Predicting neutrino mass constraints from galaxy cluster surveys

Predicting neutrino mass constraints fromgalaxy cluster surveys

Undergraduate Thesis in Astrophysics 2010

Jessica MuirMichigan State Univeristy

Adviser: Mark Voit

Abstract

Galaxy cluster surveys allow us to study the spectrum of density perturbationsin the local universe, which can in turn provide a way of constraining cosmologicalparameters. The advent of large sky surveys in the next decade will dramaticallyimprove the precision with which these measurements can be made, and in order totake full advantage of the results it is crucial that we understand how the results dependon cosmology in a precise manner. It is also necessary that we take into account eventhose parameters which have a small effect on the evolution of structure, such as Ων ,the density contribution from neutrinos. In this aim, I used the software packagesCosmosurvey and Cosmofisher to predict how well data from future surveys will beable to constrain Ων . This paper will describe how Cosmosurvey predicts the resultsof cluster surveys for sets of cosmological parameters and how Cosmofisher uses Fishermatrix techniques to describe the relations between uncertainties of large numbers ofparameters. My results indicate that data from future galaxy clusters will be able toconstrain Ων to within 0.01 of its actual value at a 95.4% confidence level. They alsoallow us to select an optimal binning scheme for survey data and to assess the effect ofuncertainties in the mass-luminosity relation for galaxy clusters on Ων constraints.

Contents

1 Introduction 1

2 Understanding and describing structure formation 42.1 The matter power spectrum P (k) . . . . . . . . . . . . . . . . . . . . . . . . 42.2 The evolution of P(k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 The effect of neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Relating P(k) to galaxy clusters . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Methods 123.1 Cosmosurvey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Cosmofisher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Results 194.1 Checking for optimal binning . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 Effects of the cluster model . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5 Conclusion 24

1 Introduction

Over the course of the last few decades, a widely accepted picture of the universe has been

pieced together. It describes the universe as having a flat geometry, with a total mass-energy

density equal to or very close to the critical density. That is, Ωtot = 1. In this model, the

universe came into being with the Big Bang 13.7 billion years ago [19], followed by a period

of rapid inflation. It has expanded and cooled ever since, and is now again expanding at an

accelerating rate.

The universe is made up of three main components: radiation, vacuum (or dark) energy,

and matter. Though its origins remain mysterious, dark energy appears to behave like a

cosmological constant Λ, with an equation of state parameter of w = −1.0. Its contributions

presently make up the majority of the mass-energy density of the universe, with ΩΛ ≈ 0.7.

Radiation consists of photons and relativistic particles, and though Ωrad was more prominent

in the early universe, its contributions to the current Ωtot are negligible. Matter makes up

1

the remaining 30% of the universe’s mass-energy density. Baryonic matter only represents

about 5% of ΩM , with the rest in the form of cold dark matter (CDM). [3]

This model, known generally as Concordance Cosmology is the result of the combined

findings from many fields of research. Measurements of the Cosmic Microwave Background

(CMB) have indicated a flat geometry and constrained the time of baryon-photon decoupling,

type Ia supernovae have provided a measure of the universe’s expansion, and studies of big

bang nucleosynthesis (BBN) have delineated tight constraints on the amount of baryonic

matter in the universe. [6]

Another set of constraints, which will be the focus of this paper, comes from galaxy

cluster surveys, and the consequent knowledge of the history of structure formation which

they convey. Galaxy clusters, which can contain anywhere from hundreds to thousands

of galaxies within a radius of about 6h−1Mpc (where h is H0/100kms−1Mpc−1) [3], are the

largest gravitationally collapsed objects in the universe. In addition to their galaxies, clusters

contain about ten times more baryonic matter in the form of intracluster gas [3], as well as

large halos of dark matter. The dark matter, which makes up most of the cluster mass,

creates deep potential wells which the gas falls into. This causes the gas to heat up and emit

x-rays, which can be detected from earth using specialized telescopes. The temperature of

the gas in a galaxy cluster is related to the depth of its potential well, so studying the spectra

and luminosity of the emitted x-rays can give a measure of its mass. [18] Surveys of galaxy

cluster x-ray emission can therefore yield a distribution of galaxy cluster masses which can

be used to make inferences about how those structures formed.

The next decade promises to further improve our knowledge of cosmology, as new sky

surveys will greatly increase the amount of data available about galaxy clusters. One such

survey is eROSITA (extended Roentgen Survey with an Image Telescope Array), which is to

be launched aboard the Russian Spectrum-Roentgen-Gamma satellite in 2012. It will have

the capability to measure x-rays of energies up to 10keV, and according to the eROSITA web-

2

page, will have “an unprecedented spectral and angular resolution.” [1] One of the project’s

goals is to use these capabilities to detect the hot gas in 50 to 100 thousand galaxy clusters.

Beyond eROSITA, the proposed Wide Field X-Ray Telescope has the potential to increase

these numbers by another order of magnitude. [12] As current survey data contains at most

a few hundred clusters with good x-ray luminosity and redshift measurements, both of these

surveys represent a drastic increase in statistics, and will accordingly allow us to make more

precise determinations of cosmological parameters. This will in turn require a more ex-

act understanding of how those parameters map onto observables, and will allow definitive

statements to be made about constraints on parameters which have even small effects – for

example, the mass density of neutrinos.

Neutrinos are neutral leptons originally proposed in 1930 by Wolfgang Pauli to explain

the continuous nature of the beta decay energy spectrum. [9, 14] Existing in three flavors,

νe, νµ, and ντ , they interact only through the weak force and gravity, and were thought to

be massless until the late 1990’s. At that time, studies of solar and atmospheric neutrinos

confirmed the existence of a phenomenon called neutrino oscillation. [5] This means that

neutrinos of one flavor type can turn into those of another, which can only occur if neutrinos

have at least three mass eigenstates, ν1, ν2, and ν3. [2] Studies of oscillations give a direct

measurement of the mass differences between these states, and therefore require neutrinos

to have a minimum mass. This has important consequences for cosmology, as neutrinos

have a number density of 113 per flavor per cubic centimeter. [10] Because of this, mass

contributions from them can have non-negligible effects on the composition and evolution of

the universe.

The goal of this paper will be to investigate how well future surveys will be able to con-

strain neutrino mass, as well as other cosmological parameters. It will open with an overview

of the formation of structure in the universe and how its observational signatures reflect cos-

mological parameters. I will then explain how, through use of pre-existing computational

3

software, I was able to predict how the observables from galaxy cluster surveys depend upon

various parameters. I will close with the examination of likelihood curves for simulated sur-

veys and the implications they have for our ability to measure the mass contributions of

neutrinos.

2 Understanding and describing structure formation

On large scales, the universe is homogeneous and isotropic, but on scales smaller than a

few hundred Mpc, this is clearly not the case, as evidenced by the presence of structure

in the form of galaxy clusters, galaxies, and stars. These structures were seeded by small

density fluctuations in the early universe which grew over time. Galaxy clusters are tracers

of these perturbations, and their masses are a measure of the amplitude of the density

fluctuations. In order to understand how we can use galaxy clusters to gain insight into the

values of cosmological parameters, we must first understand how structure can be described

and quantified and how it evolved throughout the history of the universe. We can then

discuss the ways in which that structure is sensitive to cosmology and how it can be related

to observables from galaxy cluster surveys. (Note: the derivations in this section are from

and can be found in more detail in [18].)

2.1 The matter power spectrum P (k)

The magnitude and distribution of fluctuations in matter density can be described with a

power spectrum, P (k). The power spectrum’s definition follows from that of linear pertur-

bations to the matter density of the universe – that is, fluctuations in density that are small

compared to the average matter density of the universe. A perturbation at a location in

space ~x can be written as

δ(~x) =ρM(~x)− 〈ρM〉〈ρM〉

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where ρM(~x) is the matter density at point ~x and 〈ρM〉 is the mean matter density of the

universe. To relate the spatial function δ(~x) to the relative amplitudes of perturbations on

different length scales, we can take its Fourier transform to get

δ(~k) =

∫δ(~x)e

~k·~xd3x

Because the universe is isotropic, we can assume that δ(~x) is as well, which allows us to

concisely describe the relative amplitudes of perturbations of different length scales by the

power spectrum:

P (k) ≡⟨|δ(~k)|2

⟩(1)

2.2 The evolution of P(k)

In scenarios involving only cold (non-relativistic) matter and gravity, the growth rates of

linear perturbations are theoretically well understood and are found to be independent of

the spatial scale of the perturbation. [6,18] This means that the rate at which the amplitude

of a perturbation grows is the same for all k values. In terms of the matter power spectrum,

this implies that P (k) would retain the same shape while moving upwards on a plot. In-

flationary models of the early universe predict that the quantum fluctuations which seeded

the primordial density fluctuations yielded a power spectrum of the form P (k) ∝ knS , where

ns ≈ 1. [18] Considering only these two ideas would lead one to expect that the matter power

spectrum of today should remain a straight line.

This is not the case; the power spectrum we observe is like the one plotted in Figure 1.

Though the spectrum is in fact nearly linear with P (k) ∝ k at low k values (spatially large

perturbations), the trend stops at a turn-off point, after which the amplitude falls for high k

(spatially small) perturbations. The reason for this is that there are factors which influence

the growth of perturbations other than gravity and non-relativistic matter, and they must

5

Figure 1: The observed shape of the matter power spectrum.

be taken into account.

As was described in the introduction, the universe has three components: matter, radi-

ation, and dark energy. Because they depend on the size of the universe in different ways,

their contributions to the total energy density of the universe have varied throughout its

evolution [3]. This had profound effects on structure formation.

At any time, scales at which structure formation happens can be divided into two broad

classes. Perturbations with length scales larger than the horizon size of the universe grow

in a scale independent way, simply increasing the amplitude of their portion of the power

spectrum without changing its shape. This is because areas of enhanced density expand

slightly slower than the rest of the universe, while areas of reduced density expand slightly

quicker. Other interactions do not come into play; because the perturbation are larger than

the horizon size, portions of them are further apart than the distance light could have possibly

traveled in the lifetime of the universe. More dynamic effects appear for scales within the

horizon size. [19]

In very early times, radiation dominated the mass-energy density in what is known as

6

the radiation era. Initially, the maxima of the photon-baryon density distribution coincided

with those of CDM. When a perturbation’s length scale entered the horizon size, the effects

of radiation pressure could take effect and the radiation and baryonic components of the

universe (which were coupled at the time) underwent acoustic oscillations. [3] The matter

component of the universe is pressureless, and because dark matter by definition does not

interact electromagnetically, the CDM did not participate in these oscillations. That is not

to say it was not affected by them: Remember, at this point in the universe’s history, the

oscillating radiation made up the majority of the total mass-energy density and the photon-

baryon acoustic oscillations effectively pushed photons out of CDM potential wells. Because

CDM was gravitationally attracted to the energy carried by these photons, this damped the

growth of matter perturbations which were within the horizon size.

As the universe expanded, the fraction of the total density in the form of radiation

decreased, and that of matter increased. When it cooled to about 9000 K [3], the matter era

began. In this phase of the universe’s evolution, the photon-baryon component continued

to oscillate, but because matter was the dominant contributor to density, the movement of

radiation was not enough to damp the accumulation of CDM. As a result, perturbations in the

CDM distribution could begin to grow again. Later, when baryonic matter decoupled from

radiation during recombination, it fell into the growing potential wells already established

by the CDM and eventually formed the visible parts of galaxy clusters.

The damping during the radiation era is what is responsible for the curved-over shape of

the observed matter power spectrum. The low-k region of Figure 1 represents length scales

that were larger than the horizon size of the universe during the entirety of the radiation

era. Perturbations of these scales underwent scale independent growth and retained the

power spectrum shape from the primordial fluctuations present after inflation. The bend

in P (k) marks the k value corresponding to the horizon size of the universe at the time of

matter-radiation equality, and all larger k’s related to perturbations which were damped by

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photon-baryon acoustic oscillations during the radiation era. The smaller the length scale,

the earlier perturbations of that scale entered the horizon size, and the longer their growth

was subjected to damping effects. This is why P (k) is more suppressed at higher k values.

It is clear that the matter power spectrum is highly dependent on cosmology. For example,

the slope of the linear part of the spectrum at low k depends precisely on the value of the

spectral index ns from the relation P (k) ∝ kns . Also, the position of the pivot point depends

on how large the universe was at matter-radiation equality, which depends on ΩM , Ωrad,

and h, among other things. [11] In addition to this, the power spectrum amplitude reflects

how much perturbations have grown through time, which is affected by the expansion of the

universe and therefore both ΩΛ and w from the dark energy equation of state. It follows that

having a good understanding of how these and other cosmological parameters leave imprints

on P (k) makes it a useful tool for deducing their true values.

2.3 The effect of neutrinos

As the aim of this paper is to use the matter power spectrum (by way of galaxy clusters) to

restrict the mass of neutrinos, let us explore their part in the evolution of structure. There

was a period of time in the early universe, well before matter-radiation equality, when the

universe was hot and dense enough for neutrinos to remain in thermal equilibrium through

weak force interactions with their surroundings. As a result, neutrinos of each flavor (νe,νµ,

and ντ ) were present in approximately equal number density. Then, as the universe expanded

and the temperature dropped below about 1 MeV [10], the weak interaction rate fell enough

so that neutrinos decoupled from the rest of the primordial plasma. They have streamed

through the universe ever since, establishing a cosmic neutrino background analogous to

the CMB. It is calculations of the pre-decoupling thermal equilibrium and the decoupling

dynamics which yield the neutrino number density value of 113 neutrinos and anti neutrinos

per flavor per cubic centimeter. [10] Using this number density, individual masses for neu-

8

trinos can be related to the neutrino mass density with the following equation, where mi is

the mass for the neutrino mass eigenstate νi:

Ων =ρνρc

=

∑imi

93.14h2eV(2)

All massive particles were relativistic in the early universe and became non-relativistic

as the universe cooled at later times. Neutrinos are unique in the fact that due to their

extremely light masses, they remained relativistic much longer than the rest of the matter in

the universe. [11] While relativistic, they acted as radiation, and in that role they influenced

the time of matter-radiation equality. After this time, an important feature distinguished

neutrinos from photon radiation: their number density (and therefore Ων) has the same

dependency on the size of the universe as CDM and baryonic matter. This means that in

the matter dominated era, as neutrinos free streamed out of potential wells in the matter

density distribution, they reduced the growth rate on the high-k end of the matter power

spectrum. [6] More precisely, the movement of massive neutrinos suppresses P (k) at length

scales smaller than their free streaming length. [10]

2.4 Relating P(k) to galaxy clusters

Through the effect of gravity, linear density perturbations grew in amplitude over time,

eventually becoming non-linear and undergoing gravitational collapse into structures such

as stars and galaxies. Structure formation in the universe occurred hierarchically, with

smaller objects forming first then clumping together to form larger ones. As the largest

gravitationally relaxed structures in existence, galaxy clusters are therefore also the most

recently formed. Because they are less affected by other astrophysical phenomena than

smaller structures, they provide a good probe of the matter power spectrum. But how can

we translate our observations of galaxy clusters into an understanding of the cosmology

9

discussed above?

The first step of relating P (k) to cluster mass distributions is to use it to find the variance

of matter distributions on various mass scales. For this purpose, we derive a quantity called

σM . We can describe variations of the amount of mass inside a sphere of arbitrary radius

rM centered at position ~r using the formula

δM(~r)

〈M〉=

∫δ(~x)W (|~x− ~r|)d3x (3)

where W (|~x− ~r|) is a spherical window function which is equal to 1 where |~x− ~r| ≤ rM and

is 0 where |~x− ~r| > rM . Here 〈M〉 is the mean amount of mass inside a sphere of radius rM

averaged over the volume of the universe, and δM = (ρM(~x)−〈ρM〉)(43πr3

M) is the difference

between the amount of mass inside an individual sphere centered at ~r and 〈M〉. δM(~r)/〈M〉

can be thought of as δ(~x) smoothed over the sphere.

By taking the Fourier transform of this, we can relate it to P (k) and can find the variance

of these fluctuations on mass scale 〈M〉:

σ2M ≡ σ2(M) ≡

⟨∣∣∣∣ δM〈M〉∣∣∣∣2⟩

=1

(2π)3

∫P (k)|Wk|2d3k (4)

where Wk = 3(sin(krM) − krM cos(krM))/(krM)3 is the Fourier transform of W (|~x − ~r|).

Therefore, σM represents the average fractional variation from 〈M〉 inside a sphere of radius

rM . Note that this means that for a given value of M , the radius of a sphere which, on

average, contains that amount of mass is given by rM = 3√

(3M)/(4πΩMρc0). The value of

σM is normalized using the parameter σ8, which is the σM value for the amount of mass

contained in a sphere with a radius of 8h−1 kpc at the current mean density of the universe.

Studying a plot of σM , like the one shown in Figure 2a, can lead to a more intuitive

understanding of what it represents. For small values of M , σM is large, while it is small for

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Figure 2: (2a) The observed shape of σM . (2b) The effect of massive neutrinos on σM .

large mass scales. This corresponds to the idea that the universe looks lumpy on small scales

and smooth on large scales. Because it is calculated from P (k), the shape and amplitude

of σM also reflect the effects of cosmology on structure formation. For example, because

increasing the value of Ων suppresses perturbation growth on small scales with respect to

large ones, this causes the plotted shape of σM to swivel around its normalization value. This

effect can be seen in Figure 2b, which is zoomed in to show a mass range which is relevant

to galaxy clusters.

In summary, P (k) describes how the amplitude of matter density perturbations depend

on length scale and σM denotes the variance in mass in a sphere of a given size, with that

size specified by the mass M which would be contained in the sphere if the universe were of

a uniform density. Both of these functions depend on the composition of the universe and

can be calculated theoretically for an arbitrary set of cosmological parameters.

From mass density variations, we can proceed towards galaxy clusters by assuming that

the perturbation growth can still be described by the rates derived from a linear (first order)

analysis, even after their amplitudes become non-linear. Given a theoretically predicted

P (k), this can be used to project the evolution of σM over time so it can be written with

the dependency σM(M, z), where z is redshift. A model of cluster formation containing

11

calculations of the probability for a perturbation of mass M to undergo gravitational collapse

can then translate σM into a cluster mass function nM(M, z). This function describes the

number of galaxy clusters with mass M or greater within a co-moving volume element at a

given redshift.

Once a predicted cluster mass function is produced, we can compare it to the results

from galaxy cluster surveys and judge the accuracy of the cosmological parameter values

we used to calculate P (k). Many such comparisons can allow us to make a mapping of the

parameters’ likelihood functions, which show the probability that value for a parameter is

the true value, given a set of observations. We can then use these functions to constrain

various parameters based on survey observations.

3 Methods

Above I described how galaxy cluster surveys can be used to constrain the values of cosmo-

logical parameters. There is a caveat to this: there are a number of cosmological parame-

ters which affect P (k) and therefore the predicted mass and redshift distribution of galaxy

clusters. Some of these are degenerate or have very similar effects. In essence, the more

parameters we allow to vary, the looser the constraints we can place on any single parameter

become. [6] In order to accurately predict the constraints that survey data can place on Ων , it

is vital that we take into account the complex relationships between cosmological parameters

and observables into. To do so, I used the software packages Cosmosurvey and Cosmofisher,

which allowed me to assess the dependencies of survey results on cosmology and to use that

information to make likelihood plots for hypothetical sets of data. This section will detail

that process.

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Figure 3: Simulated cluster survey results produced by Cosmosurvey.

3.1 Cosmosurvey

Cosmosurvey, developed by D. Ventimiglia, consists of a library of Fortran 90 programs

which model structure formation and galaxy cluster surveys. [16] Given a set of parameters

describing cosmology, a hypothetical cluster survey, and the relation between the mass and

x-ray luminosity of galaxy clusters, it produces a set of expected cluster counts binned in

x-ray luminosity and redshift. These results can be plotted in a histogram like the one shown

in Figure 3. It accomplishes this by using information about how σM depends on cosmology

to predict its value for the input set of cosmological parameters. Then, as was described

in section 2.4, it models the formation of clusters from density perturbations to produce a

cluster mass function nM(M, z). A cluster model of the form L = L0(M)β is used to relate

the mass of clusters M to their x-ray luminosity L, with the scatter of the relation defined as

σλ. The parameters of the survey are then used to predict how many clusters per luminosity

and redshift bin will be detected.

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3.1.1 Adding Ων as an input parameter

When I began working with it, Cosmosurvey used a fitting formula to produce σM values as

input for its cluster models. It was equipped to handle a number of cosmological parameters,

with the notable exception of Ων . It was therefore the first step of my project to add Ων as

an input parameter to Cosmosurvey.

To do this, I provided a look-up table partial derivatives of σM with respect to relevant

cosmological parameters, which are listed in Table 1. Cosmosurvey was then modified by

M. Voit so that it would retrieve values from the look-up table and use them to make a first

order approximation of σM for small variations of the parameters about their fiducial values.

The production of the σM look-up table accrued in several stages. First, I used the

software package CMBFast to model structure formation and calculate the matter power

spectrum for a fiducial set of cosmological parameters. [7] For all the parameters other than

Ων , I chose my fiducial values to match those given by [8]. They were determined from the

combined constraints of CMB measurements from WMAP five-year data, baryon acoustic

oscillations, and supernova surveys. The fiducial value of Ων was chosen to be 0.005 because

this value lies in the middle of the range allowed by current constraints, which declare that

0 < Ων < ∼ 0.01. [11]. I also calculated two additional power spectra for each parameter,

varying it by ±3% while holding the others at their fiducial values. In these variations,

ΩΛ was set equal to 1 − ΩM in order to ensure that the universe being considered was

geometrically flat. Likewise, to keep the total amount of matter consistant, ΩCDM was set

equal to ΩM − Ωb − Ων .

The next step was to convert the calculated power spectra into σM values. To do this,

I wrote a numerical integration routine which made use of Equation 4 to convert power

spectra output from CMBFast into σM functions. I used this routine to calculate σM values

in the mass range 1012M < M < 1017M for each power spectrum. Partial derivatives of

σM with respect to each parameter were calculated about the fiducial values using the first

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Parameter Fiducial value DescriptionΩν 0.005 Neutrino density contributionΩM 0.279 Matter density contributionΩb 0.046 Baryon density contribution

ΩCDM 0.228 CDM density contribution: ΩM − Ωb − Ων

ΩΛ 0.712 Dark energy density contribution: 1− ΩM

w -1.0 From dark energy equation of statens 1.0 Primordial power spectrum indexσ8 0.817 Normalization for σMh 0.701 Hubble parameter (H0/100)

Table 1: Fiducial values used in CMBFast calculations to produce a look-up table of σMpartial derivatives.

order approximation:

∂σM∂θi

=σ+M − σ

−M

θ+i − θ−i

(5)

Here, θi is the parameter being considered, θ±i is the value of that parameter varied by ±3%

from its fiducial value, and σ±M is the value of σM calculated using θ±i . When the values of

these derivatives were stored in a look-up table and interfaced with Cosmosurvey, it could

then use them to extrapolate σM for arbitrary sets of cosmological parameters. After this

modification, any of the parameters listed in Table 1 could be used as input to Cosmosurvey.

3.2 Cosmofisher

Once the effect of changing Ων could be predicted by Cosmosurvey, I turned to the software

package Cosmofisher to evaluate how tightly data from future galaxy cluster surveys will

be able to constrain its value. Cosmofisher, also written by D. Ventimiglia, is a Python

code designed “to help create, manipulate, visualize, and gain value from Fisher information

matrices.” [15] This section will expand upon this statement by giving an overview of the

Fisher matrix technique for understanding constraints from data as well as the functionality

of Cosmofisher. Then it will describe how I used Cosmofisher to predict constraints on Ων .

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3.2.1 Fisher matrix technique

Fisher matrices encode information about Gaussian uncertainties for sets of many variables,

and for this reason they are often used in the analysis of cosmological constraints. [4] Their

usefulness derives from the fact that they contain important information about a dataset’s

likelihood function about its maximum likelihood point in N -dimensional parameter space,

yet are able to do so by using only N(N − 1) numbers, as will be described below.

A quantitative evaluation of the quality of a fit to data can be described using its χ2

value, where for the fitting parameter θ, a set of N observables m, their variances σ, and

their values predicted by the fit n(θ):

χ2(m, θ) =N∑i=1

[mi − ni(θ)]2

σ2i

(6)

The larger χ2 is, the more deviations there are between the observables of m and their fitted

values n(θ). It therefore measures the “badness of the fit”, with a smaller χ2 representing a

better fit. Likelihood, which describes the probability of a fit parameter’s value being correct

given a set of observables is defined for observables with Gaussian uncertainties as:

L(θ|m) = exp

[−χ

2(m, θ)

2

](7)

In a general case, where multiple parameters θ are considered and the uncertainties of ob-

servables are not necessarily Gaussian, L(θ|m) ≡ −lnL(θ|m) provides an analog for χ2, with

its minimum occurring at the parameter value θML which gives the maximum likelihood. [17]

A dataset’s Fisher matrix approximates L(θ|m) to be Gaussian about θML, and describes

its curvature at that point. As a more steeply curved likelihood function will result in

tighter constraints, this encodes the amount of information the dataset provides about each

16

parameter θi. [17] The components of the Fisher matrix are:

Fij(θ) =

⟨∂L∂θi

∂L∂θj

⟩(8)

The inverse of a Fisher matrix is called the dataset’s covariance matrix, with F−1ij = Cij =

ρijσiσj. Here, σi and σj characterize the Gaussian uncertainties of θi and θj about their

maximum likelihood values and ρij is a correlation coefficient ranging from 0 to 1. If the

effects of θi and θj on observables are independent, ρij = 0, while ρij = 1 implies that they

are completely correlated. [4]

Likelihood plots are two-dimensional mappings of lines of constant χ2 and can therefore

represent constraint information for only two parameters at a time. If we would like to

produce likelihood plots of two parameters out of many for a multi-dimensional parameter

space, assuming we do no know the values of the other parameters, we must integrate over

their probabilities. This process is known as marginalization, and is facilitated by the form of

the Fisher matrix. To marginalize over a variable θi, we must simply remove the ith row and

column from the covariance matrix C, then invert what remains to get a new Fisher matrix.

Once we have marginalized over all the undesired parameters, the remaining two-dimensional

Fisher matrix can be used to plot confidence ellipses. [4]

The logarithmic nature of the Fisher matrix components makes it easy to combine con-

straints from multiple sets of data; we can simply add the two Fisher matrices. The utility,

combined with the ability of Fisher matrices to store and and create visual representations

of likelihood information make them particularly useful for describing cosmology, which is

described by many interrelated parameters.

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3.2.2 Functionality and use of Cosmofisher

Cosmofisher has two independent functions: It can construct Fisher matrices and produce

likelihood plots for input Fisher matrices. The plotting routine encapsulates the marginal-

ization and confidence ellipse creation described in the previous section. The way in which

it constructs Fisher matrices merits a more in-depth discussion.

For my project, I needed to create Fisher matrices for the predicted results of future

galaxy cluster surveys. Cosmofisher is able to do this by accessing Cosmosurvey. Rather

than comparing fit predictions to actual data, Cosmofisher treats the parameters’ fiducial

values as their true values, and assesses the fit of an arbitrary set of parameters by comparing

its predicted cluster counts per redshift and x-ray luminosity bin against those of the fiducial

set. It takes as input a definition file containing a Cosmosurvey command which gives survey

parameters and desired fiducial values for cosmological and cluster model parameters. This

input file also indicates which parameters should be included in the Fisher matrix. (If a

parameter is not included in the Fisher matrix, its uncertainty is considered to be zero and

it is held fixed at the fiducial value.) When run, Cosmofisher uses the Jacobian function from

the numerical python package, Numdifftools [13], to call Cosmosurvey many times, varying

parameters slightly each time and collecting information on how the fit depends on those

variations. Because a Fisher matrix is essential the Jacobian of L with respect to the varying

parameters (see Equation 8), the end-product of this routine is a Fisher matrix, which can

then be used for other purposes, such as the creation of likelihood plots.

Table 2 shows the values I used as input to produce a Fisher matrix with Cosmofisher.∗

Once this was completed, I was able to generate likelihood plots to study the constraints of

Ων and to look for any degeneracies it might have with other parameters. I also produced

∗The fiducial value for Ων is set to 0.0049 because of a glitch in Cosmofisher, which produces a singularmatrix if the same Ων is used for both the creation of the σM look-up table and the Fisher matrix. Theglitch does not appear to affect the accuracy of Fisher matrices made for other values of Ων and is currentlyunder investigation.

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Parameter Param. Type Fid. value Description? Ων cosmological 0.0049 Neutrino density contribution? ΩM cosmological 0.279 Matter density contribution

Ωb cosmological 0.046 Baryon density contribution? w cosmological -1.0 From dark energy equation of state

ns cosmological 1.0 Primordial power spectrum index? σ8 cosmological 0.817 Normalization for σM

z1 survey 0.1 Minimum redshiftz2 survey 2.5 Maximum redshiftnz survey 10 Number of redshift binsl1 survey 44.0 log10 of minimum luminosity in ergs/sl2 survey 46.0 log10 of maximum luminosity in ergs/snl survey 10 Number of luminosity binsdΩ survey 12.0 Survey solid angle in steradiansfl survey 1.24 ×10−13 Survey flux limit in ergs s−1cm−2

? L0 cluster model 0.0 Mass-luminosity normalization? β cluster model 1.8 Mass-luminosity power law index? σλ cluster model 0.1 Scatter in M - L relation at z=0

ψ cluster model 0.0 Power law index for evolution of σλγ cluster model 0.55 Linder growth index

Table 2: Values used to create original Fisher matrix. Parameters marked with a ? are thoseincluded in the Fisher matrix.

additional Fisher matrices to investigate how these constraints depended on the cluster model

parameters and how the survey results are binned. This will be discussed in more detail in

the following section.

4 Results

Having produced a Fisher matrix using the parameters listed in Table 2, I generated likeli-

hood plots of Ων with respect tho the cosmological parameters ΩM , σ8, and w. These can

be seen in Figure 4. It is important to keep in mind that the values at which the confidence

ellipses are centered are not significant, as I specified them as input fiducial parameters.

What we are concerned with is the size of the ellipses, as these give the predicted constraints

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on the plotted parameters.

Also, because we know that Ων > 0, the parts of these likelihood plots which correspond

to negative Ων are unphysical. The reason that the confidence ellipses extend into this

region is that they are produced by extrapolating knowledge of how the quality of fit varies

around Ων ’s fiducial value (its maximum likelihood point). Because the fiducial value is very

small, the uncertainties for Ων are comparable to it and may cause the ellipse to contain

non-physical, negative values. A more sophisticated technique of estimating constraints

could specify the prior knowledge that Ων must be positive and could use that to tighten

constraints. For the purpose of this project, however, it is sufficient to ignore the negative

values and to focus on the upper constraint on Ων

Examination of Figure 4 reveals that the three likelihood plots all give similar constraints

for Ων , with the 1σ confidence level at about 0.01, and the 2σ confidence at about 0.014.

This means Ων is constrained to within 0.005 and 0.009 ≈ 0.01 of its fiducial value with

68.3% and 95.4% confidence respectively.. Though there is a small amount of tilt, the axis

of the confidence ellipses in all of these plots are very nearly aligned with the horizontal

and vertical axis. This indicates that any degeneracy that Ων may have with ΩM , σ8, or w

is small compared to their uncertainties. Because of this, it is unlikely that restricting the

horizontal expanse of an ellipse using the constraints on any one of these parameters from

another dataset will significantly tighten our estimate of Ων

4.1 Checking for optimal binning

I also investigated the effects of changing the number of redshift and luminosity bins into

which the simulated cluster data was sorted. Finer binning allows for a more detailed shape

of the cluster mass function to be measured, but increasing the number of bins will decrease

then number of clusters in each of them. An optimal binning scheme should balance the

trade-off between precision of fit and the need for robust statistics. By running Cosmofisher

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Figure 4: Likelihood plots for Ων vs. (4a) ΩM , (4b) σ8, and (4c) w. The dark and lightellipses correspond to the 68.3% (1σ) and 95.4% (2σ) confidence levels respectively.

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with different values of nz and nl (see Table 2), I created Fisher matrices for surveys with

varying numbers of redshift and luminosity bins and compared the resulting likelihood plots.

These plots can be found in Figure 5.

From these plots, we can observe that reducing the number of redshift bins to five rather

than ten loosens the constraints on Ων , as does increasing it to twenty. This indicates that

the optimal number of redshift bins for constraining Ων is about ten, though this number it

gives slightly looser constraints on ΩM , σ8, and w than when nz = 20.

The dependence of constraints on the number of luminosity bins is slightly more compli-

cated. Using twenty bins instead of ten improves the constraints for all of the parameters

investigated. The ellipses for thirty bins have nearly the same range in Ων as those for

twenty bins, but become significantly more narrow for ΩM , σ8, and w. Dividing the clusters

into forty luminosity bins then slightly increases the uncertainty for these variables while

tightening the constraints on Ων . Therefore, of the values tested, nl = 40 gives the best

constraints on Ων . It would be an intersting exercise in the future to look at the dependence

of these constraints in more details, calculating Fisher matrices for intermediate numbers of

luminosity bins to try and pinpoint the optimal binning.

4.2 Effects of the cluster model

As was mentioned in section 3.1, Cosmosurvey relates the x-ray luminosity of galaxy clusters

to their mass using a relation L = L0(M)β with a scatter of σλ. Though I did not create

likelihood plots showing the cluster model parameters L0, β, or σλ, there exist uncertainty

in their values, so they were included in the Fisher matrices shown above. In the last stage

of my project, I investigated the effects of those uncertainties. To do so, I generated a Fisher

matrix with the fiducial values shown in Table 2, but specifed in the input to Cosmofisher

that the values of the cluster model paramters should be held fixed. Figure 6 shows the

resulting likelihood plots, overlaid onto those from Figure 4 for comparison. Fixing the

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Figure 5: Comparison of likelihood plots for Ων for different numbers of redshift (5a, 5c, 5e)and luminosity bins (5b, 5d, 5f).

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Figure 6: Comparison of likelihood plots for Ων with and without allowing for uncertaintiesin the cluster model.

cluster model parameters (that is, saying that we know exactly how the mass and luminosity

of galaxy clusters are related) drastically tightens the constraints on ΩM , σ8, and w, but it

appears to very weakly affect the constraints on Ων , if at all.

5 Conclusion

The constraints on neutrino mass from galaxy cluster surveys are expected to improve in the

near future as the result of a dramatic increase in available data. In order to quantify these

expectations, I have used the software packages Cosmosurvey and Cosmofisher to predict

the constraints on Ων for simulated cluster surveys. By doing this, I was able to explore

how these constraints depended on the number of redshift and luminosity bins the survey

results were sorted into. From my results, I estimate that using ten redshift bins and forty

luminosity bins will give optimal constraints on Ων . I also found that uncertainties in the

cluster model relating galaxy cluster mass to x-ray luminosity do not strongly affect our

ability to constrain Ων .

My results predict that galaxy cluster surveys like the ones simulated will be able con-

strain Ων to within about 0.01 of its actual value at a 95.4% confidence level. For comparison,

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analysis like the one in [11] are able to combine information from CMB, supernovae, baryonic

acoustic oscillation, and current galaxy cluster measurements to place a constraint on the

summed neutrino masses of∑

imi < 0.48. Using Equation 2, this translates to the relation

Ων < 0.01. It appears that even with their drastic increase in statistics, future cluster sur-

veys alone will not be able to improve upon our current bounds on neutrino mass. However,

combining future survey data with constraints from other cosmological datasets, such as

measurements of the CMB and large scale structures, is expected to tighten the constraints

on Ων . Though they are not definitive, the results of this project are promising for the

prospect of using future galaxy cluster surveys to refine our determination of neutrino mass.

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