Undergraduate thesis for B.S. in astrophysics, obtained May 2010 from Michigan State University. Advisor: Mark Voit

1. Predicting neutrino mass constraints from
galaxy cluster surveys
Undergraduate Thesis in Astrophysics 2010
Jessica Muir
Michigan State Univeristy
Adviser: Mark Voit
Abstract
Galaxy cluster surveys allow us to study the spectrum of density perturbations
in the local universe, which can in turn provide a way of constraining cosmological
parameters. The advent of large sky surveys in the next decade will dramatically
improve the precision with which these measurements can be made, and in order to
take full advantage of the results it is crucial that we understand how the results depend
on cosmology in a precise manner. It is also necessary that we take into account even
those 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 packages
Cosmosurvey and Cosmofisher to predict how well data from future surveys will be
able to constrain Ων . This paper will describe how Cosmosurvey predicts the results
of cluster surveys for sets of cosmological parameters and how Cosmofisher uses Fisher
matrix techniques to describe the relations between uncertainties of large numbers of
parameters. My results indicate that data from future galaxy clusters will be able to
constrain Ων to within 0.01 of its actual value at a 95.4% confidence level. They also
allow us to select an optimal binning scheme for survey data and to assess the effect of
uncertainties in the mass-luminosity relation for galaxy clusters on Ων constraints.

2. Contents
1 Introduction 1
2 Understanding and describing structure formation 4
2.1 The matter power spectrum P (k) . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 The evolution of P(k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 The effect of neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Relating P(k) to galaxy clusters . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Methods 12
3.1 Cosmosurvey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Cosmofisher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4 Results 19
4.1 Checking for optimal binning . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2 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

3. 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−1 Mpc (where h is H0 /100kms−1 M pc−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

4. 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

5. 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
ρM (x) − ρM
δ(x) =
ρM
4

6. 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)ek·x d3 x
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) ∝ k nS , 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

7. 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

8. 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
7

9. 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) ∝ k ns . 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

10. trinos can be related to the neutrino mass density with the following equation, where mi is
the mass for the neutrino mass eigenstate νi :
ρν i mi
Ων = = (2)
ρc 93.14h2 eV
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

11. 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)
= δ(x)W (|x − r|)d3 x (3)
M
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 )( 4 πrM ) is the difference
3
3
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 :
2
δM 1
σM ≡ σ 2 (M ) ≡
2
= P (k)|Wk |2 d3 k (4)
M (2π)3
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
3
average, contains that amount of mass is given by rM = (3M )/(4πΩM ρc0 ). 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
10

12. 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

13. 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.
12

14. 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.
13

15. 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 1012 M < M < 1017 M for each power spectrum. Partial derivatives of
σM with respect to each parameter were calculated about the fiducial values using the first
14

16. Parameter Fiducial value Description
Ων 0.005 Neutrino density contribution
ΩM 0.279 Matter density contribution
Ωb 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 state
ns 1.0 Primordial power spectrum index
σ8 0.817 Normalization for σM
h 0.701 Hubble parameter (H0 /100)
Table 1: Fiducial values used in CMBFast calculations to produce a look-up table of σM
partial derivatives.
order approximation:
−
∂σM σ + − σM
= M+ − (5)
∂θi θi − θi
±
Here, θi is the parameter being considered, θi is the value of that parameter varied by ±3%
± ±
from its fiducial value, and σM is the value of σM calculated using θi . 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 .
15

17. 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):
N
2 [mi − ni (θ)]2
χ (m, θ) = 2
(6)
i=1
σi
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:
χ2 (m, θ)
L(θ|m) = exp − (7)
2
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 θM L which gives the maximum likelihood. [17]
A dataset’s Fisher matrix approximates L(θ|m) to be Gaussian about θM L , 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

18. parameter θi . [17] The components of the Fisher matrix are:
∂L ∂L
Fij (θ) = (8)
∂θi ∂θj
−1
The inverse of a Fisher matrix is called the dataset’s covariance matrix, with Fij = Cij =
ρij σi σj . 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.
17

19. 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 singular
matrix if the same Ων is used for both the creation of the σM look-up table and the Fisher matrix. The
glitch does not appear to affect the accuracy of Fisher matrices made for other values of Ων and is currently
under investigation.
18

20. 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 redshift
z2 survey 2.5 Maximum redshift
nz survey 10 Number of redshift bins
l1 survey 44.0 log10 of minimum luminosity in ergs/s
l2 survey 46.0 log10 of maximum luminosity in ergs/s
nl survey 10 Number of luminosity bins
dΩ survey 12.0 Survey solid angle in steradians
fl survey 1.24 ×10−13 Survey flux limit in ergs s−1 cm−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 σλ
γ cluster model 0.55 Linder growth index
Table 2: Values used to create original Fisher matrix. Parameters marked with a are those
included 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
19

21. 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
20

22. Figure 4: Likelihood plots for Ων vs. (4a) ΩM , (4b) σ8 , and (4c) w. The dark and light
ellipses correspond to the 68.3% (1σ) and 95.4% (2σ) confidence levels respectively.
21

23. 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
22

24. Figure 5: Comparison of likelihood plots for Ων for different numbers of redshift (5a, 5c, 5e)
and luminosity bins (5b, 5d, 5f).
23

25. Figure 6: Comparison of likelihood plots for Ων with and without allowing for uncertainties
in 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,
24

26. 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 i mi < 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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