertainty of (2.27) 2.7 Reionisation and the peak in the EE spectrum The CMB is polarised at the rv 10% level. This polarisation is of interest because, as mentioned in the introduction, it is only generated at certain times in the Universe's history. It is useful to define basis functions for angular power spectra for the polar isation as well as for the temperature. Following Zaldarriaga and Seljak (1997) we first decompose the 2x2 polarisation pattern on the sky into Q and U Stokes' parameters, which are also co-ordinate dependant. We then project these onto the 20 spin-2 spherical harmonie functions, (2.28) The spin-2 spherical harmonies, ± 2 Yim ((), In terms of the coefficients a±2,zm, we can define the linear combination -(a2,lm + a-2,Zm)/2 (2.29) i(a2,lm- a-2,Zm)/2. (2.30) The letters E and B are used in analogy to electric and magnetic fields: the former is curl-free and the latter is divergence-free. The definition of the angular power spectra Cf can be extended in terms of these a~, in the same formas for the temperature. 1 +l 2l + 1 2.: (af:;a{;,.), (2.31) m=-l +l -l--1 2.: (azm T* azmE) ' and (2.32) 2 + 1 m=-l 1 +l 2l + 1 2.: (a{;;af:n), (2.33) m=-l 21 where TE denotes the cross power spectrum between temperature andE-mode polarisation. The other cross spectra are expected to be zero due to their parity. The TE and EE spectra have been measured by a number of groups (starting with Kovac et al. 2002) and they exhibit the same kind of acoustic oscillations as the temperature spectra, although out of phase. They are predicted to exist by the cosmological model, and with no extra free parameters, so their existence is a stringent test of the theory. The CMB becomes polarised when an electron scatters radiation with a local quadrupole anisotropy, or equivalently, if in the electron's rest frame there is a nonzero 8 2 . A di pole anisotropy, for instance, is not enough to generate polarisation because if hot and cold radiation are incident on the electron from directions 180° apart, they will not preferentially "shake" the electron in a particular direction. However, if the incident radiation is a quadrupole, with full variation from hot to cold over 90°, then the outgoing radiation will be polarised along the axis corresponding to the colder incident radiation (Hu and White 1997). Most of the power in a plot of e.g. C1EE is from scattering at recombination. However, in reionised mo dels there is an increase in power in the EE and TE spectra expected at low l due to a rescattering process as predicted by Zaldarriaga (1997). This happens because of the effect just described: the newly free electrons interact with the cosmological quadrupole and generate polarisation. This new peak (Fig. 2-2) was first observed in the first-year WMAP data release, in the cross-spectrum between the temperature and polarisation (Kogut et al. 2003) 22 ,...... , N 10° ~ ::t -...... - T=0.32 1::: 0.16 N 2 0.08 '-..... 1 o- UJ UJ_ 0.04 ,...... ,u 0.02 0.01 .. + 4 ~ -...... - 1 o- / / ! / / / 1o- 6 ! 1 10 100 1000 multipole 1 Figure 2-2: The effect of changing the Thompson optical depth to reionisation. The CMB EE power spectrum is plotted for an instant reionisation madel with seven values of T between 0 and 0.32; this corresponds to redshifts of reionisation between 0 (i.e. no reionisation) and 28. These plots are obtained using the cosmo 2 2 logical Boltzmann code CAMB with rtdmh = 0.12, rtbh = 0.022, H0 = 70km s-1 Mpc1 23 Another way that reionised electrons affect the observed CMB is that the 2 rescattering damps out structures by a factor of e- r on length scales that are sub-horizon at reionisation, where {Zrec dt T =Jo dz Œrxe(z) dz, (2.34) which was introduced in section 2.2. This damping effect can also be seen in Fig. 2-2. The result for observations is that the scalar amplitude As and T are 2 correlated parameters. Sin ce the height of the low-l peak scales with 7 , however, this degeneracy can be lifted by measuring the peak well. The process of reionisation is not well understood. From the WMAP data it seems unlikely that any substantial ionisation had occurred by redshift 30; also, observations of Gunn-Peterson absorption indicate that only ,....., 1% of atoms were neutral (Fan et al. 2006) at redshift 6. Although for the current WMAP data little more than a single number describing this phase can be reliably extracted, namely T, it is hoped this will change with future experiments (Hu and Holder 2003) Several authors (e.g. Kaplinghat et al. 2003; Haiman and Holder 2003) have studied the feasibility of using this peak to learn more about the process of reionisation, in particular the constraints on the ionisation history xe(z). Hu and Holder (2003) and Mortonson and Hu (2007) take the approach of decomposing Xe(z) into amplitudes in bins in redshift and determining which linear combinations of these amplitudes are most constrained by the data. They find that the constraints on T are quite robust when allowing this new freedom. 24 In the current work we take a similar approach. Rather than adding freedom to the ionisation history, however, we add freedom to the primordial power spectrum. 2.8 Deviations from powerlaw spectra, and constraining them from data Now that we have discussed the relevant astrophysics for the CMB, we turn our attention back to cosmological inflation. Inflation, as introduced in section 2.5, is a successful model for explaining several properties of the observed universe, however, its main prediction has been for the form of the seeds of cosmological structures. In addition to creating Gaussian, adiabatic fluctuations, inflation provides a mechanism for generating these fluctuations on scales larger than those within causal contact in standard big-bang cosmology, such as those observed on large angular scales in the CMB. It also causes the modes of a given wavenumber to be temporally in phase with one other at the time of horizon re-entry, in a way that is difficult to generate with other models. This phase coherence is needed to explain the acoustic peaks in the cosmological power spectrum (Hu and Dodelson 2002). As discussed, fun etions V ( generically predict powerlaw spectra with n 8 ;S 1. Fortunately, specifie forms of these potentials can be observationally distinguished, because they typically predict unique relationships between n 8 and r, which is the ratio of the amplitude of tensor fluctuations to that of scalar fluctuations. For instance, although current data provide only upper limits on r, the allowed region in the ns-r parameter space from WMAP datais small enough to rule out the exactly scale-invariant 25 Harrison-Zeldovich spectrum with ns = 1 and r = O. ln this simple scenario, inflation does not end because the field remains at a local minimum, and does not evolve. By contrast, the simplest dynamical model, a quadratic potential with V (t.p) = m~t.p 2 /2 and t.p away from its minimum, is allowed by the data. Despite its phenomenological success, inflation is a model that is not currently embedded in an accepted theory of high energy physics. The physical nature of the field t.p, as well as that of its potential V(t.p), is unknown, other than that it is not part of the standard model of particle physics. For instance, the field t.p could represent the distance between two branes in cosmological brane theory, or the modulus of a curled-up dimension in string theory (e.g. Quevedo 2002). Because the form of V (t.p) is not constrained by confirmed fundamental physics, deviations from the nearly scale-invariant powerlaw are not necessarily unexpected. An example is a step in the inflaton potential, which could be obtained with a hybrid field model. In this scenario the potential energy density during inflation is due to contributions by two weakly-coupled scalar fields, and the overall energy density briefly stops evolving at the transition between them ( Covi et al. 2006). The result is a localized oscillatory feature in the power spectrum that can be searched for in the WMAP data. Even within the standard single-field slow-roll inflationary context, non smooth features in the primordial power spectrum could be obtained from the physics preceding inflation. Due to the very large expansion factor of space, 60 which is an overall factor of at least e , modes that are observable today would have been extremely small at the earliest stages. In particular, sorne observable 26 scales would have had physical wavelengths smaller than the Planck length (or alternatively at sorne other high-energy scale at which new physics might come into effect) (Martin and Brandenberger 2001). Including a cutoff at this scale in the calculation of the generation of fluctuations can lead to an imprint of oscillations on the primordial power spectrum, the amplitude of which depends on the energy scale of the new physics involved (Easther et al. 2002). Allowing for such oscillations due to these trans-Planckian effects in the model power spectrum gives a modestly better fit to the WMAP data (Martin and Ringeval 2005). The stage following inflation, during which the field reaches the bottom of the potential and decays into other particles, can also give rise to new features in the power spectrum (e.g. Barnaby and Cline 2006). These can be tested with the WMAP data as well (Hoi et al. 2007). For the study of reionisation, we are interested in large scales. This regime is also of particular interest for early-universe theory, because these are the scales that exit the horizon first. At the very largest angular scales, in particular l = 2, 3, 4, several authors have noted peculiar features in the WMAP maps. In the first year maps the power at these three multipoles was lower than expected, with the probability of obtaining such a low value at the percent level; however, this depends on the portion of the sky used in the measurement as well as the fore ground subtraction technique, and may not be statistically significant (Efstathiou 2004). With the new analysis performed for the three-year WMAP data release, only the quadrupole remains noticeably low. This may also be due to features in 27 the inflaton potential, on scales that are horizon-sized today ( e.g. Contaldi et al. 2003). Another perhaps significant feature on these scales is the peculiar alignment observed between the axes of the moments up to l = 6 (de Oliveira-Costa et al. 2004; Schwarz et al. 2004) which may not agree with the assumption of statistical isotropy. This alignment was found to be quite strong in the first year maps, incompatible with statistical isotropy on the sky at rv 99% confidence (Land and Magueijo 2005) but was found to be slightly less significant with the third-year data release (Land and Magueijo 2007). It has been noted that these alignments could be due to galactic foregrounds, as well as extragalactic but local effects, such as weak lensing of the CMB dipole by an appropriate arrangement of nearby structure (Yale 2005). Because so much information about the dynamics of inflation might be contained in the primordial power spectrum, a fair amount of work has focussed on reconstructing the power spectrum and looking for features. ( All of the examples cited so far with real data use similar techniques to what will be described in the next section.) One standard approach is to let the reconstructed parameters be those that describe the potential responsible for inflation, both in the slow-roll context and in more general models, with the goal of learning about inflation directly. (Leach and Liddle 2003; Cline and Hoi 2006; Peiris and Easther 2006; Powell and Kinney 2007). Sorne authors (Hu and Okamoto 2004; Leach 2006) have asked which modes in the power spectrum can be best constrained within cosmic variance, both with 28 ~·. and without allowing other cosmic parameters to vary, using a principle component analysis approach. The result is that the modes near the acoustic peaks are most well-constrained, particularly with the inclusion of polarisation information; this is due to the microphysics that happens on those scales. We take a similar approach, also determining the effect of freedom in the form of the primordial power spectrum on the measurement of the cosmological parameters. Hu and Okamoto provide the broadening of constraints on other parameters. However, their freedom is allowed over the entire range of k and it is unclear whether their change in the constraint on 7 is explicitly due to the physics of reionisation or an indirect effect from the degeneracies with other parameters. We thus restrict the new freedom explicitly to low l. 29 CHAPTER3 Application to current data Recent datasets have made it possible to do precise tests of cosmological models. In this chapter we describe a reconstruction of the primordial power spectrum of curvature fluctuations .0. 2 ( k) from WMAP data, and discuss the effect of allowing this power spectrum to vary upon the measurement of severa! parameters including T, the optical depth to reionisation due to Thompson scattering. 3.1 Choice of parameterisation There are several approaches to reconstructing the primordial power spec- trum. Sorne examples in the literature include finding the amplitudes of wavelets (Mukherjee and Wang 2003), parameter-free direct deconvolution approaches (Shafieloo and Souradeep 2004; Tocchini-Valentini et al. 2006), and linear inter polation in bandpowers (Bridle et al. 2003; Spergel et al. 2007). We use a simple binning approach, similar to the latter, in which we parameterise the primordial power spectrum by its value at m values of comoving wavenumber ki, i = 1 ... m. We replace the standard power spectrum with the following modified version in the cosmological Boltzmann code Code for the Anisotropies in the Microwave Background (CAMB) (Lewis et al. 2000), which is based on CMBFast. The primordial spectrum at a particular scale k maps into the power in the CMB at a range of multipoles l, with the dominant contribution to Cz typically 30 coming from length scales of (3.1) This is the relationship for the angular scale of an abject of comoving length scale k on the last-scattering surface, set by equation 2.17; reionisation happens at a lower redshift and must be scaled by the ratio of the angular diameter distances. 1 We take the parameters ki to lie between 5 x 10-4 and 5 x 10-3 Mpc- ; this maps into multipoles '"" 2-30 for the primary CMB. This is the multipole range at which the reionisation peak in the CfE curve occurs for reasonable cosmological parameters (Zaldarriaga 1997). The widths of the transfer functions over which the integral in eq. 2.16 is taken (as plotted in e.g. Tegmark and Zaldarriaga (2002)) set the maximum useful resolution within this range. We choose 5 reference points, logarithmically equally spaced in k such that ln ki = ( i - 2)c ln k 2 for i = 3, ... , 6, and with the constant 4 1 3 1 c chosen such that k2 = 5 x 10- Mpc- and k6 = 5 x 10- Mpc . We set the 4 1 lowest control point at k1 = 10- Mpc- . For the wavenumbers higher than k6 , corresponding to the locations of the acoustic peaks and Silk damping, we allow for a spectral index ns in addition to an amplitude. We parameterise the the power spectrum by its value at each of these m = 6 control points, Ai = tl2 (ki)· For wavenumbers between these control points, we interpolate the power spectrum according to (3.2) 31 1 1 where k* = 0.05Mpc- and As= Am(km/k*)n·- , and for modes below k1 we extrapolate based on A1 and A2. With this method of log-log interpolation, the spectrum is a power law in k between any two ki; if the Ai are appropriately tuned, it reduces to the standard powerlaw for all wavenumbers, eq. 2.22. Also, the spectrum is continuous at km, but with ns and As defined at this value of k*, it is described for k > km by the standard wavenumber used in CAMB. (The WMAP collaboration, however, uses a reference value of k* = 0.002 Mpc-1 in its definitions of As and ns)· We thus have seven parameters describing the power spectrum: (A 1, ... , A6, n 8 ), with As in equation 3.2 obtained from A6 and n 8 • We shall call this form of the power spectrum the "broken" power spectrum madel. Four sample broken spectra, corresponding to four randomly-generated points in this seven-dimensional parameter space, are shawn in the top panel of Fig. 3-1. This parameterisation is complementary to that of Bridie et al. (2003): because they wish to constrain the power on scales smaller than the horizon size at reionisation, all but two of their 16 bandpowers are placed at wavenumbers greater 3 than 5 x 10- , which is our k6 . 3.2 CMB power spectra The bottom two panels of Figure 3-1 show how the freedom in the primordial power spectrum translates into the observable CMB power spectrum via the equations of the evolution of perturbations, which were discussed in Chapter 2. The scaling from eq. 3.1 is used to line up the horizontal axes for the features in the CMB temperature power spectrum with those in the primordial power spectrum. 32 For l < 40 the temperature power spectrum err traces the broken power spectrum quite closely, with sorne smoothing. This is because the temperature anisotropy on these large length scales is super-horizon at the time of recombi nation, and these perturbations arise only from the gravitational redshifting of photons out of the potential wells sourced by the broken primordial power spec trum. No other causal process acts on such large length scales this early in the Universe's history, as discussed in section 2.4. Since there is no extra freedom allowed for l > 40, the shape of the output spectrum is unchanged from the powerlaw ACDM spectrum at these multipoles: the pattern of the acoustic peaks is unchanged and continues to fit well the WMAP data points, which are also plotted. For the case of the polarisation power spectrum, CfE, it is more difficult to visually discern the effect of the broken primordial power spectrum. The reionisa tion peak is visible for all models shawn, and its amplitude increases strongly with increasing Zre (or, equivalently, with increasing T for the instantaneous-reionisation madel considered). The effect of changing each of the power spectrum amplitudes by a large amount is shawn in Fig. 3-2. 3.3 Using Monte Carlo Markov chains to constrain cosmology To find limits on T with the new freedom allowed in the madel we use Monte Carlo Markov chains (Gilks et al. 1996), which are a method of performing Bayesian inference. For a given dataset D, we have Bayes' theorem for a madel with parameters p, P(fi1D) ex P(j})P(DIP} (3.3) 33 ·----~=~:~~---~~--· ························ ··············· - -~ .. ~.. : ·..... · 1 o-to...______~------~------J 0.0001 0.0010 0.0100 ,...., N ~ .3 1::: N ...... J:_ ü -:::- 1000 =+ 10 100 1000 multipole 1 ,...., ! 10.000 1.000 .....::::::::.:.-- ··+····· .... ·.. 10 100 1000 multipole 1 Figure 3-1: Primordial (top panel), CMB TT (centre panel), and CMB EE (bot tom panel) power spectra for several 11-parameter "broken" spectrum models, showing how the freedom allowed in the primordial spectrum propagates into the CMB. In addition to the seven parameters describing the primordial power spec 2 2 trum, the four other parameters allowed to vary are {r2bh , f2ch , h, Zre}, which for these models are {0.028, 0.092, 0.89, 13.3} (solid line), {0.050, 0.212, O. 72, 10.1} (dotted line), {0.039, 0.185, 0.75, 7.4} (dashed line) and {0.038, 0.165, 0.77, 4.9} (dot-dashed line). Note that for l < 20 the values of C1EE increase with increasing Zre. The vertical grey bars are the binned uncertainty from the WMAP 3-year data from Hinshaw et al. (2007) (TT data) and from Page et al. (2007) (EE data). These particular models are not good fits to the data but are shown for demon strative purposes; they have values of -2ln L between 3586 and 3774. 34 j 1.5x1o* ~ 1.0x1o~ I ,5,QX1QJ 10 lOO 1000 Muttipole Moment 1 Figure 3-2: The effect on the err and crE spectra of doubling or nulling each of the first 5 power spectrum parameters. Note that for the TE case the y-axis plots only one power of l. where P(j)) is the prior on the model parameters; P(Dif)) is the likelihood, the probability of obtaining the dataset given the parameters; P(P1D) is the posterior, the distribution of the parameters given the data; and the proportionality constant depends only on D and not the parameters. If, for instance, one wishes to calculate the mean value of the N -dimensional model parameter distribution E(j)) = J dN p P(J)1D)p, or the expectation value of sorne function of the parameters E(f(j))) = J dN p P(P1D)f(j)), such as the variance or a higher moment, one way to proceed would be to calculate the likelihood at a number of fixed locations in parameter space and numerically integrate. However, such a grid-based approach is likely to be computationally wasteful, since the likelihood function will be close to zero at many locations in the parameter space. The problem is exacerbated with a large-dimensional parameter space and/ or 35 with a very strongly peaked likelihood function. Bayes' theorem also provides no prescription for finding the normalisation constant JdNp P(f})P(DIP). An alternative approach is Monte Carlo integration over the posterior. Here the goal is to obtain a set of points, {Pi}?=u distributed in the parameter space according to the posterior distribution, allowing the expectation value of a function f to be evaluated via a simple sum, ~ 1 n E(f(P)) = dNpf(f})P(PJD) ~;; ~f(fti). (3.4) J 2=1 The set of points only needs to be found only once, and the expectation value of many functions can then be evaluated. We use a modified version of the publicly-available CosmoMC package (Lewis and Bridie 2002) to generate a set of such points in cosmological parameter space, corresponding to the posterior distribution of various models given the dataset from the WMAP satellite. CosmoMC performs a random walk in parameter space whose steps are chosen according to a set of rules that, if successful, cause convergence to the target posterior. Because the ( i + 1)st point in the sequence depends only upon the ith point and not the rest of the history of points, the sequence of points chosen satisfies the definition of a Markov chain. The default implementation uses the Metropolis algorithm (Metropolis et al. 1953), in which a new point in the chain, y, is proposed according to a proposal distribution q(YIPi). The proposai distribution is in principle arbitrary and can depend on the current position of the chain; for instance it could be a multivariate Gaussian centered on ]Ji. This new point is then accepted into the chain with 36 probability -+ • ( P(yJD) ) a(Y1Pi) =mm 1, -+ • (3.5) P(ffiJD) If it is not accepted then ffi+ 1 is chosen to be equal to ffi and the process is 1 repeated for ffi+ 2 . CosmoMC uses CAMB to calculate the CMB power spectrum, and then uses the WMAP likelihood code, described below, to obtain the likelihood. The general Metropolis algorithm for CosmoMC with WMAP data, then, is (Gilks et al. 1996) • Set i = 0 and choose a starting point Po • Loop over i until stopping criterion reached - Propose a new point y according to the proposa! distribution q(YI.Pi) - Use CAMB to obtain the CMB power spectra C1x from the parameters y (this takes sever al seconds, or very fast depending on the parameters) - Pass these power spectra to the WMAP likelihood code and obtain the value of P(DIY) (this also takes several seconds) - Choose a random number u'""' Uniform(O, 1); if u:::; a(YI.Pi) then set Pi+I =y, otherwise set Pi+1 =Pi - (Every Mth iteration update the proposa! density) - Increment i • end. 1 This form of a is used in the Metropolis algorithm, to which the more gen eral Metropolis-Hastings algorithm reduces in the case of a symmetric proposai distribution. 37 Although any non-pathological proposai density will result in eventual conver gence to the target distribution, in practice the particular choice of the proposal density is important to obtain quick convergence-a proposai too narrow will propose many points near the current one, resulting in a high acceptance rate and highly-correlated chains that take a long time to traverse the distribution; while a proposal too broad will result in the likelihood being calculated at locations far outside the region of interest, most of which will not be accepted into the chain, also resulting in a long time to traverse the distribution. By default, CosmoMC takes the proposai density to be a multivariate Gaussian with width 2.4 times the standard deviation of each parameter; this factor is chosen because it optimises this tradeoff for the case of a Gaussian target distribution (Dunkley et al. 2005). These standard deviations are periodically calculated from the chains during execution, along with the rest of the full covariance matrix between parameters, Ca(3· This gives a continually updated proposal density, centered around the current point Pi, (3.6) where the Greek index following the dot indicates the component of the vector of parameters. Often, several chains are run in parallel; in this case the information is shared between them. Not all directions are proposed on every step simultaneously, however. In the method of calculating CMB power spectra used by CAMB, most of the computing time is spent calculating the transfer functions; the final step, obtaining the Ct's 38 0 1 00 200 300 400 500 chain step number Figure 3-3: Samples from a Markov chain for the 11-parameter model, showing (top) the angle subtended by the sound horizon, a slow parameter, and (bottom) 3 1 the amplitude of the primordial power spectrum at k4 = 1.5 x 10- Mpc- . Note that the fast parameter changes position much more often than the slow parame ter. from the transfer functions and the primordial power spectrum using eq. 2.16, is a relatively quick integration over a single variable, k. In order to take advantage of this, CosmoMC differentiates between the "slow" parameters, those which describe the transfer functions, and the "fast" parameters, those which describe the primordial power spectrum. The algorithm takes several steps in the subspace of fast parameters for each set of slow parameters. This distinction is particularly important for applications like the current one, in which the space of parameters describing the primordial power spectrum is significantly higher-dimensional than the two or three dimensions found in standard models. The steps taken by a slow and a fast parameter in a sample chain are shown in Fig. 3-3. One of the advantages of MCMC is that the time taken to reach convergence typically scales with the dimension of the parameter space, as compared with the 39 exponential scaling with dimension in grid-based methods. In order to determine when the chains have converged, CosmoMC takes advantage of the fact that several chains are often run in parallel. It periodically calculates the mean of the variances for each parameter across the parallel chains, as well as the variance of the means across the chains. The ratio of these quantities, R, should approach 1 as the chains converge to the target distribution and have explored it suffi.ciently (Gelman and Rubin 1992); when the largest value of IR- 11 for alllinear combi nations of the parameters has dropped below a small critical value, the chains can be considered to have converged to the distribution. This is also sometimes used as a stopping criterion: the number of points required to obtain a small value of R is high enough that the last half are very close to the target distribution. WMAP likelihood code. The most computationally intensive step in the above algorithm is the evaluation of the likelihood of the WMAP data given a set of C{s, where X stands for TT, TE and/orEE. The code we use, which the WMAP collaboration makes public, is described in Hinshaw et al. (2007) for the temperature data and Page et al. (2007) for the polarisation. For the low-l data uses fits directly to the degraded sky maps without calculating angular power spectra; this consists of calculating the likelihood, proportional to exp( -m7(S + N)m/2) for a map m, signal covariance matrix Sand noise covariance N. Since inverting a large matrix becomes prohibitive, for higher multipoles a Monte Carlo method, called MASTER (Hivon et al. 2002), is used. The total 2ln L is the sum of each of the components: the 2ln L for both the 40 temperature and polarisation, for the low-l and high-l cases. For the maximum likelihood madel supplied, the value of 2ln L is 3541 for 3526 datapoints. For the six-parameter ACDM madel, CosmoMC varies six parameters in its 2 2 10 default implementation: {flbh ,rldmh , 100B8 ,T,ln(10 As),ns}, which are the physical baryon density, the physical dark matter density, 100 times the angle subtended by the sound horizon at recombination, the optical depth to reionisa tion, and the amplitude and spectral index of the fluctuations, respectively. The last two of these describe the power spectrum. For the extended "broken" power spectrum madel, we include the first four of these and add seven parameters as de scribed in section 3.1. The result is an 11-dimensional parameter set consisting of 2 2 10 {Slbh , Sldmh , 100 88 , T, B1, B2, B3, B4, Bs, B6, n 8 }, where Bi = ln(10 Ai)· We take flat priors on these paramaters; for the amplitude parameters this translates into log priors for the Ai. The Ai are thus forbidden from going negative (by contrast, the reconstructions in Bridie et al. (2003) and Spergel et al. (2007) have flat priors on the amplitude parameters and their reconstructed amplitudes are negative in sorne bands.) Note that in the 11-parameter case the spectral index n 8 refers only 1 to the modes with k > k6 = 0.005 Mpc- , while in the ACDM case it applies for all wavenumbers. We modified CosmoMC torun the Metropolis algorithm for this expanded parameter set. We typically obtain over a million points in parameter space from eight chains run in parallel. With each chain allotted its own 2 GHz processor, it takes several days to obtain this many accepted points for the ACDM madel, and longer for extended models. This is however substantially more points than the 41 minimum needed for accurate measurements of low moments such as the mean and variance of the posterior distribution of a given parameter. 3.4 Results For the ACDM madel the chains accept approximately 30 - 40% of proposed points and the IR- li statistic drops below 0.01 after rv 7000 points per chain. In the case of the 11 parameter madel it takes '"" 150000 points to reach a comparable level of convergence, and approximately 20% of proposed points are accepted. The change in 2ln L is 5.4 7 for 5 extra degrees of freedom between these two models. Figure 3-4 shows the marginalised constraints on the amplitudes Ai. The 1 9 agreement with the WMAP best-fit powerlaw values As(0.002 Mpc- ) = 2.40 x 10- 1 and n 8 (0.002 Mpc- ) = 0.961 (Spergel et al. 2007) is shawn. The amplitudes are more tightly constrained at the largest values of k. The amplitude on the largest scale A1 is not effectively constrained away from 0 at 2a. These error bars correspond to the posteriors marginalised over all of the parameters, in particular each other; they do not provide complete information about the freedom allowed in the spectrum because the parameters are correlated. Figure 3-5 shows the contours of the 2-D marginalised posteriors in the Bi-Bj plane for each pair of amplitude parameters. Other than B 1, each pair of parameters is clearly well-constrained by the data, indicating that the chosen spectral resolution is not too fine. However, the parameters describing power in adjacent bins are also clearly anti-correlated, because the contours are stretched along the direction of the line Bi = -Bi+l· This is expected because an increase in one of the parameters should partly cancel a similar decrease in a neighboring 42 1 1 o-s- l T ------t--t--t--t--~--_ 1 o-1o- - - 1 2 1 0 - '--r---'----'----'--'--'-'-'-'-..____,___....___'---'--L..I....JL...l-...___1--'-----"--.L...... L....l.....I-J...J...l- 10-S 10-4 10-3 1 k (Mpc- ) Figure 3-4: The constraints on the six power spectrum amplitude parameters, Bi = ln(1010 Ai), in the 11-parameter model. The 68% and 95% confidence limits for the amplitude in each bin, marginalised over all other cosmological parame ters, shown in black and grey, respectively. The crosses indicate the values of the parameters corresponding to the maximum-likelihood point in the Markov chains. The dashed line is the best fit powerlaw to the WMAP data. The first 2a error bar reaches to low values and is not effectively constrained away from zero. 43 one, resulting in little change to the resulting CMB power spectrum. The degree of correlation, measured from the chains using the statistic (3.7) -the calculation of which is a useful application of eq. 3.4-is approximately -0.5 for adjacent parameters and +0.2 for next-to-adjacent parameters. This checkerboard-like pattern in correlations is shown in Fig. 3-6. The last amplitude parameter B 6 is not as correlated with the others because it is constrained by much more data, all modes with l > 80. The broadening of the constraints on the other four parameters which are allowed to vary in the model is shown in Figure 3-7. The standard error onT has increased from 0.030 to 0.033, 11% above the ACDM value. This broadening can be seen in the posterior shown in the top-left panel of that figure. This is comparable to the broadening that occurs if fewer WMAP wavebands are included in the analysis, or if information is removed for sorne l's (Page et al. 2007). The standard error on the matter densities Db and Ddm have increased by 33% and 29%, respectively. These parameters are mainly determined by the heights of the acoustic peaks, and adding freedom to the Sachs-Wolfe plateau likely makes them less well determined. For comparison Bridle et al. (2003) find increases of order 200% for the first year WMAP data release in their 16 bins at higher wavenumbers than considered here. We also obtain the constraint on the spectral index broadening by rv200%, but for the broken power spectrum it is only 1 constrained by modes with k > 0.005 Mpc- . 44 0.8l.O[Zll .àe 0.6 a. 0.4 0.2 -7.40-1.97 3.46 6.89 "· 0.8l.Olljj .cl 0.6 e a. 0.4 ~~:~~x2.5 0J 0.2 2.0 -7.40-1.97 3.46 8.89 1.75 2.77 3.79 4.61 B, B, 0.8l.Olljj ~ 0.6 m" :::~x ar:::~(:'\ Q. 0.4 3.0 ~ 3.0 N 0.2 2.5 2.5 -7.40-1.97 3.46 8.89 1.75 2.77 3,79 4.81 2.24 2.96 3.69 4.42 B, B, B, 0.8l.Olljj ~ 0.6 a. 0.4 0.2 -7.40-1.97 3.46 8.89 1.75 2.77 3.79 4.81 2.24 2.96 3.69 4.42 1.46 2.16 2.85 3.55 B, B, B, "· 0.8l.Olljj .àe 0.6 a. 0.4 0.2 -7.40-1.97 3.46 8.89 1.75 2.77 3.79 4.81 2.24 2.96 3.69 4.42 1.46 2.16 2.85 3.55 2.97 3.27 3.57 3.86 B, B, B, B, B, 0.8l.Olljj ~ 0.6 a... 0.4 0.2 -7.40-1.97 3.46 8.89 1.75 2.77 3.79 4.81 2.24 2.96 3.69 4.42 1.46 2.16 2.85 3,55 2.97 3.25 3.54 3.82 2.86 3.02 3.19 3,35 B, B, B, B, e, B, Figure 3-5: The constraints from the MCMC on the six power spectrum amplitude parameters that are allowed to vary in the model. The thick and thin lines indicate 68% and 95% confidence limits, respectively; the crosses indicate the values of the parameters corresponding to the maximum-likelihood point in the chains. The constraint on the seventh parameter, the spectral index above 0.05 Mpc-I, is not shown. As in Fig. 3-4 the first power spectrum parameter is not constrained to be above zero for the 2CT contour. 45 Figure 3-6: The matrix indicating the degree of correlation between pairs of ampli tude parameters (Bi, Bj), as defined in eq. 3.7. The rows and columns correspond to the amplitude parameters B 1 through B6 , and the pairings in the bottom trian gle of this matrix correspond to the panels in Fig. 3-5. 46 The cosmological parameters are thus quite robust to adding freedom on these large scales within the constraints provided by current data, although there is sorne broadening. In the next chapter we investigate how this should change for future experiments. 47 0.8 _ci 0.6 ~ o._ 0.4 -1.94x10-5 9.82x10-2 1.98x10-1 T 0.8 q," _ci 0.6 ô 1.040 0 e o._ 0.4 1.030 1.028 1.040 1.052 (100)8, 0.13 0.13 0.12 0.8 [email protected] 0.11 .ri 0.6 .c :x "<: 'X ~ 0.10 J 0.10 e 0 o._ 0.4 0.09 0.09 0.08 0.08 -1.94x10-5 9.82x10-2 1.98x10-1 1.028 1.039 1.050 0.07 0.10 0.13 T (100)9, odmh2 0.025 0.025 0.025 0.024 0.024 0.024 0.8 0.023 0.023 0.023 _ci 0.6 -". "<:. .c e a 0.022 a 0.022 ê 0.022 o._ 0.4 0.021 0.021 @Y 0.021 (@ 0.020 0.020 0.020 ~5 2 1 -1.94X10- 9.82Xl0- 1.98X10- 1.028 1.039 1.050 0.07 0.10 0.13 0.0195 0.0224 0.0253 T (100)8, odmh2 Obh2 Figure 3-7: The constraints on the four non-power spectrum parameters which are allowed to vary in the model, with (grey) the ACDM model and (black) the 11-parameter model described in the text. The plots along the diagonal show the 1-d marginalised (unnormalized) posteriors for the indicated parameters, and the off-diagonal plots show the contours of the joint 2-D posteriors. 48 CHAPTER4 Forecasting for future data Although the Monte Carlo approach of the previous chapter had the advan- tage that it explored the entire likelihood space available to a given model, it was restricted in the sense that it only did so for a single dataset, that from the WMAP mission. While the WMAP measurements are cosmic variance-limited for the temperature data for multipoles up to l ,....., 1000 (and in sorne cases, such as l ,....., 10, are measured much better than cosmic variance) the polarisation measurements are not. Since the optical depth to reionisation is most strongly constrained with the addition of polarisation information, in this chapter we will perform forecasting to determine the constraints of T for future experiments with lower uncertainty on the polarisation. This future data is expected to come from several sources. The all-sky Planck Surveyor mission (planned for launch in 2008; Lamarre et al. (2003)) is expected to be approximately 10 times more sensitive in RMS noise-per-steradian units for polarisation than WMAP, and the next generation of satellites which will attempt to find the B-mode polarisation will be even more sensitive (Weiss 2006). 4.1 The Fisher information matrix for CMB experiments Fisher matrix methods are often used in the context of cosmology for forecast- ing the ability of future experiments to test models often with the aim of helping 49 guide experimentalists in determining design parameters for future experiments (e.g. Jungman et al. 1996). These methods have been for sorne time to help guide experimentalists determine design parameters for future experiments. For instance, much of the work in this chapter is based upon techniques used in the mid-1990s to guide the then next-generation of experiments in providing the strongest constraints of cosmological observables (Zaldarriaga et al. 1997). More recently, these methods have been used to predict what can be learned about the currently less-constrained components of the cosmological model, such as the equation of state of dark energy, using data such as upcoming weak lensing surveys and extragalactic supernova surveys (Heavens et al. 2007), and the ionisation history of the uni verse, using the same future experiments as we consider here. The idea is to explore the curvature of the forecasted log-likelihood function about the maximum-likelihood point. This introduction mainly follows that of Dodelson (2003). The observed power on the sky at a given multipole l, Ct,obs, can be estimated as the average over m of the squares of the 2l + 1 spherical harmonie coefficients azm· As mentioned in Chapter 2, according to infiationary predictions, and confirmed by recent data to high accuracy (Spergel et al. 2007), each azm is Gaussian distributed with mean 0 and variance Cz. The quantity Ct,ohs is an unbiased estimator for Cz in the case of no noise. As the sum of squares of 2l + 1 independent and identically distributed random variables, it is chi-square distributed with mean Cz and variance )2/(2l + 1)Cz. 50 Approximating this chi-square distribution as a Gaussian near the peak, for a CMB experiment we can form the usual chi-square statistic (4.1) 2 so that the likelihood is L = exp( -x /2). The dependence of C1 upon the set of cosmological parameters, p, has been made explicit. The sum is taken over all values of l probed by the experiment, and in the idealisation of a noise-free, all-sky experiment, the weighting factors are the cosmic variance (4.2) OC.! cv = J( 21 : 1) C, As will be discussed below, we can absorb the effects of noise and incomplete sky coverage into the definition of these weights, 5C1, leaving eq. 4.1 valid for a more realistic experiment. Taking the second derivatives of x2 with respect to the parameters, we obtain Since, in the case of forecasting for future data, we have no values of Ct,obs readily available, we would like to average over all possible datasets. The second term on the right hand side will be zero upon taking this average, because the Ct,obs are distributed about the mean value Cz. We define the Fisher matrix to be one-half this average, (4.4) 51 where we have dropped from the notation the dependence of Cz upon on the parameters j1. The value of forming the Fisher matrix in this way is that it can be used to estimate the widths of the distributions of parameters. Although the likelihood function is not necessarily quadratic in the parameters P, we can expand it to second order about the fiducial point fk.d, taken to be the maximum-likelihood point, such that 8x2 /op a = 0, and so (4.5) with summation implied over a and (3 from 1 toN, the number of parameters. We are interested in the set of points in the parameter space at which x2 will change by 1, thus giving us the 10" bound on the likelihood function. To find these points we can solve the equation .6.x2 = x2 (p')- x 2 (fk.d) = 1 for jJ' to obtain (4.6) Now, choosing the point fJ' to lie along a certain direction in parameter space from fk.d, say along co-ordinate Pm we set (3 = a and see that the x2 changes by 1 if .6. _ , _ (F-1)1/2 Pa -Pa - Pa,fid - cm· (4.7) Thus, the diagonal element of the inverse Fisher matrix sets the uncertainty on the corresponding parameter. 2 This is a general result: the error estimate (F-1)Y;, obtained by expanding x to second or der, is established as the minimum of all unbiased estimators according 52 to the Cramer-Rao inequality. Any unbiased estimator will have at least this uncertainty on the parameter Pa· Note that it is the derivatives of the Cz's with each parameter, and not the Cz's themselves, which set the uncertainty on that parameter. Since the Fisher matrix is the full matrix of curvature of x2 in the basis directions in the parameter space, its inverse can be considered a covariance matrix between parameters, which can be used to find degeneracies between parameters as well as the absolute error on each one. Real-world effects. We now consider modifying the above for experiments with sorne real-world effects. The weights 6Cz, the denominator in the expression for x2, were simple in the cosmic variance-limited case, eq. 4.2. For an experiment with noise, incomplete sky coverage, and a finite experimental bearn, these weights instead become ( 4.8) 1 2 where fsky is the fraction of the sky measured, w- is the noise weighting, and B is a factor to account for the bearn. We will discuss each of these in turn. The factor !sky appears as a prefactor because observing less of the sky is similar to increasing the cosmic variance limit. This factor will be less than 1 even for a satellite experiment because the dust in the galactic plane is much brighter than the CMB fluctuations and cannat be completely characterized with multiwave- length maps. Also, the pixels containing resolved extragalactic point sources must be removed from the map if they are too bright to allow a confident measure of 53 the CMB anisotropy behind them. For these reasons, WMAP uses a value of fsky ~ 0.8 (Hinshaw et al. 2007) To quantify the noise in the experiment we define the weight-per-solid-angle (Dodelson 2003), (4.9) where a is the average noise per pixel in the measurements normalized to the CMB background temperature, and b..w is the size of each pixel in steradians. This is a measure of the experimental noise that is independent of the pixelling scheme. The noise is taken to be white here; if the amplitude of the noise depends on l then the constant factor w is replaced by the effective noise power spectrum across multipoles. Finally the factor B-2 is a window function to account for the smearing effect of a finite experimental bearn. The bearn function is the equivalent to the point spread function in optical astronomy: the real CMB sky is convolved with it to obtain the observed CMB sky. As a result, it sets an upper limit to the multipoles on which anisotropies can be observed. The convolution in real space is a simple product in the spherical harmonie domain, and in the approximation that the bearn function is a Gaussian on the sky with angular size ab, the CMB anisotropies are observed through a multiplicative window function 2 -u2 l2 B = e b (4.10) In the case of WMAP, the bearn sizes range between 0.2 and 0.8 degrees for different wavebands (Page et al. 2003) and B 2 falls significantly away from 1 for 54 multipoles of several hundred. The bearn function thus down-weights the influence of high-multipole data in the constraint on a given parameter, and it effectively sets the value of l up to which to sum in eq. 4.4. Additional spectra. Up to this point we have considered the case of a single kind of power spectrum, taken to be the temperature power spectrum. For the current application, we are interested in the case in which the TE and EE spectra are measured as well. In this case the expression for x2 contains terms from all component power spectra Cf where X represents TT, TE, and EE. If there is only a single power spectrum in consideration, the C1's can be taken to be uncorrelated with each other for different values of l, however, if the different spectra are taken into account there is a non-negligible correlation between them. 2 The division with the weighting (6C1)2 in the definition of x , eq. 4.4, is then generalised to the inverse covariance matrix between the spectra. The Fisher matrix becomes lmax acx acy 1 Faf3 = L L -i-Cov- (Cf, Cj)-i-· (4.11) l=2 X,Y Pa P(3 The covariance matrix between different power spectra at a given l is given by, in the (err, crE, CzEE) basis (Zaldarriaga et al. 1997), CTT)2 CTECTT ( l,eff l l,eff x y) 2 .!(CEE+ CTT)CTE CTECEE (4.12) Cov(Cz 'Cz = (2l + 1)fsky 2 l,eff l,eff l l l,eff CEE)2 ( l,eff where only the top triangle of the symmetric matrix is displayed. The effective Cf with noise included, C(eff is obtained by adding to Cf the effective noise at 55 that multipole. Because for current data the temperature measurements are cosmic variance limited, we neglect this and set C1~~ = Cfr. However, for the E-mode polarisation we allow a noise term, resulting in (4.13) where here the term w refers to the noise in a map of Q or U Stokes' parameters. Equation 4.12 reduces to eq. 4.8 if considering only the (TT,TT) matrix element. Equation 4.11, together with the expression for Cov(Cf, cr), can be seen as the generalisation of eq. 4.4. 4.2 Application Our goal is to find the constraints on cosmological parameters when freedom is added to the power spectrum, similar to what was clone in Chapter 3 in the case of WMAP data. We calculate the Fisher matrix in eq. 4.11 for the same broken power spec- trum madel as was described in section 3.1. For simplicity we choose our pa- rameter set to consist of the 7 power spectrum parameters, plus T. We evaluate the derivatives about the fiducial madel (equivalently the point in parameter space p6d) given by those values of the parameters corresponding to a standard 9 powerlaw with As = 2.3 x 10- and n 8 = 0.95, bath of which are evaluated at 1 k* = 0.05 Mpc- . With different points in the parameter space the results were found to be quite consistent. For the parameters that we do not vary, we set ndmh2 = 0.12, nb = 0.022 and h = 0.70; we also only consider scenarios with instant reionisation. 56 Following Zaldarriaga et al. (1997) we take w = 3 x 1014 for WMAP polarisation maps and w = 2 x 1016 for those of Planck. These numbers are based upon the noise levels from combining measurements from several wavebands; not all wavebands are included because they are needed to remove foreground contamination as well. Also since the noise is not white in practice, these are meant as arder-of-magnitude values. Noting that the terms that enter the Fisher matrix are the derivatives of the Cz's with respect to the parameters f/, weighted by the inverse covariance, we numerically calculate the derivatives 8Cf j8pa, evaluated at Pa,fid· ln arder to do this we perturb each of the parameters by 2% above and below its fiducial value, and use CAMB to find the perturbed C1's. To form the Fisher matrix we use the covariances in 4.12; to obtain the inverse covariance we invert this (3 x 3) matrix for each l. We choose to sum to lmax = 1000, which is the same upper multipole which was used in the calls to the WMAP likelihood function in the previous chapter. We expect most of the discriminatory power to come from l < 50 for constraining T for law-noise experiments, however. 4.3 Results The inverse Fisher matrix for the broken power spectrum madel, normalized 18 to have values of 1 along its diagonal, is shawn in Fig. 4-2. We have set w = 10 , which is 100 times the value for Planck (i.e. RMS sensitivity better by a factor of 10). This can be interpreted as a correlation matrix between parameters. As 1 expected, T is degenerate with B 6 , the amplitude for k > 0.005 Mpc , due to the 57 27 scaling of the data with A8 e- for high multipoles. The checkerboard-like pattern in correlations between the amplitude parameters seen in Fig. 3-6 is also visible here. Fig. 4-1 shows a plot of a 7 vs. w for both the ACDM madel and the broken power spectrum madel. There are several regimes. In the high-noise case, for 13 w < 10 , the polarisation measurements are significantly down-weighted due to their high noise, so there is no effectively no information from them and the only information available is from err. The estimate of the err or on T plateaus at a 7 = 0.07 for the broken power spectrum and 0.04 for the powerlaw. We obtain this plateau because if only the temperature measurements are available then the degeneracy between T and As is somewhat broken by having measurements on both sides of the horizon scale at reionisation. Adding freedom in the power spectrum amplitude at these scales broadens this constraint, however. 18 The other limit in Fig. 4-1 is the very low-noise limit, with w > 10 . There is a plateau in a 7 here because this is the transition where the polarisation measure ments become cosmic-variance limited, and decreasing the experimental noise does not add more information. Here the la uncertainty on T is approximately 0.002 for the powerlaw madel and 0.003 for the broken power spectrum. The noise levels for both WMAP and Planck are between these two regimes. An experiment with a value of fo larger than that of Planck by a factor of 10 would reach the inherent limit on measuring T. Figure 4-3 illustrates the results of the calculation. It shows the effect on the CMB power of perturbing both T and each of the amplitude parameters by an 58 0.010 w Planck 0.001 1 010 Figure 4-1: Constraints on tau for two cosmological models, the broken power spectrum (upper curve) and ACDM (lower curve), as a function of the inverse noise in the polarisation measurements, w. The only variables allowed to vary are the power spectrum parameters and T. The noise levels for WMAP (left) and Planck (right) are shawn as verticallines. 59 1.0 0.6 0.2 -0.2 -0.6 -1.0 Figure 4-2: The inverse Fisher matrix between parameters for the eight-parameter model normalized to have values of unity along the diagonal. The noise has been set to Planck levels. The 6 x 6 top-left corner can be compared to Fig. 3-6 which was obtained with the WMAP likelihood. 60 l:: N ...... w 1o-15 w ....---.u ..-- + '--"' l:: N ...... w 1o-15 wu- ....---. ..-- + '--"' 10 1 10 Multipole Multipole Figure 4-3: The effect of perturbing the EE power spectra by the la values of the parameters. The top left panel shows the effect of perturbing the optical depth about the fiducial model by +aT (top line) and by -aT (bottom line). The top right panel shows the effect of perturbing the power spectral amplitude parameters by +aBi and -aBi, with each shade corresponding to a different i. These are for Planck noise levels. The bottom two panels show the same quantities but for an experiment with RMS sensitivity higher than Planck by a factor of 10. The grey envelope is the overall cosmic variance limit for a noise-free experiment, which sets the size of these bands. 61 amount equal to their la values when w is set to both the Planck level and 100 times the Planck level. The background is the cosmic variance window, equation 2.27, which is broadest for small values of l. The main result of this chapter is contained in Fig. 4-1: that relaxing the assumption of the form of the power spectrum of fluctuations on large scales will, for very low-noise experiments, relax the constraint onT by rv 50%. This is slightly more broadening than is found in the ionisation model-independent calculations by Mortonson and Hu (2007). Also, this amount of broadening is slightly less than that found by allowing freedom across the entire range of k which the CMB probes (Hu and Okamoto 2004), rather than restricting freedom to the region where reionisation is important. The peak from reionisation thus can be used to reliably extract information about the optical depth, which is needed to measure the overall amplitude of fluctuations in the CMB accurately, regardless of possible unknown effects on the primordial power spectrum. 62 CHAPTER 5 Conclusions In this wor k we presented a reconstruction of the primordial power spectrum on large scales. We did not detect deviations from the WMAP best-fit powerlaw. We found that the constraints on the optical depth to reionisation broaden from current data by 10% when adding freedom on large scales, and this is expected to increase to a broadening of approximately 50% as future constraints on the CMB polarisation improve. The Planck Surveyor satellite is scheduled to launch in 2008 and is expected to provide a more sensitive measurement of the CMB EE power spectrum by a factor of"' 100. It is likely that Planck will also provide rather stronger constraints on the reionisation history. The optical depth is an important quantity to measure accurately not only because it provides information about the process of reionisation, but also because uncertainties in T propagate into uncertainties on other parameters. For example, weak lensing observations are expected to provide stronger constraints on w, the equation of state parameter for clark energy. 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