Precision Measurement of Angular Separation in Cosmology

Precision Measurement ofAngular Separation in Cosmology

Master thesis

attainment of the academic degree Master of Science

Faculty of Physics

University of Bielefeld

submitted by

Tatiana Esau

Supervisor and rst assessor: Prof. Dr. D. J. Schwarz

Second assessor: Dr. D. Boriero

Bielefeld, April 10, 2014

Contents

0.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1 Introduction 51.1 Brief overview of cosmology . . . . . . . . . . . . . . . . . . . 51.2 Brief overview of astronomy . . . . . . . . . . . . . . . . . . . 6

1.2.1 Radio astronomy . . . . . . . . . . . . . . . . . . . . . 81.2.2 Active Galactic Nuclei . . . . . . . . . . . . . . . . . . 9

1.3 Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.1 Very Large Array . . . . . . . . . . . . . . . . . . . . . 161.3.2 Very Long Baseline Array . . . . . . . . . . . . . . . . 161.3.3 RadioAstron . . . . . . . . . . . . . . . . . . . . . . . . 171.3.4 JIVE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Cosmography 192.1 Redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Trigonometric parallax . . . . . . . . . . . . . . . . . . . . . . 202.3 Luminosity distance . . . . . . . . . . . . . . . . . . . . . . . . 212.4 Cosmic parallax . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 The LTB and FLRW models 253.1 Basics of general relativity . . . . . . . . . . . . . . . . . . . . 253.2 A short introduction to LTB models . . . . . . . . . . . . . . 263.3 A short introduction to FLRW models . . . . . . . . . . . . . 27

4 Cosmic parallax in the LTB metric 314.1 The derivation of the cosmic parallax in the LTB metric . . . 31

5 Cosmic parallax in the FLRW metric 375.1 Cosmic parallax in at FLRW metric for comoving sources . . 37

1

2 CONTENTS

5.2 Cosmic parallax in the ΛCDM model forgravitationally bound sources . . . . . . . . . . . . . . . . . . 41

5.3 Estimation of the secular parallax . . . . . . . . . . . . . . . . 48

6 Estimation of cosmological parameters 536.1 Likelihood estimation method . . . . . . . . . . . . . . . . . . 536.2 Fisher Information Matrix . . . . . . . . . . . . . . . . . . . . 566.3 Fisher matrix analysis of angular sizes

in the ΛCDM model . . . . . . . . . . . . . . . . . . . . . . . 57

7 Conclusion 65

8 Appendix 67

0.1. ABSTRACT 3

0.1 Abstract

This thesis is handed in to obtain the degree "Master of Science" in physicswith the emphasis laid on cosmology. The goal of this work is to investigatethe application of the precise position measurement of astrophysical objectson the sky in cosmology. The thesis is structured as follows: chapters 1 to 3briey summarize theoretical foundations used in this work, chapters 4 to 6illustrate the main ideas of this master thesis and the nal chapter sums upthe obtained results.

4 CONTENTS

Chapter 1

Introduction

1.1 Brief overview of cosmology

The word "cosmology" is derived from Greek "cosmologia", and signies theteaching of the world. This science studies the universe as a whole. Thegoal of cosmology is the mathematical description of the origin, developmentand the basic underlying structures of the universe. For thousands of years,humans had the desire to know more about the cosmos. Almost since thebeginning of mankind, every religion and every society have been taking andstill take the cosmology as the foundation of their doctrine. Only in the20th century cosmology was recognized as a part of exact sciences, thanksto the development of Einstein's theory of relativity. In cosmology, dierentsciences as astronomy, astrophysics, mathematics, nuclear and elementaryparticle physics work together. Currently, cosmology is the most rapidly de-veloping science. Even though a little more than one century ago, cosmologydid not exist as a scientic discipline, nowadays, the Big Bang Theory isa generally accepted theory of the description of the universe. Two basicassumptions underlie the Big Bang Theory: General Relativity and the cos-mological principle. The theory of relativity states that the laws of natureare universal which allows to describe the universe with the same laws thatare valid close to the earth. The cosmological principle states that the uni-verse is spatially homogeneous and isotropic on the large scale. By usingthe statement of the cosmological principle and General Relativity one ob-tains the Friedmann equations which describe the evolution of the spatiallyhomogeneous and isotropic universe. To solve these equations, one starts

5

6 CHAPTER 1. INTRODUCTION

from the present state of the universe. Subsequently, one can go backwardsin time to obtain a statement about the early universe. The solutions de-pend in particular on the measured values of the Hubble constant and thedensity parameters which describe the mass and the energy of the universe.With the help of astronomical observations, the cosmological parameters canbe determined. There are three strong evidences in favor of the Big BangTheory:

* Abundances of light chemical elements in the universe. After aboutthe rst three minutes of the existence of the universe, the formationof the four light elements 2H, 3He, 4He and 7Li, was completed. Thetheoretical values are in excellent agreement with the measured ones,apart from 7Li.

* Cosmic Background Radiation (CMB). The cosmic microwave back-ground radiation is a relic radiation from earlier stages of the hot BigBang. CMB originates from the time about 300,000 years after theBig Bang, when radiation was decoupled from matter by the fallingtemperatures and the universe became transparent. In 1946, CMB waspredicted theoretically and its temperature was estimated to be about5K. After about twenty years, the CMB was measured by a coincidence.The measured temperature of the relic radiation amounts to 2.725 K.

* Expansion of the universe. In 1929 E.Hubble observed nearby galaxiesand noticed that their spectra redshifted. He showed that the redshiftlinearly increases with the distance and plotted the distance of thegalaxies against their speed. Thus he determined the Hubble constantH0. However, he did not connect these with the expansion of theuniverse. The expansion of the universe was discovered by Lemaîtrewho explained the redshift of the galaxies by the Doppler eect.

Thanks to projects like COBE, WMAP and Planck, the cosmological pa-rameters, including the Hubble constant, could be measured very precisely[1, 2, 3, 4].

1.2 Brief overview of astronomy

Astronomy is one of the oldest sciences which studies the dierent objects inthe universe using scientic methods. Although it is a very old science, it is

1.2. BRIEF OVERVIEW OF ASTRONOMY 7

ourishing as never before. Due to the fast development of technologies inthe last century, new disciplines with a fairly arbitrary division were createdas a part of astronomy. The main instrument of astronomy is the studyof electromagnetic radiation. Most of the direct information coming fromthe cosmos reaches us in this form. The astronomy experienced a majorstep forward in the Renaissance era which comes along with the discoveryof the rst telescope. However, despite this promising discovery, people wererestricted to the visible region of the electromagnetic spectrum and had noway to do research oside our planet. Over several thousands of years, ourknowledge about the universe was based on the observations in the visiblerange of light. The Earth's atmosphere is transparent only in two wavelengthranges: the optical window is in the range from 290 nm to 1 µm and the radiowindow is in the range from about 5 cm to 20 m, see gure 1.1.

Figure 1.1: Permeability of the Earth's atmosphere. The gure is taken fromwww.ipac.caltech.edu (NASA)


The discovery of the radio window has brought out a new eld of theastronomy, the radio astronomy. Since the current research is no longerjust ground-based, the entire spectrum of electromagnetic waves is available.The continuous development in the past several years of large ground-basedtelescopes and space telescopes has changed the understanding of the universefundamentally. Thanks to the discovery and application of the maser andthe supercooled detectors, it has become possible to increase the accuracy ofradio telescopes [5, 6, 7].

1.2.1 Radio astronomy

In 1888 Heinrich Hertz discovered the electromagnetic radiation at radiofrequences which led to the observation of radio signals from the sky by CarlJansky in 1931. This discovery opened up a new eld of science the radioastronomy. Over several years, it turned out that this eld is very promising.The measurements in the radio wave range are particularly suitable for thecreation of high-resolution maps of distant galaxies. Many of the most distantgalaxies have been discovered in this way. The rst radio telescope was builtduring the second world war by an amateur astronomer Grote Reber. In1946 his observations were conrmed by an astronomer team from England.Unfortunately, with their telescope it was not possible for them to determinethe position of the observed objects exactly. Since the wavelength of theradio waves is much longer than the wavelength of the optical light, oneneeds to build much bigger antennas to get a sharp image. This problemcould be solved by having several small telescopes connected together. Suchan equipment was called a radio interferometer. The rst interferometer wasbuilt in 1954 by British and Australian astronomers. At the end the 50's andthe beginning of the 60's, astronomers made a catalog of radio sources overthe whole sky. One of the most famous catalog was the Third CambridgeCatalogue, which can be abbreviated to "3C". Many objects in this cataloghave not been regarded as anything unusual. Most of these objects wereconsidered to be normal stars of our galaxy, until A.Sandage uncovered aspecial case in 1960. He discovered and investigated the "star" 3C48. The3C48 was very bright, thus it was believed that the star could not be locatedoutside of our galaxy, while its spectrum was completely unidentied. A littletime later, several objects with the same properties like 3C48 were discoveredand only in 1963 M. Schmidt has been able to explain the spectrum of theradio source 3C273. He noted that all the lines were shifted in the entire


spectrum relatively to their usual position, which could be explained by theDoppler eect. The distance to 3C273 was calculated according to the Hubblelaw and it was found that the object was at a great distance from the MilkyWay. However, astronomers have known that a normal star could not produceenough energy to appear bright to us at such a large distance. Hence, suchsources are not stars, which has also been conrmed in later studies. Suchsources are named quasi-stellar objects or quasars. Quasars were identiedas the most distant objects. Meanwhile, quasars were detected up to redshiftz = 7. They are among the most luminous objects in the universe.

Nowadays, it has been found that a quasar is a nucleus of an activemassive galaxy, which is point-like in the optical images and emits a lot ofenergy in the other parts of spectrum. In the 1960s, using radio interferom-etry, many dierent radio objects were observed. Today they are known asradio galaxies [4, 6, 7].

1.2.2 Active Galactic Nuclei

By means of radio astronomy, it was observed that some galaxies have anactive nucleus. An active galactic nucleus (short AGN) is a region in thecenter of a galaxy that emits strong radiation over the full electromagneticspectrum. The most of active nuclei have a stronger luminosity than theirentire host galaxy. A lot of these objects have been found at large z, whichmay indicate that the activity of the nucleus is a characteristic of the earlylife of the galaxy, and it makes the study of AGNs complicated. They aremost common in galaxies with a large mass. In addition, the galaxy mustbe fused to the center, so the most of AGNs occur in elliptical galaxies. Thehallmark of galaxies with an AGN is a changing luminosity, the period canvary from months to only a few hours. From this, one can conclude that theemission region is not larger than our solar system, which implies that thelikely source of energy for AGNs is a super massive black hole in the center ofthe galaxy. There also almost always are large amounts of gas in the centralregions of the galaxy which form an accretion disk. It is believed that thematter in an accretion disk swirls around before it falls into the black hole.This means that the gravitational potential energy of matter that falls intoa black hole transforms into the kinetic energy. Collisions between particlesconvert their kinetic energy into thermal energy, which is the cause of theintense radiation of AGNs. Thus the reason of the strong radiation of theAGN is the hot gas in the accretion disk, which is placed around the black


hole.The nucleus of such galaxies often emits relativistic jets. The jets consist

of matter radiation and move away from the center of AGNs almost with thespeed of light. Depending on the viewing angle, one can observe one or twojets directed in opposite directions. These jets extend far beyond the areawhere the stars of the galaxy are located. Sometime, the jets come up againstthe intergalactic gas and form so-called hotspots which can be observed verywell.

The AGNs can be classied depending on the presence (or absence) ofbroad lines in the spectrum, depending on the band of the spectra whichcan be observed or on the intensity of the nuclear luminosity and also onthe radio properties of the object. This means that one and the same objectcan be found in two or more classes of AGNs, depending on its physicalcharacteristics. There are following classes of active galaxies:

Seyfert galaxies show broad emission lines from the galactic nucleus.The most host galaxies of Seyfert AGNs are either spiral or irregular galaxies.They were originally subdivided into Seyfert 1 (S1) and Seyfert 2 (S2) classes,where S1 has very broad emission lines and more likely emits low-energy X-rays, while S2 has only moderate lines.

Low luminosity AGN. In the spectrum of radiation of this class it lacksthe "Big Blue Bump" which could be seen in the UV-range in many AGNs.Hence, it is common to think that these objects do not possess the innerregion of the accretion disk which would produce the UV Bump.

Low-ionization nuclear emission-line regions (LINERs). Thisclass shows only weak nuclear emission-line regions. The AGNs of this groupare less luminous than a S2. It is questionable whether this group reallybelongs to AGNs.

Radio-quiet quasars. These can be regarded as more luminous versionsof Seyfert nuclei. The distinction of these classes is arbitrary and is usuallyconnected with a limit of optical magnitude. The quasars can be in dierenthost galaxies such as spirals, irregulars or ellipticals. There is also somecorrelation between the mass of the host galaxy and the luminosity of thequasar. The greater the mass is, the more powerful the quasar is.

Radio-loud quasars have the same behavior as radio-quiet quasars. Thedierence is the emission of a jet. The spectrum shows broad and narrowemission lines and also strong X-rays and radio emission.

Blazars (BL Lac objects and OVV quasars) are quasars which havevery weak emission lines. The luminocity of these AGNs can rapidly vary


during only a few days. Quasars which have the same variablity but strongeremission lines are called optically violently variables (OVVs). Blazars are themost luminous objects in the universe.

Fanaro-Riley Type I (FRI) and Fanaro-Riley Type II (FRII).These two groups were introduced by the astronomers Fanaro and Riley todistinguish between radio galaxies. Fanaro and Riley dened a ratio, RFR,by the distance of the two brightest areas of the surfaces of the galaxy andthe largest radius of the entire source. Thus, the sources are classied asfollows: FR Type I, if RFR less than 0.5; FR Type II, if RFR larger than 0.5.An example of FRI and FRII is shown in gure 1.2.

Figure 1.2: Examle of Fanaro-Riley type I galaxy (FRI) and Fanaro-Rileytype II galaxy (FRII), respectively. The rst gure is a radio map of M84elliptical galaxy and the second one is a radio map of the quasar 3C 175.Both maps are based on data obtained with VLA . The maps are taken fromhttp://ned.ipac.caltech.edu/level5/Cambridge

Later, these astronomers discovered that there is a correlation between thesurface brightness and spectral radio luminosity of the source. According toa frequency-dependent radio luminosity, radio resources belong either to one


or to the other FR class. In 1993 it was noted by Young that for a givenradio luminosity, there exists an optical luminosity limit which divides twotypes of sources into FRI and FRII, see gure 1.3.

Figure 1.3: Radio luminosity is plotted against absolute visual magnitude of thehost galaxy for the FRI sources (1) and FRII sources (2). The gure is taken from[8].

Figure 1.4 is a schematic representation of all described classes above. Inthis gure one can see that the belonging to one or the other class dependson the observing angle.

1.2. INTERFEROMETRY 13

Figure 1.4: Schematic representation of the classication of the AGN. The class ofobject which we see depends on the observing angle. The gure is taken from [26].

The big possibilities for the study of active galactic nuclei open up the dis-covery and further development of radio interferometry [2, 4, 5, 7, 9, 26].

1.3 Interferometry

The spatial resolution of a telescope is proportional to λ/D, where λ is thewavelength of the observed light and D is the diameter of the telescope.Thus it is clear that the resolution of a single radio antenna cannot be very


good. A method to improve the resolution of radio telescopes was developedin the 50's. It has been found that the resolution can be increased by aninterconnection of several telescopes. This method is called interferometry,since it is based on the known physical phenomenon, the interference of light.The further development of technology in the 60s nally made possible torealize an interferometer without a cable connection. The distance betweentelescopes can amount to several thousand kilometers. This method hasbecome known as Very Long Baseline Interferometry. This observationaltechnique is one of the most used principles in radio astronomy.

The easiest case to explain is by using the example with only two tele-scopes, see gure 1.5.

P‘

‘

B

B

θ

Figure 1.5: Schematic representation of the VLBI. The gure is taken fromwww.see.leeds.ac.uk with some small changes.

Since two telescopes cover only a small part of the hypothetical large tele-scope, they provide a very poor image compared to many telescopes. Never-theless, useful information about the angular structure on the ne scale canbe obtained. Consider a radio signal from a point-like object which reachesthe telescopes which are placed at a distance B (B is called the baseline) fromeach other. If the direction of the source of cosmic radiation is perpendicular


to the base of the interferometer the signals from the two antennas yield thesame phase signal. That means that one observes an interference maximum.The signal is amplied and registered. After some time, due to the rotationof the Earth, the source of the light changes its position relatively to thebaseline of the interferometer.

Figure 1.6: Directivity pattern of aradio interferometer. The gure istaken from www.astro.puc.cl.

Then the radio waves from the sourcewill reach the antennas with a dierentphase. The path to radio antenna 1 isgreater on the segment P ′ and the mag-nitude of the resultant signal is reduced.If the segment P ′ is equal to a half ofthe wavelength (or 3λ/2, 5λ/2 etc.), theradio waves are coming in with the oppo-site phase and the signals from antennas1 and 2 completely cancel each other. Sothe power of the received signal will vary,increasing and decreasing, i.e., the radi-ation pattern of the radio interferometeris a series of petals, see gure 1.6.

To estimate the resolution of aninterferometer, we choose the baselineequal to the multiple of the wavelengthat which the measurement is performed.That means

B = nλ. (1.1)

The waves of the observed object arrive at the antennas on dierent paths.So the path dierence P ′ for the constructive interference is given by theformula

P ′ = B sin Θ = mλ, (1.2)

where m ∈ N, Θ is the angle between the direction of incoming rays anda normal to the line on which the antennas are located. If we assume thatm = 1 and we consider small angles Θ we obtain

sinΘ ≈ Θ ≈ λ

nλ=

1

n[rad]. (1.3)

This means the greater the baseline is, the better the resolution is. Therecurrently exists a number of functioning VLBI systems. In the next sectionswe will discuss them briey [6, 7, 9, 10].


1.3.1 Very Large Array

Figure 1.7: VLA. The gure is taken fromhttp://www.aoc.nrao.edu (National Ra-dio Astronomy Observatory).

Two connected telescopes providevery poor images compared to alarger number of them. For this rea-son, a special arrangement of manytelescopes is used. An example ofsuch techniques is the Very LargeArray (VLA) which is a radio as-tronomy observatory and is locatedin the desert of New Mexico, USA.The VLA contains 27 large indepen-dent antennas (each with the diam-eter of about 25 m) extending overabout 40 km. The antennas of theVLA are placed Y-shaped. Due toits large area the VLA has a good sensitivity. The best possible angularresolution which can be reached with the VLA is about Θ ≈ 0.05′′ at awavelength of 7 mm [4, 6, 30].

1.3.2 Very Long Baseline Array

Figure 1.8: VLBA. The gure is takenfrom http://images.nrao.edu (NationalRadio Astronomy Observatory).

VLBA is an interferometer whichconsists of ten telescopes of whicheight are located in the U.S., one inHawaii and one in St. Croix, Vir-gin Islands. Each of the 10 anten-nas has a diameter of 25 m, andthey span a baseline of about 8000km which provides an angular reso-lution Θ . 1 mas. The operatingfrequences of the VLBA are rang-ing from 330 MHz to 43 GHz (from0.9 m to 7 mm). The VLBA iscontrolled by the Science OperationsCenter of VLA in New Mexico. Thesignal is amplied, digitized and saved on a fast recorder for each antenna.These data are sent to the Science Operations Center in New Mexico and


are evaluated there [6, 7, 31].

1.3.3 RadioAstron

Figure 1.9: RadioAstron (RadioastronSpace VLBI Mission). The gure is takenfrom http://www.asc.rssi.ru/radioastron

RadioAstron is a space telescope,whose operating principle is basedon the VLBI method. It consistsof an antenna which is moving onan elliptical orbit with an orbital pe-riod of 9.5 days , and several ground-based telescopes. The maximal dis-tance from the Earth is 350,000 km.

The space telescope was launchedin July 2011 and it is planned thatthe telescope remains active for atleast ve years. The goal of thismission is to study the AGNs, blackholes and neutron stars, and alsoto collect information about the in-terstellar plasma. Due to such agreat baseline, the resolution of Ra-dioAstron is very high, Θ ≈ 7 µas.The operating frequences of the Ra-dioAstron range from 18 GHz to25 GHz (from 92 cm to 1.35 cm)[32, 33].

1.3.4 JIVE

The Joint Institute for VLBI in Europe (JIVE) is located in the Netherlandsand is a member of the EVN (European VLBI Network). The EVN wasformed in 1980 by a consortium of ve radio astronomy institutes in Europeand performs observations of cosmic radio sources with a high angular reso-lution. In 1993 JIVE was established by the consortium for VLBI in Europe.The task of JIVE is to operate the EVN VLBI Data Processor and to de-velop it further. Currently, the JIVE and the consortium include 9 instituteswith 14 radio telescopes in 8 European countries and also institutes withtelescopes in Russia, Ukraine, China, Italy, Poland, South Africa, Australia,


Chile, Finland, Germany, the Netherlands, Puerto Rico, Spain, Sweden andthe UK [36, 37].

Chapter 2

Cosmography

The standard model of cosmology assumes that the universe can be describedby the hot Big Bang. However, the Big Bang Theory does not provide anunique description of the present universe. First, the cosmological parametershave to be determined by the observations and after this, we can choose one orthe other cosmological models. In this chapter, we will introduce importantcosmological quantities and give a short explaination about how the distanceof celestial objects can be measured.

2.1 Redshift

The cosmological observations show that almost everything in the universemoves away from us. The further an object is away, the faster it moves awayfrom us. It was seen through spectral lines of light of distant objects whichwere shifted into the longer wavelength range. This eect is known as theredshift, which is denoted z and is basically the Doppler eect. Thus, theredshift is given by

z =λobs − λem

λem, (2.1)

where λem is the wavelength of the emitted light and λobs is the wavelengthof the observed light [1].

19

20 CHAPTER 2. COSMOGRAPHY

2.2 Trigonometric parallax

A possibility to determine the distance between an observer and a source isthe trigonometric parallax. The term "parallax" is derived from the Greekand can be translated as "alteration". The parallax in astrophysics and incosmology is an apparent change of the placement of a nearby object againstthe background of the much more distant objects when the observer movesfrom one end of the baseline to the other. That means that the objectswhich are closer to the observer appear to move relatively to the backgroundobjects. If the length of the baseline is known, the measurement of theparallax allows us to calculate the angular distance to the object.

Observed object A

Observed object B

Sun

Earth

Apparent track of object B (line)

Apparent track of object A (circle)

Distant background objects

Distant background objects

_

l

Figure 2.1: Graphical representation of the trigonometric parallax that is inducedby the motion of the Earth around the Sun (annual parallax). Figure 2.1 is takenfrom [6] with small changes.

An object is at a distance of one parsec, which is the abbreviation for "par-allax arcsecond", if it has a parallax of one second of arc and the length ofbase is equal to 1 au 1. That means that the angular distance of an object

1au is the abbreviation of Astronomical Unit and 1 au = 149597870700 m

2.2. LUMINOSITY DISTANCE 21

is given by

dA =l/2

tan(θ/2)∼=l

θ, (2.2)

where θ is the parallax 2 and l is the baseline.There are many causes which can induce the trigonometric parallax. We

consider two of these causes, the rst one is the motion of the Earth aroundthe Sun which leads to the annual parallax, and the second is the motionof the solar system around the galactic center, which generates the so-calledsecular parallax. Figure 2.1 shows the graphical representation of the trigono-metric parallax which is caused by the motion of the Earth around the Sun.

For an object that is located at a distance of many Mpc, this method isnot suitable because the parallax is immeasurably small. The usual methodfor determining the distance of such objects is to use the luminosity distance[1, 2, 6].

2.3 Luminosity distance

This method is based on the application of the so-called standard candles.Standard candles are the objects which are assumed to have the same abso-lute luminosity L (also the emitted energy ux per unit area and solid an-gle) everywhere in the universe. The standard candles include, for example,Cepheids and supernovae of type Ia. By comparing the absolute luminosityof standard candles with the measured radiation ux density S per time andper unit area there can be made conclusions to the distance of the source.

The luminosity distance dL is an apparent distance of an object assumingthat the light intensity decreases with the square of the distance, that means

d2L =

L

S. (2.3)

By the expansion of the space the emitted photons lose energy ∝ 1 + z andby the expansion of the universe the time between the arrival of photonsincreases ∝ 1 + z. Thus, the received energy ux is given by

S =L

d2ph(1 + z)2

, (2.4)

2in astronomy θ has only small values and therefore, tan θ ∼= θ


where dph is the physical distance to the observed source. Therefore, theluminosity distance is given by

dL = dph(1 + z). (2.5)

From the formula (2.5) it can be seen that for nearby objects (z 1) it isdL ≈ dph and that more distant objects appear to be further than they reallyare dL > dph [1, 2, 3].

2.4 Cosmic parallax

The next notion, which we will work closely with, is the "cosmic parallax".The cosmic parallax (also CP) is analogous to the classical trigonometricparallax, except that in this case, the parallax is induced by the expansionof the space.

center C

observer

observer

Figure 2.2: Overall scheme describing the possible cosmic parallax of two sourcesa and b in Lemaître-Tolman-Bondi (LTB) space-time, where θ := θb − θa. Figure2.2 is taken from [11].

2.4. COSMIC PARALLAX 23

The cosmic parallax in addition to the redshift drift, are the parts of thereal-time cosmology, which is a new eld of cosmology. This eld was createdin the recent years thanks to improved astrometric and spectroscopic tech-niques. The real-time cosmology means that some cosmological quantitiescan be measured during about one human generation, that means during10− 15 years.

The standard cosmology is built on two main assumptions, the general rel-ativity and the cosmological principle. The General relativity is well proven,so there are no doubts about its truth. The cosmological principle, how-ever, is an assumption which says that the universe is spatially homogeneousand isotropic. The evidence of isotropy is the cosmic microwave background,whereas the homogeneity cannot be veried directly. The study of the cos-mic parallax provides an opportunity to investigate the homogeneity of spaceindirectly.

The redshift drift is the temporal variation of the redshift which is pro-portional to the local expansion rate. To observe this eect, high-precisionspectroscopy is required. For example, in [35], the necessary precision to mea-sure the redshift drift for the sources in the area 2 < z < 5 was estimated.The result indicates that a 42m telescope is able to detect the redshift driftover a period of 20 years, using 4000 h of observing time. The CP is thetemporal change of the angular separation between two sources. The obser-vations of this eect relies on high-precision astrometry. Some examples ofthe high-precision astronomy were considered in the section 1.3.

To consider the general case of the cosmic parallax one uses models whichare based on the Lemaître-Tolman-Bondi (also LTB) metric. LTB modelsdescribe an anisotropic universe for every observer, except for the central one.In each anisotropic expansion the angular separation between two sources isnot constant in time. That means that the cosmic parallax is induced bythe dierential expansion rate. The overall scheme describing the cosmicparallax is shown in gure 2.2. Details for LTB spacetime are presented inthe next chapter [11, 12, 13, 18].

Chapter 3

The LTB and FLRW models

A metric is the basis of general relativity. By describing the distance betweentwo space-time points a metric represents the geometry of the space-time.In this chapter the important foundations of the Lemaître-Tolman-Bondi(LTB) and Friedmann-Lemaître-Robertson-Walker (FLRW) space-time willbe presented.

3.1 Basics of general relativity

The physical phenomenon of gravitation is described by the Einstein's eldequations. The fundamental idea of these equations is that the gravity in-uences the geometry of the space. The Einstein's eld equations can bewritten as

Rµν −1

2gµνR + gµνΛ =

8πG

c4Tµν , (3.1)

where c is the speed of light, G is the gravitational constant, Rµν is theRicci curvature tensor, R is the Ricci curvature scalar, Tµν is the energy-momentum tensor, gµν is the metric and Λ is the cosmological constant. Thedenition of the Ricci scalar R is

R = gµνRµν (3.2)

and of the Ricci curvature tensor Rµν is

Rµν = gτµ(∂ρΓρτν − ∂τΓρρν + ΓθτνΓ

ρρθ − ΓθρνΓ

ρτθ), (3.3)

25

26 CHAPTER 3. THE LTB AND FLRW MODELS

where Γθρν is a Christoel symbol, which is given by

Γρµν =1

2gρσ(∂νgσµ + ∂µgσν − ∂σgµν), (3.4)

and the relation of the energy-momentum tensor for a ideal uid is given by

Tµν = (ρ+ p/c2)uµuν + pδµν , (3.5)

where ρ is the mass density, p is the pressure and uµ is the 4-velocity of theuid [3, 16, 34].

3.2 A short introduction to LTB models

The Lemaître-Tolman-Bondi metric is a spherically symmetric metric, whichis the most general posibility to describe the space-time without the assump-tion of homogeneity, and it can be written as

ds2 = −c2dt2 +[R′(t, r)]2

1 + β(r)dr2 +R2(t, r)dΩ2, (3.6)

where dΩ2 = dθ2 + sin2 θdφ2 and r, θ, φ are the spherical coordinates, β(r) isa position dependent spatial curvature function, R(r, t) is a function whichdepends on the time and the radial coordinates and primes correspond to ∂r.

In this space-time two dierent Hubble parameters exist, the rst one inthe radial and the second one in the transverse direction, which are denedas follows:

H‖ =R′

R′, (3.7)

H⊥ =R

R, (3.8)

where dots correspond to ∂t.By inserting the LTB metric into the Einstein's eld equation one obtains

the equation of motion, which, for a universe with dust and dark energy, isgiven as:

3.2. A SHORT INTRODUCTION TO FLRW MODELS 27

H2⊥ + 2H⊥H‖ −

β(r)c2

R(t, r)2− β′c2

R(t, r)R(t, r)′= 8πGρ+ Λc2, (3.9)

and the acceleration equation which is given as:

6R(t, r)

R(t, r)+2H2

⊥−2β(r)c2

R(t, r)2−2H⊥H‖+

β′c2

R(t, r)R(t, r)′= −8πG(ρ+3p/c2)+2Λ.

(3.10)The LTB universe appeares anisotropic for any observer except the cen-

tral one. If the homogeneity is assumed, LTB models must go into the FLRWmodels which describe an isotropic and homogeneous universe (see the chap-ter 3.3) [11, 12, 15, 16, 17, 22, 34].

3.3 A short introduction to FLRW models

The Friedmann-Lemaître-Robertson-Walker space-time describes a homoge-neous and isotropic universe and is a special case of the LTB space-time.That means that, with the assumption of homogeneity of the space, thescale function R(t, r) is separable and dened for the FLRW universe as

R(t, r) = a(t)r (3.11)

and the curvature function for the FLRW universe is dened as

β(r) = −kr2. (3.12)

Thus, the LTB metric goes into the FLRW metric and looks as

ds2 = −c2dt2 + a(t)2

(dr2

1− kr2+ r2dθ2 + r2 sin θ2dφ2

), (3.13)

where a(t) is the scale factor, k is a curvature constant (k = +1, 0,−1 forthe closed, at and open universe, respectively) and r, θ, φ are the sphericalcoordinates.If we derive the Hubble function using the denition (3.11) and (3.7), (3.8),we obtain that there is no dierence between the radial and the transverseHubble function in the FLRW space-time, i.e. H‖ = H⊥ and

H(t) =a(t)

a(t). (3.14)


By inserting the FLRW metric into the Einstein's eld equation (3.1) oneobtains the Friedmann equation which is given by(

a(t)

a(t)

)2

=8πGρ(t)

3− kc2

a(t)2+

Λc2

3(3.15)

and the acceleration equation is

a(t)

a(t)= −4πG

3(ρ(t) + 3p(t)/c2) +

Λ

3. (3.16)

We introduce the dimensionless density parameters:− of dark energy

ΩΛ =Λ

3H20

, (3.17)

− of matter (baryonic and dark matter)

ΩM =8πGρ

3H20

, (3.18)

− of spatial curvature

Ωk = − k

a20H

20

, (3.19)

− of the radiation

Ωrad = −8πGρrad3H2

0

, (3.20)

where a0 is the scale factor today (it is chosen as a0 = 1) and H0 is theHubble rate today.Thus, the Friedmann equation can be rewritten as

H(z)2 = H20 [ΩM(z + 1)3 + ΩΛ + Ωk(1 + z)2 + Ωrad(1 + z)4]. (3.21)

3.3. A SHORT INTRODUCTION TO FLRW MODELS 29

The Lambda-Cold-Dark-Matter (ΛCDM) model is currently the most dis-cussed model and it is a special case of the FLRW model which has a curva-ture equal to 0 and Λ >0. Since the curvature in the ΛCDM model is equalto zero and we live in matter dominated epoch, the Friedmann equation isgiven by

H(z)2 = H20 [ΩM(z + 1)3 + ΩΛ], (3.22)

where z is the redshift, which can be calculated for the FLRW metric usingthe denition (2.1) and the fact that for light ds2 = 0. So we obtain

z + 1 =a0

a(t). (3.23)

Figure 3.1: CMB map of the Planck mission (with 3% of the sky replaced by aconstrained Gaussian realization). The gure is taken from [29].


The values of the cosmological parameters describing our universe can beobtained, for example, from the Planck mission. The Planck mission pro-vides a good estimation of the parameters because of its high sensitivity, itshigh angular resolution and its wide frequency range. The accuracy of thePlanck mission is much better than of all previous missions. In the table 1there are results for the cosmological parameters obtained by Planck whichwere calculated with dierent assumptions.

ParametersPlanck(CMB+lensing) Plank+WP+highL+BAO

Best t 68% limits Best t 68% limits

Ωbh2 0.022242 0.02217± 0.00033 0.022161 0.02214± 0.00024

Ωch2 0.11805 0.1186± 0.0031 0.11889 0.1187± 0.0017

ΩΛ 0.6964 0.693± 0.019 0.6914 0.692± 0.0010

Σ8 0.8285 0.823± 0.018 0.8288 0.826± 0.0012

H0 68.14 67.9± 1.5 67.77 67.80± 0.77

Age/Gyr 13.784 13.796± 0.058 13.7965 13.798± 0.037

Table 1: Cosmological parameters values for the Planck-only best-t 6 pa-rameter ΛCDM model (Planck temperature data plus lensing) and for thePlanck best-t cosmology including external data sets (Planck temperaturedata, lensing, WMAP polarization [WP] at low multipoles, high-l experi-ments, and BAO, labelled [Plank+WP+highL+BAO] in Planck Collabora-tion XVI (2013)). The table 1 is taken from [29].

In the succeeding sections, all calculations are performed with the fol-lowing values of cosmological parameters: ΩM=0.308, ΩΛ=0.692, H0=67.8[1, 3, 15, 16, 29].

Chapter 4

Cosmic parallax in the LTB

metric

One of the possibilities to apply the precise measurement of angles whichwe will consider is the investigation of the cosmic parallax. After comparingthe theoretical predictions of the cosmic parallax in the LTB model with thepractical measurements, one can look for alternative to the ΛCDM cosmolog-ical models that work well with all existing observations but do not requirethe homogeneity of space. Therefore, the cosmic parallax in the LTB spaceis a good method to investigate the structure of space. Using the CP onecan investigate the large-scale deviations from homogeneity. However, in thischapter we will only look at the general mathematical basis for the derivationof the cosmic parallax in LTB and we will not perform the estimation of themagnitudes of the possible parallax.

4.1 The derivation of the cosmic parallax in the

LTB metric

The cosmic parallax ∆tγ is dened as the temporal change of the angularseparation between two sources. That means

∆tγ ≡ γ2 − γ1. (4.1)

In the following we consider an o-center observer in a coordinate systemwith the origin in the center of the LTB metric, which describes an isotropic

31

32 CHAPTER 4. COSMIC PARALLAX IN THE LTB METRIC

but inhomogeneous universe with respect to the center C. In gure 2.2 wecan see the overall scheme describing the possible variation of the angularseparation of two sources a and b. Space expands isotropically with respectto the center C and anisotropically with respect to the o-center observer O.Here r, θ, φ are the comoving spherical coordinates. Because of the symmetryof the chosen spherically symmetric model, objects move radially outwardswith respect to the center, keeping r, θ, φ constant. First of all, we considerthe expansion with respect to the center in the at FLRW universe which weobserve by the o-center observer at the distance Xobs from the center. Thecomoving coordinates r and robs correspond to the physical distances X andXobs, respectively. Now we can set up the relation between the observer'sline-of-sight angle ξ, the coordinates of the sources X, the coordinate of theo-center observer Xobs and the angle θ. It is

cos ξ =X cos θ −Xobs√

X2 +X2obs − 2XobsX cos θ

. (4.2)

For simplicity, we choose such sources that have the same φ coordinate. First,we consider a pair of sources at locations a1, b1 which are in the same planeas the axis CO and are separated by the angle γ1. After some time ∆t duethe expansion of space, both sources move to the positions a2, b2 which liein the same plain with the angular separation γ2. The physical distances Xand Xobs change like ∆tX and ∆tXobs, which can be described by using theHubble law as

∆tX = XH(t0, X)∆t ≡ XHX∆t, (4.3)

where

X(r) ≡∫ r

g1/2r′r′dr

′ =

∫ r

a(t0, r′)dr′, (4.4)

thus for FLRW models XFRW = a(t0)r. According to the gure 2.2, we canrewrite the denition (4.1) as

∆tγ ≡ γ2 − γ1 = (ξb2 − ξa2)− (ξb1 − ξa1). (4.5)

For two sources a and b at distances X much larger than Xobs we dene aparameter s as following

s ≡ Xobs

X 1. (4.6)

4.1. THE DERIVATION OF THE COSMIC PARALLAX IN LTB 33

Using (4.2) and the denition (4.6) of the parameter s we obtain

ξ = arccos

(X cos θ −Xobs√

X2 +X2obs − 2XobsX cos θ

)= arccos

(cos θ − s√

1 + s2 − 2s cos θ

),

(4.7)where s2 → 0 because of the formula (4.6). In the next step, one can use theTaylor expansion of the rst order with respect to s. The result is shown inthe next formula.

Pf (s) = f(s)|s=0 + f ′(s)|s=0s+O(s2) = θ + s sin θ = ξ, (4.8)

where

f(s)|s=0 = arccos

(cos θ − s√

1 + s2 − 2s cos θ

)|s=0 = θ, (4.9)

and

f ′(s)|s=0 =−1√

1−(

cos θ−s√1−2s cos θ

)2

(−√

1− 2s cos θ + (cos θ−s)2 cos θ

2√

1−2s cos θ

1− 2s cos θ

)|s=0

=1− cos2 θ√1− cos2 θ

= sin θ.

(4.10)

Therefore, each term in (4.5) is given by

ξb1 = θb1 + sin θb1Xobs1

Xb1

; ξa1 = θa1 + sin θa1Xobs1

Xa1

;

ξb2 = θb2 + sin θb2Xobs2

Xb2

; ξa2 = θa2 + sin θa2Xobs2

Xa2

.

(4.11)

Since the expansion is radially with respect to the center, the angle θa1 is thesame as θa2 for every time and the angle θb1 is the same as θb2 for every time.Therefore, using (4.5) and (4.11) we obtain the following relation of the CP

∆tγ = sin θb

(Xobs2

Xb2

− Xobs1

Xb1

)− sin θa

(Xobs2

Xa2

− Xobs1

Xa1

). (4.12)


According to the Hubble law, we obtain

∆tγ = sin θb

(∆tHobsXobs1

∆tHXbXb1

− Xobs1

Xb1

)− sin θa

(∆tHobsXobs1

∆tHXaXa1

− Xobs1

Xa1

)≈ ∆tXobs

[sin θb

(Hobs −HXb)

Xb

− sin θa(Hobs −HXa)

Xa

].

(4.13)

In the FLRW metric, Hobs = HXa = HXb. Therefore, the cosmic parallaxvanishes, unless the sources are not comoving with the Hubble ow, e.g.because they are gravitationally connected. We will consider these two caseslater in the exact formalism [11, 12, 13].

The derivation which was performed above includes many assumptionsand thus it is not an exact calculation of the CP in LTB model. To obtainthe exact result we should consider the full relativistic propagation of thelight. The way of photons is determined by the geodesic equation

d2xρ

dλ2+ Γρµν

dxµ

dλ

dxν

dλ= 0 (4.14)

or alternatively, we can calculate it using the Euler-Lagrange equation

d

dλ

dL

dx− dL

dx= 0, (4.15)

where λ is an ane parameter and L is the Lagrange function which is givenby

L =1

2gµν

dxµ

dλ

dxν

dλ=

1

2

[−c2

(dt

dλ

)2

+(R′)2

1 + β(r)

(dr

dλ

)2

+R2

(dθ

dλ

)2].

(4.16)Using the Euler-Lagrange equations and the 4-velocity identity

gµνdxµ

dλ

dxν

dλ= 0, (4.17)

we obtain four dierential equations of the second order.

1: x = t

c2 d2t

dλ2+

R′R′

1 + β(r)

(dr

dλ

)2

+RR

(dθ

dλ

)2

= 0, (4.18)

4.1. THE DERIVATION OF THE COSMIC PARALLAX IN LTB 35

2: x = r

d2r

dλ2+

(R′′

R′− β′

(2 + 2β)

)(dr

dλ

)2

+2R′

R′dr

dλ

dθ

dλ−(1+β)

R

R′

(dθ

dλ

)2

= 0, (4.19)

3: x = θd

dλ(R2 dθ

dλ) = 0⇒ R2 dθ

dλ= const, (4.20)

The formula (4.20) can be interpreted as the conservation of angular momen-tum J = R2 dθ

dλ.

4: gµνdxµ

dλdxν

dλ= 0

−c2

(dt

dλ

)2

+(R′)2

1 + β(r)

(dr

dλ

)2

+R2

(dθ

dλ

)2

= 0. (4.21)

Now we consider a light ray that hits the observer at an angle ξ, see gure4.1. We choose the initial condition in the way that the photons hit theobserver that lies at r = r0 and θ = 0, at the time t0. An auxiliary photonthat moves radially with respect to the center on the x-axis has a followingnormalized 3-velocity vector:

~v =

√1 + β

R′(1, 0, 0). (4.22)

The spatial components of the vector uµ can be found if we build a tangentto the photon path at the time t0. After the normalization, giju

iuj = 1, weobtain

~u =1

c

dλ

dt

(dr

dλ,dθ

dλ, 0

). (4.23)

Using the inner product of ~v and ~u we obtain a relation for the angle ξ

cos ξ = gijuivj = − R′

c√

1 + β

dλ

dt

dr

dλ. (4.24)


center C

observer

ξ

Figure 4.1: Ray of light, hitting the observer O at an angle ξ. Figure 4.1 is takenfrom [20] with some small changes.

The four dierential equations of the second order can be redened intove dierential equation of the rst order and can be solved with the choseninitial condition. If it is possible to make assumptions for the function β(r)and for the scale function R(r, t) by other theories, one can calculate theCP in the exact formalism using the solution of the dierential equations. Asimpler way to estimate the CP is using the formula (4.13) which was derivedthrough the Euclidean approximation. Such an estimation was performed,for example, in [11], where the estimation of the distance between the centerand the o-center observer comes from [20, 21].

If provided that the estimated values of the CP have a measurable mag-nitude, the homogeneity of the space can be indirectly veried by the appli-cation of the proposed theory. Any signicant deviation from homogene-ity will encourage the search for alternative models to the standard one[11, 12, 13, 16, 18].

In the next chapter, we will work with the cosmic parallax in the FLRWuniverse and for this, we will use the formulas derived here.

Chapter 5

Cosmic parallax in the FLRW

metric

In this chapter the possibility to apply the precise measurements of an anglefor the determination of the cosmic parallax in FLRW models, and especiallyin the ΛCDM model, will be discussed. First, the cosmic parallax in the atFLRW universe will be examined by the application of an exact mathematicalformalism, then the cosmic parallax in the ΛCDM model for gravitationallybound sources will be calculated.

5.1 Cosmic parallax in at FLRW metric for

comoving sources

First of all, to determine the CP in at FLRW models we go back to theformula (4.13) which was obtained by an approximation. As it has alreadymentioned, in the FLRW universe Hobs = HXa = HXb, and therefore, thecosmic parallax is equal to zero in this formula. Admittedly, in the deriva-tion of the formula (4.13) a lot of assumptions were made, thus, we wantto investigate the cosmic parallax in the at FLRW models, if the exactcalculation will be considered.

Because of the singularity of the spherical coordinates at the point r = 0,we rst consider an o-center observer who is placed at the distance r0 fromthe center of the chosen coordinate system. Figure 5.1 shows rays of lighthitting the observer O who is at a distance r0 from the origin. For simplicity,we choose the coordinate system in the way that one of the axes goes through

37

38 CHAPTER 5. COSMIC PARALLAX IN THE FLRW METRIC

the center C, the observer O and one of the sources.Analogous to the LTB universe, the Euler-Lagrange equation (4.15) can

be used to calculate the light path in at FLRW models. The Lagrangefunction for such models is given by

L =1

2gµν

dxµ

dλ

dxν

dλ=

1

2

[−c2

(dt

dλ

)2

+ a(t)2

(dr

dλ

)2

+ r2a(t)2

(dθ

dλ

)2].

(5.1)

center C

γ

observer O

θ

Figure 5.1: Rays of light hitting the observer O at an angle γ.

Using the Euler-Lagrange equation and the 4-velocity identity, we obtain thefollowing relations

1: x = t

c2 d2t

dλ2+ a(t)a(t)

[(dr

dλ

)2

+ r2

(dθ

dλ

)2]

= 0, (5.2)

2: x = rd2r

dλ2+ 2H

dr

dλ

dθ

dλ− r

(dθ

dλ

)2

= 0, (5.3)

5.1. COSMIC PARALLAX FOR COMOVING SOURCES 39

3: x = θ

d

dλ(r2a(t)2

(dθ

dλ

)) = 0⇒ r2a(t)2

(dθ

dλ

)= const=:J, (5.4)

4: gµνdxµ

dλdxν

dλ= 0

−c2

(dt

dλ

)2

+ a(t)2

(dr

dλ

)2

+ r2a(t)2

(dθ

dλ

)2

= 0. (5.5)

The last equation will be used to normalize the vector ~u. We assume thatthe observer is at rest and the sources move away from the observer due tothe space expansion. The coordinate system is chosen in the way that thesource a1 moves away from the observer along the axis x and the coordinateφ is constant. The spatial direction of ~v and ~u can be obtained, like in theLTB universe, if we construct the tangent to the photon path. Thus, thespatial components of the unit vector for the source a1 in at FLRW modelsare given by

~v =1

a(t)

dλ

dra

(dradλ

, 0, 0

). (5.6)

and the spatial components of the unit vector for the source b1 are given by

~u =1

c

dλ

dt

(drbdλ

,dθ

dλ, 0

)(5.7)

Using the inner product of vj and ui, we obtain a relation for the angle γ inat FLRW models:

cos γ = gijuivj =

a(t)

c

(drbdt

). (5.8)

Using the denition of the FLRW metric (3.13), the fact that for every atmodel k = 0 and that for the light ds2 = 0, the relation (5.8) can be rewrittenas

a(t)

c

drbdt

=

√1− a(t)2r2

c2

(dθ

dt

)2

. (5.9)

The term(dθdt

)2can be expanded as

(dθdλ

)2 (dλdt

)2and according to the 4-

velocity identity (5.5), we obtain


(dθ

dλ

)2(dλ

dt

)2

=

(dθ

dλ

)2[

c2

a(t)2(drdλ

)2+ r2a(t)2

(dθdλ

)2

]. (5.10)

Now, the equations (5.9) and (5.10) can be inserted into the equation (5.8)and the result is given by

cos γ =

√√√√1− a(t)2r2

(dθ

dλ

)2[

1

a(t)2(drdλ

)2+ r2a(t)2

(dθdλ

)2

]. (5.11)

After simplifying the formula (5.11) we obtain

cos γ =

√√√√√√√1−

1

( drdλ)2

r2( dθdλ)2 + 1

. (5.12)

Because of the property of light to propagate on a straight line, the followingapproach is used

r(λ) =−r0m

sin θ(λ)−m cos θ(λ), (5.13)

where ro is the coordinate of the observer with respect to the origin, θ isthe angle between the source, the origin and the observer, λ is a parameter,which is choosen as λ ∈ [0; 1] and m is the gradient which is given by

m =−rq sin θq

r0 − rq cos θq, (5.14)

with the initial conditions r(0) = r0, r(1) = rq and θ(0) = 0, θ(1) = θq. Thus,the relation (5.13) describes a straight line which goes from the observer tothe source. The derivative of the equation (5.13) is

dr(λ)

dλ=r0m(cos θ +m sin θ)

(sin θ −m cos θ)2

(dθ

dλ

). (5.15)

5.1. COSMIC PARALLAX FOR GRAVITATIONALLY BOUND SOURCES 41

Using the equations (5.4), (5.13), (5.14) and (5.12), the relation of the angleγ is given by

cos γ =

√√√√√1−

1(cos θ+m sin θ)2

(sin θ−m cos θ)2 + 1

. (5.16)

Using the initial condition θ(0) = 0 and the denition (5.14), we obtain thefollowing relation for the angle γ

cos γ =

√1−

[1

1m2 + 1

]=

1√1 +m2

= const. (5.17)

In this relation it can be seen that the angle between two sources in eachat FLRW model has a dependence on the initial angle of the source θq,its radial coordinate rq and the distance r0 between the observer and thecenter of the coordinate system, which means that the angle γ is constant intime. According to the denition of the CP (4.1) and to the fact that γ isconstant in time, it is obvious that the cosmic parallax in such models doesnot exist. These ndings correspond to our expectations because otherwise,the isotropy of space would be violated. The exception constitute the sourceswhich are connected by gravity.

In the next section, the cosmic parallax of two sources, which are gravi-tationally bound, will be calculated [15, 16, 34].

5.2 Cosmic parallax in the ΛCDM model for

gravitationally bound sources

As it was shown above, there is no cosmic parallax eect in a homogeneousand isotropic universe for two comoving sources that are not bound gravita-tionally. In this section, a CP in the ΛCDM model will be estimated, usingthe assumption that the sources are gravitationally bound, e.g. are of typeFRII (Fanaro-Riley Class II). The FRII sources are radio galaxies with anactive galactic nucleus (AGN) which are very luminous at radio wavelengths.In FRII sources one can observe two jets which are directed oppositely eachother. These sources have weak jets but bright hotspots at the end of thelobes. The goal is to investigate the cosmic parallax between the hotspots


with respect to us. The graphical representation of the CP of two boundsources in the ΛCDM model is shown in gure 5.2.

In a FLRW universe the formula of the CP is simpler than in the LTBuniverse. Since, per denition, the cosmic parallax is the temporal change ofthe angular separation between two sources, and each point in the ΛCDMmodel can be considered as a central one, the relation of the CP (4.1) can berewritten as following

∆tγ = γ2 − γ1 = θ2 − θ1. (5.18)

According to gure 2.2 and to gure 5.2 we can see that in the FLRW spacethere is not dierence between θ and γ. (Remember that θ1 := θb1− θa1 andθ2 := θb2 − θa2)

center C = observer O

Figure 5.2: Graphical representation of the CP in the ΛCDM model. Rays of lighthit the observer O at an angle γ1 at time t1 and at an angle γ2 at time t2.

To obtain the relation of an angle, the equation (3.13) must be trans-formed with respect to the angle θ. Using the assumption that we only


observe small angles, see gure 2.1, we obtain

θ =l

dA=l(1 + z)

a0r, (5.19)

where l is a distance between two sources, which are on the same shell (whichmeans that they have the same redshift with respect to the observer) and dAis an angular distance. The denominator a0r for the ΛCDM is given by theFriedmann equation as

a0r = c

∫ z

0

dz′

H(z′)=

c

H0

∫ z

0

dz′√ΩM(1 + z′)3 + ΩΛ

, (5.20)

where H0 = h× 100[

kms Mpc

]is the Hubble parameter today, h is dimension-

less Hubble parameter and ΩM is the matter density parameter (dark plusbarionic) today.

By inserting the formula (5.20) into (5.19) we obtain the relation of theangle θ in units of [as]

θ(z) = 68.755

(l

Mpc

)h

(1 + z)∫ z0

dz′√ΩM (1+z′)3+ΩΛ

. (5.21)

Thus, according to the equation (5.21) the CP is

∆tθ = α

(1 + z2)∫ z20


− (1 + z1)∫ z10


, (5.22)

where z2 = z1 + ∆tz and α = 68.755(

lMpc

)h

To estimate the cosmic parallax one must nd out how the redshiftchanges with respect to time. Using the denition of the redshift, we ob-tain the following formula for the change of the redshift in time:

∆tz = z2 − z1 =a(t0) + ∆t0a(ts) + ∆ts

− a(t0)

a(ts), (5.23)

where ts is the emission time of light from the source and t0 is the observationtime. Using the Taylor expansion, we get

∆tz = ∆t0

(a(t0)− a(ts)

a(ts)

)+O

(∆t0t0

)2

= H0∆t0

(1 + z − H(z)

H0

). (5.24)


The results of the calculation of the redshift change ∆tz for dierent z andfor ∆t0=10 years are presented in table 3 in the appendix.

Alternatively, because of the small values of ∆tz one can use the Taylorexpansion of equation (5.22):

Pf (∆tz) = f(∆tz)∣∣∣∆tz=0

+ f ′(∆tz)∣∣∣∆tz=0

∆tz +O(∆tz)2 (5.25)

Thus, one obtains

f(∆tz)∣∣∣∆tz=0

= α

(1 + z + ∆tz)∫ z+∆tz

0dz′√

ΩM (1+z′)3+ΩΛ

− (1 + z)∫ z0


∣∣∣∆tz=0

= 0

(5.26)

f ′(∆tz)∣∣∣∆tz=0

= α

∫ z+∆tz

0dz′√

ΩM (1+z′)3+ΩΛ

− (1+z+∆tz)√ΩM (1+z)3+ΩΛ(∫ z+∆tz

0dz′√

ΩM (1+z′)3+ΩΛ

)2

∣∣∣∣∣∆tz=0

∆tz

= α

∫ z

0dz′√

ΩM (1+z′)3+ΩΛ

− (1+z)√ΩM (1+z)3+ΩΛ(∫ z

0dz′√

ΩM (1+z′)3+ΩΛ

)2

∆tz

(5.27)

Using the equation (5.25), we obtain

∆tθ = α∆tz

∫ z

0dz′√

ΩM (1+z′)3+ΩΛ

− (1+z)√ΩM (1+z)3+ΩΛ(∫ z

0dz′√

ΩM (1+z′)3+ΩΛ

)2

+O(∆tz)2 (5.28)

The integrals in the denominator of the equation (5.22) are not solvableanalytically. Therefore, for gravitationally bound sources which are posi-tioned on the same shell and at the distance of 1Mpc from each other, thecosmic parallax in the ΛCDM model has to be calculated numerically, usingthe values of the redshift change ∆tz from table 3 in appendix.


The results of the cosmic parallaxes with respect to z are shown also intable 3. The same results can be obtained by using the formula (5.28). Agraphical representation of the CP with respect to z is shown in gure 5.3.

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

0 1 2 3 4 5 6 7 8 9

Δθ/1

0 y

ears

[µ

as]

z

Figure 5.3: Graphical representation of the cosmic parallax with respect to z inthe ΛCDM model for two gravitationally bound sources at a distance of 1Mpc.

The logarithmical representation of the absolute values of the cosmic parallaxwith respect to the redshift is shown in gure 5.4. From the values in table3 we can see that the magnitude of the cosmic parallax after 10 years is verysmall and not measurable for most redshifts. The best possible accuracy ofthe modern technique, for example, the accuracy of RadioAstron is approx-imately 7µas, that means that only the sources lying at z < 0.003 can beused for the investigation of the CP. The period of 10 years was chosen, basedon the idea of real-time cosmology which was briey introduced in chapter2.4. In the formula (5.28), using the relation (5.24), we can see that thetime dependence of the CP is linear. That means that by doubling the ob-servation time, the values of the parallax will be doubled and we will obtainmeasurable values of the CP for the sources, positioned at the more distantredshifts. In the gure 5.5 the angle under which an object of xed physicalsize is seen at dierent redshifts in the ΛCDM model is shown. According tothe gure 5.5, we would expect the change of the sign of the resulting CP,


approximately from z = 1.5. The sign change does not happen, since fromapproximately z = 1.5, ∆tz also changes the sign [15, 16].

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

100

0 1 2 3 4 5 6 7 8 9

Δθ/1

0 y

ears

[as]

z

Figure 5.4: Logarithmical representation of the CP with respect to z in the ΛCDMmodel for two gravitationally bound sources, placed at distance 1 Mpc.

100

150

200

250

300

350

400

450

500

550

0 1 2 3 4 5 6 7 8 9

Θ(z

) [a

rcse

c]

z

Figure 5.5: Angle with respect to z in the ΛCDM model for two gravitationallybound sources, placed at distance 1 Mpc.


In addition, if considering sources of the type FRII, one has to keep inmind that the jets dilate nearly with the speed of light which can have a greatinuence on the magnitude of the CP. This inuence will be estimated in thenext step. The distance between the hotspots changes mostly according to

l2 = l1 + 2 t c = l1 + 6.132× 10−6Mpc. (5.29)

where l2 is the distance between two hotspots after 10 years and 10yr× c =3.066× 10−6Mpc which means that the formula (5.22) can be rewritten as

∆tθ =68.755h

Mpc

l2(1 + z2)∫ z20

dz′√Ωm(1+z′)3+(1−Ωm)

− l1(1 + z1)∫ z10

dz′√Ωm(1+z′)3+(1−Ωm)

. (5.30)

0.0001

0.01

1

100

10000

0 1 2 3 4 5 6 7 8 9

Δθ/1

0 y

ears

[µ

as]

z

black, CP with l1 expands

blue, CP with l1 const

Figure 5.6: The black line shows the logarithm of the cosmic parallax, taking intoaccount the expansion of the jets with respect to z in the ΛCDM model for twogravitationally bound sources which are at initial distance l1 = 1 Mpc from eachother. The blue line is the logarithm of the cosmic parallax with respect to z inthe same model for two gravitationally bound sources which are at distance 1 Mpcfrom each other, which remains constant over the time.


The results of the calculation of the cosmic parallax, taking into accountthe expansion of the jets, are shown in appendix in table 4 and a graphicalrepresentation of this is given in gure 5.6.

If comparing the values from table 3 and table 4 in appendix, we cansee that the dierence between the values of CP amounts several orders ofmagnitude, see also the gure 5.6. This means that the eect of the extensionof the jets dominates the CP and hence, such a source of the type FRIIcannot serve the purpose. For the investigation of the cosmic parallax oneneeds sources which are bound by gravity and which do not change their size.

5.3 Estimation of the secular parallax

In addition to the eect of the expansion of jets, the cosmic parallax isinuenced by an annual and secular parallax. In this section, the secularparallax will be estimated and the results will be compared with the CPfrom section 5.2. The secular parallax is a parallax which is caused by themotion of our solar system in the Milky Way. In a lot of recent astrometricalarticles the Galactic parameters have been estimated, see e.g. [25]. Theseestimations give us the distance r to the galactic center as 8.4± 0.6kps anda circular rotation speed V of 254± 16km/s.

In gure 5.8 we can see a schematic represenation of the secular parallax.In this scheme the points D and C are the gravitational bound sources whichare at rest, the point A gives the position of the solar system in the MilkyWay at time t0 and B shows the position of the solar system in the MilkyWay after 10 years. From gure 5.8 we can see that the secular parallaxreads

∆tθs = θ1 − (θ2 + θ3). (5.31)

Due to straightforward geometry, the angular change between the movingsolar system and the sources, which were assumed at rest, can be calculated.As an example, we carry out the calculation of the secular parallax for twosources which lie at the redshift z = 0.1 and which have the distance of 1Mpcfrom each other. For this purpose we need to know the physical distance ofthe chosen redshift. The physical distance for the ΛCDM model is calcu-lated by using the FLRW metric and the following denition of the physicaldistance

Xph = a0r =a0c

H0

∫ z

0

dz′√Ωm(1 + z′)3 + ΩΛ

. (5.32)

5.3. ESTIMATION OF THE SECULAR PARALLAX 49

Since the integral is not analytically solvable, we solve it numerically by usingthe program "Mathematica". The results are shown in table 5 in appendix.

In the next step one can calculate all the parameters which remain con-stant for dierent Xph. The rst one is la which tells us how far the solarsystem travels on the arc with respect to the initial position after 10 years

la = t V = 8.010144× 1010km = 0.002596pc. (5.33)

Knowing the length of the arc which the solar system travels on, one cancompute the angle α, see gure 5.8. It is given by

la = πrα

180⇒ α = 1.7707× 10−5 = 63.745mas. (5.34)

Thus, the length l is

sin(α

2

)∼=α

2=

l

2 r⇒ l = 0.002565pc. (5.35)

From gure 5.8 one can see that the angle β = α2is given by

β = 8.854× 10−6 = 31.874mas. (5.36)

The lengths AK and BK are given by

AK = l cos β = 2.565× 10−3pc, (5.37)

BK = l sin β = 3.964× 10−10pc. (5.38)

Using equation (5.31) and gure 5.8 we obtain the following relation for thesecular parallax after 10 years

∆tθs = 2 arctan

(CF

FA

)−[arctan

(CE

EB

)+ arctan

(DE

EB

)], (5.39)

where FA =√X2ph − CF 2 and Xph is the physical distance to the sources,

Xph = CA = DA, CF is the half of the distance between two sources and itis chosen to be equal to 0.5Mpc, CE = 0.5Mpc−AK, EB = FA+BK andDE = 0.5Mpc + AK. So the formula (5.39) is


∆tθs = 2 arctan

5× 105√X2ph − 0.52

− arctan

5× 105 − 2.565× 10−3√X2ph − 0.52 + 3.964× 10−10

− arctan

5× 105 + 2.565× 10−3√X2ph − 0.52 + 3.964× 10−10

,

(5.40)

where the Xph is expressed in units of pc.

The secular parallax can be calculated, using formula (5.40). The resultsof the secular parallaxes and the values of the physical distances which de-pend on the redshift for the ΛCDM model can be found in table 5. Thegraphical representation of the secular parallax, dependent on the redshift ofthe chosen sources, is shown in gure 5.7.

1e-06

0.0001

0.01

1

100

10000

1e+06

1e+08

0 1 2 3 4 5 6 7 8 9

Δtθ

s /1

0 y

ears

[µ

as]

z

red, secular parallax

blue, CP for bound sources

Figure 5.7: The red line is the graphical representation of the secular parallax de-pending on z for the ΛCDM model and the blue line is the graphical representationof the absolute values of the cosmic parallax of two bound sources at a distance of1Mpc from each other depending on z for the ΛCDM model.

5.3. ESTIMATION OF THE SECULAR PARALLAX 51

A

B

A

C

A

D

A

E

A

r

A

A

O

A

α

β

F

A

K

A

Figure 5.8: Schematic representation of the secular parallax. The lengths in theimage do not have the correct ratio.


In gure 5.7 one can see that the further away the source is, the smallerthe secular parallax is. On the other hand, the cosmic parallax at largedistances is too small to be measured. This means that the cosmic parallaxcan be studied at the sources that are located at small distances from us(z < 0.003), taking into account the secular parallax.

Chapter 6

Estimation of cosmological

parameters

In this chapter, the precise measurement of angles will be applied to estimatecosmological parameters. For this purpose we rst consider the mathematicaltools which allow the estimation of parameters.

6.1 Likelihood estimation method

The maximum likelihood estimation method of parameters is a popular sta-tistical method. Due to its advantages over some other estimation techniqueslike the least squares method and the torque, the likelihood estimator is themost important principle for estimating the parameters of a distribution func-tion. This method is used to create a model, based on statistical data, and toprovide estimates of the model parameters. The likelihood method is basedon the assumption that all the information about the statistical sample iscontained in the likelihood function.

We assume that we have a set of N data x = (x1, x2, ..., xN) that is de-scribed by a joint probability density function f(x;θ), where θ = (θ1, θ2, ..., θM)is a set of unknown parameters for which we need to nd estimates. Thelikelihood function is dened by the probability density function that is eval-uated with the data x, but viewed as a function of the parameters. Thus,L(θ) = f(x;θ). Since the random variables xi are independent, the likeli-hood function is given by

53

54 CHAPTER 6. ESTIMATION OF COSMOLOGICAL PARAMETERS

L(θ) =N∏i=1

f(xi;θ). (6.1)

The idea of the maximum likelihood method is that the estimation of the pa-rameter θ is given by some vector θ which maximizes the likelihood function.That means

L(x1, x2, ..., xN ; θ) = maxθL(x1, x2, ..., xN ;θ). (6.2)

The necessary condition to maximize a function is

∂∂θ1L(x1, x2, ..., xN ;θ) = 0,

∂∂θ2L(x1, x2, ..., xN ;θ) = 0,

.........................................∂∂θN

L(x1, x2, ..., xN ;θ) = 0.

(6.3)

These relations are called the likelihood equations. Once they are solved, itmust be checked by using the sucient condition that the found values arereally the maxima. For the search of the maxima it is easier to use the naturallogarithm of the likelihood function instead of the likelihood function. Thatmeans that we use the following equations instead of the equations (6.3)

∂∂θ1

lnL(x1, x2, ..., xN ;θ) = 0,∂∂θ2

lnL(x1, x2, ..., xN ;θ) = 0,

.........................................∂∂θN

lnL(x1, x2, ..., xN ;θ) = 0.

(6.4)

The disadvantage of the likelihood method is that we only obtain thevalues which give us the best possible t for a chosen cosmological modeland we do not know whether this t is good in the absolute sense.

These theoretical ideas can be carried out in a given example. Here, atwo-variables case is considered. The probability density function is chosento be a Gaussian with the standard deviation σ and the expectation value µ.The Gaussian distribution is given by

6.1. LIKELIHOOD ESTIMATION METHOD 55

f(x) =1

σ√

2πexp

−(x− µ)2

2σ2

. (6.5)

Using (6.5), the likelihood function is built as follows:

L(x1, x2, ..., xN ;µ;σ2) =N∏i=1

f(xi) =

(1

σ√

2π

)N N∏i=1

exp

−(xi − µ)2

2σ2

.

(6.6)It is more convenient to use the natural logarithm of the likelihood function.So lL is dened in the following way

lL:= lnL(x1, x2, ..., xN ;µ;σ2) = −N2

ln (σ2)− N

2ln (2π)−

N∑i=1

(xi − µ)2

2σ2.

(6.7)According to the necessary condition, equation (6.4), we obtain

∂∂µlL(x;µ;σ2) =

∑Ni=1

(xi−µ)σ2 = 0,

∂∂σ2 lL(x;µ;σ2) =

∑Ni=1

(xi−µ)2

2(σ2)2 − N2σ2 = 0.

(6.8)

After the calculation of (6.8) we obtain

µ = 1

N

∑Ni=1 xi,

σ2 = 1N

∑Ni=1(xi − µ)2.

(6.9)

Remark: In the following we will drop the tilde over µ and σ.

To verify that we actually have found a maximum of the likelihood functionfor the values of µ and σ2, the second derivative of lL can be calculated. Sowe obtain

∂2

∂µ2 lL(x;µ;σ2) = − Nσ2 ,

∂2

∂(σ2)2 lL(x;µ;σ2) = − 1(σ2)3

∑Ni=1 (xi − µ)2 + N

2(σ2)2 ,∂2

∂µ∂σ2 lL(x;µ;σ2) = − 1(σ2)2

∑Ni=1 (xi − µ).

(6.10)


By using the relations (6.9) the second derivatives are given by

∂2

∂µ2 lL(x;µ;σ2) = − Nσ2 ,

∂2

∂(σ2)2 lL(x;µ;σ2) = − N2(σ2)2 ,

∂2

∂µ∂σ2 lL(x;µ;σ2) = 0.

(6.11)

We can see from (6.11) that the second derivatives are negative, hence wereally found maxima of the likelihood function [2, 14, 19, 27, 28].

6.2 Fisher Information Matrix

The second mathematical tool is the Fisher matrix analysis. The Fishermatrix analysis leads to an estimation of the best possible accuracy of acertain model's parameters. If assuming some cosmological model µ(p) andconsidering a set of cosmological parameters pi, where i = 1, 2, ..., N , thenthe Fisher information matrix is given as

Fij:=∂2lL∂pi∂pj

. (6.12)

The Fisher information matrix for the Gaussian distribution can be calcu-lated by using the formula (6.7) and the denition (6.12). Thus, we obtain

Fij =N∑k=1

(xk − µ(p))

σ2

∂2µ(p)

∂pi∂pj−

N∑i=1

1

σ2

∂µ(p)

∂pi

∂µ(p)

∂pj. (6.13)

Due to equation (6.9) the rst term in formula (6.13) is equal to zero, there-fore, the Fisher matrix for the Gaussian distribution is given by

Fij = −N∑k=1

1

σ2

∂µ(p)

∂pi

∂µ(p)

∂pj. (6.14)

The best possible 1σ error on the parameter pi is given by the square root ofthe eigenvalues of the covariance matrix C which is dened by

C = −F−1. (6.15)

The square root of the eigenvalues of the covariance matrix also gives us theaxes lengths of the error ellipses. The eigenvectors of the covariance matrix

6.2. FISHER MATRIX ANALYSIS IN THE ΛCDM MODEL 57

show us the direction of the axes of the error ellipses with respect to thecoordinate frame. It is known from probability theory that for normallydistributed random variables 68.3% of the realizations lie in the interval ±1σand 95.4% lie in the interval ±2σ. The axes lengths of the 95% probabilityerror ellipse can be obtained by multiplying the axes lengths of the 68%probability error ellipse by the factor

√6.17 [2, 14, 18, 19, 27, 28].

6.3 Fisher matrix analysis of angular sizes

in the ΛCDM model

The Fisher matrix analysis can be used to estimate the determination ac-curacy of the parameters for a selected model. In this section the Fishermatrix analysis is performed for the estimation of cosmological parameterswith the assumption of using the data of the angular size measurement. TheFisher matrix will be calculated for dierent assumptions: the rst one is foran one-dimensional case, which means that only one cosmological parame-ter is variable, and the second one is for the case that two parameters arevariable simultaneously. First of all, the likelihood function for the ΛCDMmodel should be established. For this purpose, the formula for an angle inthe ΛCDM model (5.21) from the chapter 5.2 is inserted into the denitionof the lnL (6.7), so we obtain

lL = −N2

ln (σ2)− N

2ln (2π)−

N∑i=1

θi − 68.755p1(1+z)∫ z0

dz′√p2(1+z′)3+(1−p2)

2

2σ2, (6.16)

where θi is the i-th angle measurement, N is the number of measurements,σ is the standard deviation of performed measurements, p1 and p2 are cos-mological parameters whose accuracy should be determined and which arechosen to have ctive values of

p1 ≡(

l

Mpc

)h = 0.678, (6.17)

p2 ≡ Ωm = 0.308. (6.18)

First, we calculate the Fisher matrix with the assumption that one of theparameter can be varied and the other is xed.


Case 1: p1 is variable, p2 is constant.

Using equation (6.16) the rst derivative of lL is given by

∂lL∂p1

=1

σ2

N∑i=1

θi − 68.755p1(1 + z)∫ z0

dz′√p2(1+z′)3+(1−p2)

68.755(1 + z)∫ z0

dz′√p2(1+z′)3+(1−p2)

. (6.19)

Because of the maximum likelihood formalism the rst derivative must beequal to 0. Thus, the following relation is obtained

N∑i=1

θi =68.755p1N(1 + z)∫ z0

dz′√p2(1+z′)3+(1−p2)

. (6.20)

Thus, according to the equations (6.14), (6.19) and (6.20), we obtain theFisher matrix for the xed p2 and variable p1. It is

F11 =∂2lL∂p1

2= −N

σ2

68.7552(1 + z)2(∫ z0

dz′√p2(1+z′)3+(1−p2)

)2 . (6.21)

Using the denition (6.15) the covariance matrix is given by

C11 =σ2

N

(∫ z0

dz′√p2(1+z′)3+(1−p2)

)2

68.7552(1 + z)2. (6.22)

Thus, the best possible 1σ error on p1, which is given according to ∆p1 =√C11, is equal to

∆p1 =σ√N

∫ z0

dz′√p2(1+z′)3+(1−p2)

68.755(1 + z). (6.23)

The results of the calculation for dierent z are shown in table 6.

Case 2: p2 is variable, p1 is constant.

After an analogous calculation we obtain the estimation of accuracy for p2.It is given by


∆p2 =σ√N

2∫ z

0dz′√

p2(1+z′)+(1−p2)

68.755p1(1 + z)∫ z

0dz′((z′+1)3−1)

(p2(1+z′)+(1−p2))

. (6.24)

The results for dierent z are also shown in table 6 in appendix. Thegraphical representation of the 1σ errors of the parameters p1 and p2 areshown in the gure 6.1.

0.306

0.3065

0.307

0.3075

0.308

0.3085

0.309

0.3095

0.31

0.677 0.6775 0.678 0.6785 0.679

Ωm

h

0.306

0.3065

0.307

0.3075

0.308

0.3085

0.309

0.3095

0.31

0.677 0.6775 0.678 0.6785 0.679

Ωm

h

0.306

0.3065

0.307

0.3075

0.308

0.3085

0.309

0.3095

0.31

0.677 0.6775 0.678 0.6785 0.679

Ωm

h

0.306

0.3065

0.307

0.3075

0.308

0.3085

0.309

0.3095

0.31

0.677 0.6775 0.678 0.6785 0.679

Ωm

h

Figure 6.1: Estimated error of the parameters p1, p2 for two sources at a distanceof 1Mpc from each other and at a distance of z = 0.1 from observer (top left),z = 1 (top right), z = 4 (bottom left) and z = 8 (bottom right)

The parameters in the gure 6.1 are plotted, assuming the observationalaccuracy of the VLA FIRST survey (VLA Faint Images of Radio Sky atTwenty cm). In this survey about N = 106 radio sources were observed witha precision of σ = 5as. We assume that the radio sources, which we considerhere, are distributed as it is shown in table 2. This distribution of the radio


sources in the universe can be found in [38].

z 0.1 0.2 0.5 1 1.5 2 2.5Nz 105 105 105 2× 105 1.25× 105 7.5× 104 6× 104

z 3 4 5 6 7 8 9Nz 3× 104 104 5× 103 103 103 103 102

Table 2: Distribution of the radio sources [38], where N is the number ofradio sources at the given redshift z.

It should be noted that all further calculations of the 1σ and 2σ errorsof the parameters p1 and p2 were made, assuming that the measurementaccuracy of σ = 5as applies to each redshifts.

Case 3: p1 and p2 are variable.

In the next step we assume that both parameters p1 and p2 are variable.That means that we consider a two-dimensional case and the following 2× 2Fisher matrix is obtained

F = −∑z

Nz

σ2

(∂θ(z)∂p1

)2∂θ(z)∂p1

∂θ(z)∂p2

∂θ(z)∂p1

∂θ(z)∂p2

(∂θ(z)∂p2

)2

. (6.25)

For the calculation of the Fisher matrix for two unknown parameters we needat least two angle measurements at dierent redshifts. These redshifts canbe chosen freely. So we consider three dierent classes of bins in the redshift:

The rst class contains the following bins of redshift: [0.1,0.2], [0.5,1], [2,3],[4,5], [6,7], [8,9].

The second class contains the following bins of redshift: [0.5,1], [0.5,1.5],[0.5,2], [0.5,2.5], [0.5,3],[0.5,4], [0.5,5], [0.5,6].

The third case contains only one bin of ve dierent redshifts which areconsidered simultaneously: [1, 2, 3, 4, 5].


The Fisher matrix for the rst two classes is given as following:

F = − 1

σ2

Nlz

(∂θ∂p1

)2

|lz+Nhz

(∂θ∂p1

)2

|hzNlz

∂θ∂p1

∂θ∂p2 |lz

+Nhz∂θ∂p1

∂θ∂p2 |hz

Nlz∂θ∂p1

∂θ∂p2 |lz

+Nhz∂θ∂p1

∂θ∂p2 |hz

Nlz

(∂θ∂p2

)2

|lz+Nhz

(∂θ∂p2

)2

|hz

,

(6.26)where lz is the low redshift boundary of the given bin and hz is the highredshift boundary of the given bin. Nlz is the number of sources measured atthe low redshift of the given bin and Nhz is the number of sources measuredat the high redshift of the given bin.

In table 7 in appendix we can see the obtained Fisher matrices and thecovariance matrices for the rst class of bins. To obtain the error ellipses,the eigenvalues and the eigenvectors of the covariance matrix, as well as the1σ and the 2σ errors of the parameters p1 und p2 must be calculated. Theresults of these calculations are shown in table 8. In table 9 we can see theFisher and the covariance matrices for the second class of redshift bins and intable 10, the results of the calculation of the eigenvalues and the eigenvectorsof the covariance matrix of the second class are shown. The Fisher matrixfor the third class contains ve summands in each entry. The results of thisare shown in table 11 und table 12.

The graphical representation of the error ellipses for the three dierentclasses are shown in gures 6.2 - 6.4. The gure 6.2 was plotted, assumingthe observational accuracy of the VLA FIRST survey and the distributionof the radio sources in the universe which was described in table 2.

If one uses the accuracy of the MOJAVE survey (Monitoring Of Jetsin Active galactic nuclei with the VLBA Experiments), one obtains N ≈0.5×103 and σ ≈ 10−5as. That means that the error ellipses of this accuracyare at least four order of magnitude higher than with the values of VLA.Using the accuracy of the RadioAstron (see section 1.3.3), we obtain thatthe accuracy of the estimated parameters h and ΩM is at least two orders ofmagnitude higher than with the values of MOJAVE VLBA. That means, ifwe would plot the error ellipses with the accuracy of MOJAVE VLBA, theywould be smaller by a factor of about 104 than those which are shown here ingures 6.2 - 6.4. The conclusion is that, using the measurement of an anglewith the accuracy of the missions VLBA or RadioAstron, one can determinethe cosmological parameters with very high precision.


0.306

0.3065

0.307

0.3075

0.308

0.3085

0.309

0.3095

0.31

0.677 0.6775 0.678 0.6785 0.679

Ωm

h

green, z=0.5-1red, z=2-3blue, z=4-5violet, z=6-7black, z=8-9

Figure 6.2: 1σ and 2σ error ellipses for the rst class of the red shift bins with theaccuracy of VLA FIRST survey data.

0.306

0.3065

0.307

0.3075

0.308

0.3085

0.309

0.3095

0.31

0.677 0.6775 0.678 0.6785 0.679

Ωm

black, z=0.5-1green, z=0.5-1.5red, z=0.5-2blue, z=0.5-4purple, z=0.5-6

Figure 6.3: 1σ and 2σ error ellipses for the second class of the red shift bins withthe accuracy of VLA FIRST survey data.


0.306

0.3065

0.307

0.3075

0.308

0.3085

0.309

0.3095

0.31

0.677 0.6775 0.678 0.6785 0.679

Ωm

h

Figure 6.4: 1σ and 2σ error ellipses for the third class of the red shift bins withthe accuracy of VLA FIRST survey data.

If we analyze the results in tables 6, 8, 10 and 12, we can notice thatthere is no correlation between the values of the 1σ and 2σ errors which werecalculated by the variation of one parameter and the values that were foundby the simultaneous variation of the two parameters. The reason is that inthe case of the variation of one parameter, the calculation was made, usingthe exact knowledge of the xed parameter.

By comparing the results of calculations of three classes of redshifts, wecan see that the most accurate estimation of the parameters is obtained if themeasurements of angles are performed at the ve dierent redshifts which areconsidered simultaneously. From gures 6.2 and 6.3 one can see that by usingthe redshift bin of the rst class, the parameter p2 can be determined moreaccurately than p1. The parameter p1 can be determined more accuratelythan p2 if we use the redshift bin of the second class.

However, one should notice that the astrophysics of jets have not beenyet understood to a level that would allow us to know their precisely sizes.This stops us from obtaining super-precise measurements of the cosmologicalparameters by means of radio observations.

Chapter 7

Conclusion

Due to the rapid development of technology in recent decades, new ways havebeen found in every area of physics, especially, in astronomy and cosmology.Using modern facilities like the VLBA or RadioAstron, it has become possibleto carry out angle measurements with very high precision.

The aim of this thesis was to investigate some applications in cosmologyof the precise angle measurement theoretically. The rst possible applica-tion which was treated here was to study the homogeneity of space on thelarge scales. The homogeneity of space is impossible to check in a directway. Therefore, the precise measurement of angles provides a good chanceto investigate the homogeneity indirectly by using the cosmic parallax eect.For this purpose, two types of calculations were performed. First of all, twocomoving sources which were placed at a distance of 1Mpc from each otherwere investigated. Depending on the result of the measurement of the anglebetween these sources and the observer, there are two ways to proceed: a) Ifwe "live" in an inhomogeneous universe, we will measure a cosmic parallaxwhich is caused by the inhomogeneous expansion of space. The magnitudeof the cosmic parallax was estimated, for example, in [18]. b) If we "live" ina homogeneous universe, we expect that there is no cosmic parallax for twocomoving sources. In this case, two gravitationally bound sources can be in-vestigated to nd a change of angular separation. The theoretical predictionof the change of the angular separation in the ΛCDM model was calculated,thus, this method is an independent proof of the ΛCDM model. Signicantdeviations from the expected values of the CP in the ΛCDM would requirethe consideration of alternative models.

The second application of precise angle measurements, which has been

65

66 CHAPTER 7. CONCLUSION

discussed here, is the estimation of the cosmological parameters in the ΛCDMmodel. The accuracy of the estimation of the parameters ΩM and h wasdetermined with the precision of the VLA FIRST survey, MOJAVE and Ra-dioAstron. Due to the calculated accuracy it was found that the precise anglemeasurement is also a good method for the determination of the cosmologi-cal parameters. However, we still need to identify good standard rules of theMpc scale to make this option come true.

Chapter 8

Appendix

z ∆tz ∆tθ [arcsec]

0.001 3.72547× 10−13 −1.73666× 10−5

0.003 1.11616× 10−12 −5.78120× 10−6

0.01 3.70332× 10−12 −1.72628× 10−6

0.03 1.09625× 10−11 −5.67633× 10−7

0.07 2.48926× 10−11 −2.36418× 10−7

0.09 3.15650× 10−11 −1.81152× 10−7

0.1 3.48284× 10−11 −1.61795× 10−7

0.3 9.01465× 10−11 −4.51525× 10−8

0.5 1.27592× 10−10 −2.15330× 10−8

0.7 1.48984× 10−10 −1.14825× 10−8

0.9 1.56126× 10−10 −6.13505× 10−9

1 1.54859× 10−10 −4.38390× 10−9

1.3 1.33737× 10−10 −1.23466× 10−9

1.5 1.06595× 10−10 −2.55238× 10−10

2 −9.23694× 10−13 −6.01899× 10−12

3 −3.58296× 10−10 −4.52152× 10−9

4 −8.73328× 10−10 −1.27249× 10−8

5 −1.52359× 10−9 −2.34163× 10−8

6 −2.29492× 10−9 −3.61168× 10−8

7 −3.17722× 10−9 −5.05686× 10−8

8 −4.16272× 10−9 −6.6605× 10−8

9 −5.24518× 10−9 −8.4106× 10−8

Table 3: Change of redshift in 10 years for dierent initial values of z in the ΛCDMmodel and cosmic parallax with respect to z in the ΛCDM model for two gravita-tionally bound sources, placed at a distance of 1Mpc from each other.

67

68 CHAPTER 8. APPENDIX

z ∆tθ [arcsec]

0.001 0.2861830.003 0.0956290.01 0.0289400.03 0.0098820.07 0.0044420.09 0.0035370.1 0.0032200.3 0.0013350.5 0.0009750.7 0.0008340.9 0.0007651 0.0007451.3 0.0007121.5 0.0007062 0.0007143 0.0007774 0.0008615 0.0009536 0.0010487 0.0011468 0.0012439 0.001342

Table 4: Cosmic parallax, taking into account jets expansion of the speed of lightwith respect to z in the ΛCDM model for two gravitationally bound sources whichare at initial distance l1 = 1Mpc from each other.

69

z Xph[Mpc] ∆tθs[arcsec]

0.001 4.42400 573.64700.003 13.2650 21.998200.01 44.1450 0.5991580.03 131.818 2.25124× 10−2

0.07 304.661 1.82352× 10−3

0.1 432.074 6.39277× 10−4

0.3 1231.86 2.75855× 10−5

0.5 1945.88 6.99869× 10−6

0.7 2579.39 3.00479× 10−6

0.9 3140.79 1.66436× 10−6

1 3397.34 1.31507× 10−6

1.3 4083.69 7.57193× 10−7

1.5 4481.73 5.72833× 10−7

2 5315.37 3.43372× 10−7

3 6515.54 1.86429× 10−7

4 7346.63 1.30047× 10−7

5 7963.69 1.02099× 10−7

6 8444.59 8.56307× 10−8

7 8832.79 7.48293× 10−8

8 9154.62 6.72116× 10−8

9 9427.02 6.15520× 10−8

Table 5: Physical distance and secular parallax with respect to z in the ΛCDMmodel for two gravitationally bound sources, placed at a distance of 1Mpc fromeach other.


z√Nσ ∆p1

√Nσ ∆p2

0.1 0.001291 0.2616560.2 0.002300 0.1214770.5 0.004264 0.0416031 0.005584 0.0183302 0.005824 0.0084933 0.005354 0.0055694 0.004830 0.0041615 0.004363 0.0033286 0.003965 0.0027747 0.003629 0.0023808 0.003344 0.002084

Table 6: Best possible 1σ error on p1 and p2 with respect to z in the ΛCDM model(for p2 or p1 xed, respectively).

71zbin

Fisher

matrix

covarian

cematrix

0.1-0.2−

1 σ2

( 7.8

7501×

1010

4.9

0337×

109

4.9

0337×

109

3.3

9964×

108

)σ

2

( 1.2

4567×

10−

10−

1.7

9666×

10−

9

−1.

7966

6×

10−

92.

8855

51×

10−

8

)

0.5-1−

1 σ2

( 1.1

9149×

1010

2.5

5435×

109

2.5

5435×

109

1.3

0848×

109

)σ

2

( 1.1

8280×

10−

10−

3.7

2116×

10−

11

−3.

7211

6×

10−

11

4.03

196×

10−

11

)

2-3

−1 σ2

( 3.2

5768×

109

1.9

4555×

109

1.9

4555×

109

1.1

6669×

109

)σ

2

( 7.49

123×

10−

8−

1.2

4922×

10−

7

−1.

2492

2×

10−

72.

0917

3×

10−

7

)

4-5

−1 σ2

( 6.9

1387×

108

4.9

1040×

108

4.9

1040×

108

3.4

8890×

108

)σ

2

( 2.25

481×

10−

9−

1.6

0207×

10−

9

−1.

6020

7×

10−

93.

1747

9×

10−

9

)

6-7

−1 σ2

( 1.3

952×

108

1.0

5634×

108

1.0

5634×

108

7.9

9858×

107

)σ

2

( 7.25

192×

10−

5−

9.6

1691×

10−

5

−9.

6169

1×

10−

51.

2701

9×

10−

5

)

8-9

−1 σ2

( 9.9

8678×

107

7.7

5569×

107

7.7

5569×

107

6.0

2311×

107

)σ

2

( 7.52

281×

10−

4−

9.6

8678×

10−

4

−9.

6867

8×

10−

41.

2473

4×

10−

3

)

Tab

le7:

Fisher

matricesan

dcovarian

cematricesfortheredshiftbinsof

therstclass.


zbins

eigenvalues

eigenvec

1σerrors

2σerrors

0.1-0.2

σ22.

8967

0×

10−

8

( −0.0

6

1

) ∆p

1=σ

1.70

197×

10−

4∆p

1=σ

4.22

761×

10−

4

σ21.

2649

3×

10−

11

∆p

2=σ

3.55

700×

10−

6∆p

2=σ

8.83

400×

10−

6

0.5-1

σ24.

0700

1×

10−

10

( −0.1

3

1

) ∆p

1=σ

2.01

980×

10−

5∆p

1=σ

5.01

710×

10−

5

σ21.

1350

0×

10−

10

∆p

2=σ

1.06

540×

10−

5∆p

2=σ

2.64

630×

10−

5

2-3

σ22.

8385

9×

10−

7

( −0.5

9

1

) ∆p

1=σ

5.32

784×

10−

4∆p

1=σ

1.32

341×

10−

3

σ22.

2620

1×

10−

10

∆p

2=σ

1.50

400×

10−

5∆p

2=σ

3.73

590×

10−

5

4-5

σ24.

3816

0×

10−

9

( −0.7

5

1

) ∆p

1=σ

6.61

940×

10−

5∆p

1=σ

1.64

422×

10−

4

σ21.

0480

0×

10−

9∆p

2=σ

3.23

730×

10−

5∆p

2=σ

8.04

120×

10−

5

6-7

σ21.

9983

3×

10−

4

( −0.7

5

1

)∆p

1=σ

1.41

36×

10−

2∆p

1=σ

3.51

136×

10−

2

σ24.

5558

0×

10−

9∆p

2=σ

6.74

970×

10−

5∆p

2=σ

1.67

658×

10−

4

8-9

σ21.

9996

1×

10−

3

( −0.7

8

1

) ∆p

1=σ

4.47

170×

10−

2∆p

1=σ

1.11

107×

10−

1

σ26.

2461

6×

10−

9∆p

2=σ

7.90

330×

10−

5∆p

2=σ

1.96

313×

10−

4

Tab

le8:

Resultsforeigenvalues,eigenvectorsan

d1σ

,2σerrors

fortheredshiftbinsof

the

rstclass.

73zbin

Fisher

matrix

covarian

cematrix

0.5-1−

1 σ2

( 1.19

149×

1010

2.55

435×

109

2.55

435×

109

1.3

0848×

109

)σ

2

( 1.1

8280×

10−

10−

3.7

2116×

10−

11

−3.

7211

6×

10−

11

4.03

196×

10−

11

)

0.5-1.5

−1 σ2

( 9.09

966×

109

3.08

834×

109

3.08

834×

109

1.20

532×

109

)σ

2

( 8.4

2796×

10−

10−

2.1

5946×

10−

9

−2.

1594

6×

10−

96.

3627

7×

10−

9

)

0.5-2

−1 σ2

( 7.71

095×

109

2.54

409×

109

2.54

409×

109

1.01

934×

109

)σ

2

( 7.3

4568×

10−

10−

1.8

3335×

10−

9

−1.

8333

5×

10−

95.

5567

6×

10−

9

)

0.5-2.5

−1 σ2

( 7.40

422×

109

2.45

966×

109

2.45

966×

109

1.02

721×

109

)σ

2

( 6.6

0248×

10−

10−

1.5

8096×

10−

9

−1.

5809

6×

10−

94.

7591

2×

10−

9

)

0.5-3

−1 σ2

( 6.54

624×

109

1.96

505×

109

1.96

505×

109

7.44

840×

108

)σ

2

( 7.3

4216×

10−

10−

1.9

3702×

10−

9

−1.

9370

2×

10−

96.

4528

7×

10−

9

)

0.5-4

−1 σ2

( 5.32

846×

109

1.58

148×

109

1.58

148×

109

5.08

233×

108

)σ

2

( 9.9

2731×

10−

10−

3.0

8911×

10−

9

−3.

0891

1×

10−

91.

1580

1×

10−

8

)

0.5-5

−1 σ2

( 5.76

244×

109

1.47

316×

109

1.47

316×

109

4.38

142×

108

)σ

2

( 1.2

3569×

10−

9−

4.1

5475×

10−

9

−4.

1547

5×

10−

91.

6251

8×

10−

8

)

0.5-6

−1 σ2

( 5.56

335×

109

1.32

943×

109

1.32

943×

109

3.34

412×

108

)σ

2

( 3.5

9300×

10−

9−

1.4

2837×

10−

8

−1.

4283

7×

10−

85.

9773

9×

10−

8

)

Tab

le9:

Fisher

matrixan

dcovarian

cematrixfortheredshiftbinsof

thesecondclass.


zbins

eigenvalues

eigenvec

1σerrors

2σerrors

0.5-1

σ24.

0700

1×

10−

10

( −0.1

3

1

) ∆p

1=σ

2.01

980×

10−

5∆p

2=σ

5.01

710×

10−

5

σ21.

1350

0×

10−

10

∆p

1=σ

1.06

540×

10−

5∆p

2=σ

2.64

630×

10−

5

0.5-1.5

σ27.

1071

8×

10−

9

( −0.3

5

1

) ∆p

1=σ

8.43

040×

10−

5∆p

2=σ

2.09

407×

10−

4

σ29.

8383

9×

10−

11

∆p

1=σ

9.91

900×

10−

6∆p

2=σ

2.46

380×

10−

5

0.5-2

σ26.

1746

1×

10−

9

( −0.3

4

1

) ∆p

1=σ

7.85

790×

10−

5∆p

2=σ

1.95

186×

10−

4

σ21.

1670

9×

10−

10

∆p

1=σ

1.08

030×

10−

5∆p

2=σ

2.68

350×

10−

5

0.5-2.5

σ25.

2980

5×

10−

9

( −0.3

4

1

) ∆p

1=σ

7.27

880×

10−

5∆p

2=σ

1.80

801×

10−

4

σ21.

2132

0×

10−

10

∆p

1=σ

1.10

150×

10−

5∆p

2=σ

2.73

600×

10−

5

0.5-3

σ27.

0472

1×

10−

9

( −0.3

1

1

) ∆p

1=σ

8.39

480×

10−

5∆p

2=σ

2.08

523×

10−

4

σ21.

3987

6×

10−

10

∆p

1=σ

1.18

270×

10−

5∆p

2=σ

2.93

770×

10−

5

0.5-4

σ21.

2415

5×

10−

8

( −0.2

7

1

) ∆p

1=σ

1.11

425×

10−

4∆p

2=σ

2.76

774×

10−

4

σ21.

5732

8×

10−

10

∆p

2=σ

1.25

430×

10−

5∆p

2=σ

3.11

560×

10−

5

75zbins

eigenvalues

eigenvec

1σerrors

2σerrors

0.5-5

σ21.7

3240×

10−

8

( −0.2

6

1

) ∆p

1=σ

1.3

1621×

10−

4∆p

1=σ

3.2

6939×

10−

4

σ21.

6279

1×

10−

10

∆p

2=σ

1.2

7590×

10−

5∆p

2=σ

3.1

6930×

10−

5

0.5-6

σ26.3

1969×

10−

8

( −0.2

4

1

) ∆p

1=σ

2.5

1390×

10−

4∆p

1=σ

6.2

4440×

10−

4

σ21.

7001

2×

10−

10

∆p

2=σ

1.3

0390×

10−

5∆p

2=σ

3.2

3880×

10−

5

Tab

le10:Resultsforeigenvalues,eigenvectorsan

d1σ,

2σerrors


the

secondclass.

zbin

Fisher

matrix

covarian

cematrix

1−

5−

1 σ2

( 1.03

642×

1010

4.9

8169×

109

4.9

8169×

109

2.5

2532×

109

) σ2

( 1.8

6286×

10−

9−

3.67

486×

10−

9

−3.6

7486×

10−

97.6

4537×

10−

9

)Tab

le11:Fisher

matrixan

dcovarian

cematrixfortheredshiftbinsof

thethirdclass.

zbin

eigenvalue

eigenvec

1σerrors

2σerrors

1−

5σ

29.4

3001×

10−

9

( −0.4

9

1

) ∆p

1=σ

9.7

1080×

10−

5∆p

1=σ

2.41

210×

10−

4

σ27.8

2262×

10−

11

∆p

2=σ

8.8

4500×

10−

6∆p

2=σ

2.19

690×

10−

5

Tab

le12:Resultsforeigenvalues,eigenvectorsan

d1σ,

2σerrors


the

thirdclass.

Bibliography

[1] A.R.Liddle, Einführung in die moderne Kosmologie ( Wiley-VCH, Wein-heim, 2009)

[2] A.Liddle, J.Loveday, The Oxford Companion to Cosmology (UniversityPress, Oxford, 2008)

[3] H.Goenner, Einführung in die Kosmologie (Spektrum, Heidelberg;Berlin; Oxford, 1994)

[4] W.J.Kaufmann, Galaxies and Quasars (W.H.Freeman and Co.,1979)

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Declaration

I hereby certify that this thesis is my own work, that it has not beenpreviously submitted to an examination oce in the identical or a similarform and that all information sources used have been referenced.

Bielefeld, April 10, 2014

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Acknowledgements

I would like to give my thanks to Prof. Dr. D.J. Schwarz for this in-teresting topic of my master thesis and for his excellent supervision. I ammost grateful to Stefan de Boer and Andrey Shikhov for the proofreading.A special thanks goes to Nick Diederich for his support and interesting dis-cussions during the whole study and to all my friends who accompanied meall the time, especially, to Anna Aguf, Rene Wegner and Nataliya Schrodke.I would like to express my deep gratitude to my family for their support: tomy parents and especially to my husband.

Precision Measurement of Angular Separation in Cosmology

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