Topics in Cosmology: Island Universes, Cosmological Perturbations and Dark Energy

Home , Cosmic time, Cosmological constant, Cosmological horizon, De Sitter universe, Edward Farhi, Ekpyrotic universe

TOPICS IN COSMOLOGY: ISLAND UNIVERSES, COSMOLOGICAL PERTURBATIONS AND DARK ENERGY

by SOURISH DUTTA

Submitted in partial fulﬁllment of the requirements

for the degree Doctor of Philosophy

Department of Physics

CASE WESTERN RESERVE UNIVERSITY

August 2007 CASE WESTERN RESERVE UNIVERSITY

SCHOOL OF GRADUATE STUDIES

We hereby approve the dissertation of

______

candidate for the Ph.D. degree *.

(signed)______(chair of the committee)

______

(date) ______

*We also certify that written approval has been obtained for any proprietary material contained therein. To the people who have believed in me. Contents

Dedication iv

List of Tables viii

List of Figures ix

Abstract xiv

1 The Standard Cosmology 1 1.1 Observational Motivations for the Hot Big Bang Model ...... 1 1.1.1 Homogeneity and Isotropy ...... 1 1.1.2 Cosmic Expansion ...... 2 1.1.3 Cosmic Microwave Background ...... 3 1.2 The Robertson-Walker Metric and Comoving Co-ordinates ...... 6 1.3 Distance Measures in an FRW Universe ...... 11 1.3.1 Proper Distance ...... 12 1.3.2 Luminosity Distance ...... 14 1.3.3 Angular Diameter Distance ...... 16 1.4 The Friedmann Equation ...... 18 1.5 Model Universes ...... 21 1.5.1 An Empty Universe ...... 22 1.5.2 Generalized Flat One-Component Models ...... 22 1.5.3 A Cosmological Constant Dominated Universe ...... 24 1.5.4 de Sitter space ...... 26 1.5.5 Flat Matter Dominated Universe ...... 27 1.5.6 Curved Matter Dominated Universe ...... 28 1.5.7 Flat Radiation Dominated Universe ...... 30 1.5.8 Matter Radiation Equality ...... 32 1.6 Gravitational Lensing ...... 34 1.7 The Composition of the Universe ...... 39

v 1.7.1 Measuring the Total Matter Content ...... 39 1.7.2 Ordinary and Dark Matter ...... 42 1.7.3 Supernovae and Cosmic Acceleration ...... 45 1.7.4 Cosmic Microwave Background Anisotropies ...... 49 1.8 Shortcomings of the Standard Cosmology ...... 58 1.8.1 The Flatness Problem ...... 58 1.8.2 The Entropy Problem ...... 59 1.8.3 The Horizon Problem ...... 59 1.8.4 The Monopole Problem ...... 60 1.9 Inflation ...... 62 1.9.1 Inflation and the Problems of the Standard Cosmology . . . . 62 1.9.2 The Dynamics of Inflation ...... 64 1.9.3 The Slow Roll Parameters ...... 66 1.9.4 Models of Inflation ...... 67 1.9.5 Issues with Inflation ...... 69

2 Island Cosmology 76 2.1 Introduction ...... 76 2.2 The Model ...... 77 2.3 NEC violations in de Sitter space ...... 80 2.4 Extent and duration of NEC violation ...... 82 2.5 Likelihood – the role of the observer ...... 87 2.6 The NEC violating ﬁeld ...... 89 2.7 Assumptions ...... 91 2.8 Conclusions ...... 94

3 Perturbation Spectra 98 3.1 Introduction ...... 98 3.2 The Theory of Cosmological Perturbations ...... 98 3.2.1 The metric perturbations ...... 99 3.2.2 Gauge Issues in Cosmology ...... 101 3.2.3 The Comoving Curvature Perturbation ...... 103 3.2.4 The Power Spectrum ...... 104 3.2.5 The Equations of Motion ...... 106 3.2.6 Quantum Theory of Cosmological Perturbations ...... 108 3.3 Diﬃculties in computing perturbations from Island Cosmology . . . . 112 3.3.1 Classical vs Quantum Fields ...... 112 3.3.2 Backreaction on spacetime ...... 113 3.4 Perturbations in a Spectator Field ...... 117 3.5 A Classical Treatment of Island Cosmology ...... 120

vi 3.5.1 Introduction ...... 120 3.5.2 The de Sitter phase ...... 121 3.5.3 The Phantom Phase ...... 122 3.5.4 The Radiation Dominated FRW Phase ...... 125 3.5.5 Calculational Strategy ...... 125 3.6 Spectrum from a Classical Treatment ...... 127 3.6.1 vk in the de Sitter and FRW phases ...... 127 3.6.2 vk in the Phantom Phase ...... 128 3.6.3 Calculation of Unknown Constants ...... 135 3.6.4 Determination of the Scalar Power Spectrum ...... 138 3.6.5 Determination of the Tensor Power Spectrum ...... 138 3.7 Conclusion ...... 140

4 Dark Energy Voids 144 4.1 Introduction ...... 144 4.2 What is causing the acceleration? ...... 145 4.3 The model ...... 147 4.3.1 Linearization ...... 150 4.3.2 Potential ...... 153 4.3.3 Initial conditions ...... 153 4.4 Results ...... 155 4.4.1 Density contrasts ...... 155 4.4.2 Equation of state ...... 160 4.5 Discussion ...... 163 4.5.1 Void formation ...... 163 4.5.2 Generality ...... 165 4.6 Conclusions ...... 166

Bibliography 172

vii List of Tables

3.1 Scalar-Vector-Tensor decomposition of metric perturbations . . . . . 100

viii List of Figures

1.1 The spectrum of the CMB, as seen by COBE. http://lambda.gsfc. nasa.gov/product/cobe/firas_image.cfm ...... 4 1.2 Gravitational lensing geometry. The lens distorts the ”true” angles β (that would have been seen without any lensing effects) into the angles θ. Source [15] ...... 34 1.3 A flow chart showing the classifications of Supernovae http://rsd-www. nrl.navy.mil/7212/montes/snetax.html...... 46 1.4 top panel: A Hubble diagram made from data from both the Supernova Cosmology Project and the High-z Supernova Search Team taken from [13].bottom panel The residual of the distances relative to a ΩM = 0.3, ΩΛ = 0.7 Universe...... 50

1.5 Best ﬁt regions in the (ΩM ,ΩΛ) plane for data from both the Super- nova Cosmology Project and the High-z Supernova Search Team. The agreement of the two experiments is remarkable. Source [13] . . . . . 51 1.6 The Dipole Anisotropy in the CMB as seen by COBE http://map. gsfc.nasa.gov/m_uni/uni_101Flucts.html...... 52 1.7 The CMB spectrum once the dipole is subtracted. http://map.gsfc. nasa.gov/m_uni/uni_101Flucts.html...... 53 1.8 The WMAP three-year power spectrum (in black) together with data from other recent experiments measuring the CMB angular power spectrum. Taken from [44] ...... 56 1.9 A plot of length scale vs (logarithmic) scale factor in an FRW cosmology. The blue line shows the evolution of the Hubble scale. The red lines show the evolution of physical scales. Since the Hubble length evolves faster than the physical scales, sub-Horizon modes have never been in causal contact prior to Horizon entry...... 61 1.10 Physical scales entering the horizon at the time of last scattering have been in causal contact before...... 63

ix 2.1 Sketch of the behavior of the Hubble length scale with conformal time, η, in the Island model, and the evolution of fluctuation modes. At early times, inflation is driven by the presently observed dark energy, assumed to be a cosmological constant. As the cosmological constant is very small, the Hubble length scale is very large – of order the present horizon size. Exponential inflation in some horizon volume ends not due to the decay of the vacuum energy as in inflationary scenarios but due to a quantum fluctuation in the time interval (ηi, ηf ) that violates the null energy condition (NEC). The NEC violating quantum fluctuation causes the Hubble length scale to decrease. After the fluctuation is over, the universe enters radiation dominated FRW expansion, and the Hubble length scale grows with time. The physical wavelength of a −1 quantum fluctuation mode starts out less than HΛ at some early time ηi. The mode exits the cosmological horizon during the NEC violating fluctuation (ηexit) and then re-enters the horizon at some later epoch (ηentry) during the FRW epoch (The modes are drawn as straight lines for illustrative purposes only, they actually grow in proportion to the scale factor)...... 78

2.2 We show a classical de Sitter spacetime for conformal time η < ηP , that transitions to a faster expanding classical de Sitter spacetime for η > ηQ. The inverse Hubble size is shown by the white region. A bundle of ingoing null rays originating at point a is convergent initially but becomes divergent in the superhorizon region at point b. This can only occur if the NEC is violated in the region η ∈ (ηP , ηQ). In the quantum domain, a classical picture of spacetime may not be valid and this is made explicit by the question marks...... 83

2.3 A spacetime diagram similar to that in Fig. 2.2 but one in which the NEC violation occurs over a sub-horizon region (shaded region in the diagram). Now the null ray bundle from a to b goes from being converging (within the horizon) to diverging (outside the horizon). However, it does not encounter any NEC violation along its path, and this is not possible as can be seen from the Raychaudhuri equation. Since the ingoing null rays are convergent as far out as the point P , the size of the quantum domain has to extend out to at least the inverse Hubble size of the initial de Sitter space. Therefore the NEC violating patch has to extend beyond the initial horizon...... 85

x 4.1 The evolution of the DDE overdensity δφ at the center of the matter perturbation, r = 0, with redshift (1+z).The scale of the perturbation is σHi = 0.01, and the mass is m/H0 = 1. Initially homogenous, the DDE develops an underdensity at late times in response to the matter perturbation...... 156 4.2 Same as Figure 4.1, with the y-axis on a logarithmic scale. The DDE tends to cluster initially, but eventually forms a void. The kink in the plot signiﬁes the change-over from positive to negative perturbation. . 156

4.3 Logarithmic profiles of the matter density contrast log10 |δm| and the dark energy density contrast log10 |δφ|, at three different redshifts. Solid lines denote the DDE profiles, and dotted lines denote the matter profiles.158 4.4 DDE density contrast δφ at the center of the matter perturbation r = 0 as a function of the redshift (1 + z) for fixed mass m/H0 = 1 and different initial matter perturbations’ widths. The larger the initial matter perturbation, the stronger is the void. The curves of σHi = 0.01 (dashed) and 0.1 (solid) almost overlap. The figure zooms on late times, z < 3...... 159 4.5 Same as Figure 4.4 with the y-axis on a logarithmic scale. The shorter scales start evolving at later times than the longer scales, but their evolution is faster. The curves of σHi = 0.01 (dashed) and 0.1 (solid) almost overlap...... 159 4.6 The DDE density contrast δφ at the center of the matter perturbation, r = 0, against redshift (1+z) for σHi = .01 and three different masses. The figure zooms on late times, z < 7...... 161 4.7 Same as Figure 4.6 with the y-axis on a logarithmic scale. The perturbation’s is extremely sensitive to the mass scale...... 161 4.8 Plot of w1 vs r for three different redshifts...... 162 4.9 Plot of % deviation in w vs r at three different redshifts...... 162 4.10 Percentage variation of the local Hubble parameter at three different redshifts ...... 163 4.11 Profile of δφ¨ at three different redshifts ...... 164 4.12 The equivalent of figure 4.3, with the double exponential potential, equation (4.33)...... 166 4.13 The equivalent of figure 4.8, with the double exponential potential, equation (4.33)...... 167

xi Acknowledgements

I wish to acknowledge all the wonderful people in my life, in particular: My mother

Sharmila, who among many, many other things was my ﬁrst and best teacher of physics - who always took the time to answer all my science-related questions when

I was little, and fostered in me an inalienable spirit of inquiry; my father Shyamal, who again among many other things, was my ﬁrst and best teacher of writing, and who taught me, through the exemplary life that he led, the virtues of perseverance, dedication, sincerity and integrity; my wonderful little sister Arundhati, a multi- faceted, multi-talented and superluminary individual, an over-achieving powerhouse perennially brimming with potential, who is a constant source of joy and pride for me; my advisor Tanmay, who always saw the best in me, who was always available to help me with whatever I needed, who always did everything in his power to bring me the best opportunities, and who is that rare combination of exceptionally brilliant physicist and sterling human being; Irit, who instilled in me the conﬁdence to trust my ability to do research, and who helped me discover my taste for gravitational physics and numerical analysis; Harsh, whose depth of understanding and gift for exposition will always remain an inspiration for me; Punam, whose warmth and kindness have made my years in Cleveland exceedingly pleasant and have left me

xii with some truly delightful memories; and ﬁnally, my love Maia who is one of the greatest blessings in my life, who epitomizes love and compassion and sincerity and gentleness, who every day inspires me to be a better human being, and without whose constant encouragement (and persistent prodding!) this thesis would never have been completed.

xiii Topics in Cosmology: Island Universes, Cosmological Perturbations and Dark

Energy

Abstract

SOURISH DUTTA

This thesis is a report on the research I have done over the last three years. I begin by reviewing the Standard Big Bang cosmology and the theory of inflation. I then describe Island Cosmology, an alternative theory of cosmic origin, which is based on a different hypothesis, and can create the observed universe without requiring an inflationary phase. After fully describing this model, I move on to the subject of computing perturbation spectra. The theory of cosmological perturbations, as it applies to calculating the perturbation spectrum from inflation, is reviewed next. Computing the perturbation spectrum in Island Cosmology involves several challenges, which I then discuss and also review progress made so far in computing the spectrum. In the

ﬁnal part of this thesis I describe my research in gravitational collapse in the presence of a dynamical dark energy component (DDE). I review the very interesting result that in linear regime, the collapsing matter induces the formation of DDE voids, or regions of underdensity. I describe the physics behind the formation of these voids, as well as possible observational consequences.

xiv Chapter 1

The Standard Cosmology

1.1 Observational Motivations for the Hot Big Bang

Model

The Standard Cosmology, or the “Hot Big Bang Model” is based on three pillars of observational evidence, namely, homogeneity and isotropy, Hubble expansion and the cosmic microwave background. I brieﬂy explain each of these below:

1.1.1 Homogeneity and Isotropy

Also known as the Cosmological Principle, this is the assumption that there is neither a special location in the Universe (homogeneity) nor a special direction (isotropy).

Studies of large scale structure show that the Universe is considerably heterogeneous up to the scale of galactic superclusters, ﬁlaments and great voids. However, on scales larger than 100 Mpc the Universe appears to be homogeneous and isotropic (see, for

1 example, [1] for a comprehensive discussion of the evidence in support of homogeneity and isotropy). 1

1.1.2 Cosmic Expansion

It is found that the light from most galaxies is redshifted, and the degree of redshift is proportional to the distance of the galaxy. First discovered by Edwin Hubble in

1929, this relationship is known as the famous ”Hubble Law”:

H z = 0 r (1.1) c where z and r stand for the redshift and distance of a galaxy respectively. H0 is called the Hubble constant, which is actually a slowly varying function of time, and the subscript 0 denotes its value at the present time. In what follows, we work in units such that the speed of light c = 1.

The Hubble law clearly indicates that the Universe is expanding, or in other words, the observed redshifts of galaxies are actually Doppler shifts. The distance between any two objects (which are not interacting gravitationally or otherwise) grows in proportion to a function of time known as the scale factor a(t), which sets the scale of the geometry of space. As I explain in 1.2, I will follow the convention of taking the scale factor to be dimensionless.

1Interestingly, recent analysis of the Wilkinson Microwave Anisotropy Probe (WMAP) [2] seems to indicate that the quadrupole and the octopole of the Cosmic Microwave Background seem to have a strong correlation with the orientation of the ecliptic plane and its motion, a result that seems at odds with the Copernican Principle. However, I will not discuss this highly interesting result any further in this report.

2 Using the classical non-relativistic form of the Doppler shift z = v (where v is the radial velocity of the source with respect to us, the observers) one can re-express

Hubble’s law as follows:

v = H0r (1.2)

Obviously, owing to the non-relativistic approximation, this form of the Hubble law is valid only for small scales where the expansion velocity is small.

Also, it is easy to show [1] that the redshift and the scale factor are related as

1 (1 + z) ∝ (1.3) a

1.1.3 Cosmic Microwave Background

The Universe is currently immersed in a bath of radiation, at a temperature of T0 = 2.725 ± 0.001K which peaks in the microwave range at a frequency of 160.2 GHz, corresponding to a wavelength of 1.9mm [3]. This background of microwave radiation, called the Cosmic Microwave Background, has a spectrum that is blackbody within

3.4 × 10−8 erg cm−2 s−1 sr−1 over the frequency range from 2 to 20 cm−1 [4]. Figure

1.1 shows the spectrum of the CMB as measured by the FIRAS instrument on the

COBE. The data matches the curve almost exactly, with error bars being smaller than the width of the curve itself.

The CMB is one of the strongest conﬁrmations of the Hot Big Bang model. In fact the Hot Big Bang model provides a natural explanation for the CMB, as a relic of an immensely hot and dense phase in the history of the Universe. Very brieﬂy, the

3 Figure 1.1: The spectrum of the CMB, as seen by COBE. http://lambda.gsfc. nasa.gov/product/cobe/firas_image.cfm

4 argument is as follows:

The mere fact that the CMB is blackbody radiation strongly indicates that it must be a relic of a phase where the constituents of the Universe were in thermal equilibrium. Since the Universe is not in equilibrium now, the assumption of a prior equilibrium state is the only way one can explain the blackbody nature of the CMB.

During this early equilibrium phase, very frequent Thompson scattering between the photons and the baryons and electrons caused the mean free paths of the photons to be small - making the Universe opaque. As the Universe expanded, the plasma cooled adiabatically. Approximately 380,000 years after the Big Bang (at z = 1088), the temperature of the plasma dropped to 3000K, and electrons and protons could combine to form neutral Hydrogen atoms, or in other words, the baryonic component of the Universe went from ionized to neutral. This event is called recombination.

Shortly after this, the rate of expansion of the Universe exceeded the rate of interaction between the photons and the electrons (this is called decoupling). After decoupling, the photons started free-streaming through the Universe - redshifting and cooling with the expansion of the Universe - and today make up the CMB.

The drop in temperature by a factor of 1100 of the radiation background from

3000K when the photons started free-streaming (at z = 1088) to 2.73K today, follows directly from the adiabatic expansion of the Universe, and it can be shown that

T (t) ∝ a(t)−1 (1.4)

(In § 1.5.7 I will formally derive this relationship and show that it is not exact, but is a good approximation). In other words, the factor of 1100 drop in the temperature

5 of the radiation background is accompanied by a 1100-fold increase in the cosmic

scale factor.

Perhaps the most striking feature of the cosmic microwave background is its very

high degree of isotropy (the anisotropy is of the order of 10−5). Incidentally, this

isotropy is also a measure of the homogeneity of the Universe at the epoch of last

scattering. I will discuss the CMB anisotropy in far greater detail in § 1.7.4.

Taken together, the above observations lead to the conclusion that the Universe started out as an immensely hot and dense plasma which has been expanding and cooling ever since. This is the crux of the Hot Big Bang Model. In the rest of this chapter I will provide a brief review of this model, highlighting aspects of it which are relevant to my research. This chapter is based on the following excellent books and reviews which have helped me immensely in developing my understanding of cosmology [1, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]

1.2 The Robertson-Walker Metric and Comoving

Co-ordinates

Guided by the homogeneity and isotropy of our Universe we seek a solution that is spatially homogeneous and isotropic, and evolves in time. The Friedman-Robertson-

Walker (FRW) metric ﬁts this description perfectly, as it describes a Universe meeting the requirements of homogeneity and isotropy, and where lengths can arbitrarily expand (or contract) with time.

The general form of the metric, in terms of a spherical polar coordinate system,

6 is the following:

ds2 = dt2 − a(t)2 dr02 + S(r0)2dΩ2 (1.5)

Here r0 is the radial co-ordinate, θ and φ are the polar and azimuthal angles, and t denotes time. a(t) is the scale factor introduced in 1.1 but in this particular form of the metric it has dimensions of length. Also dΩ2 = dθ2 + r02dφ2 and the function

S(r0) is determined by the curvature of space:

 R sin(r0/R) positive curvature   0  S(r ) = r0 ﬂat space (1.6)    R sinh(r0/R) negative curvature

Here, R = a(t0) is the scale factor at the present epoch. Note that S(r) only determines the curvature of space locally and not globally, which is what one would expect from a local theory such as General Relativity. The global topology need not be the same as the local topology. For example, a locally

ﬂat topology might be part of either an inﬁnite plane, or a 3-torus.

Making a transformation to a diﬀerent radial coordinate r = S(r0), and rescaling the scale factor by R to make it dimensionless, the FRW metric takes the following form:

dr2 ds2 = dt2 − a2(t) + r2dΩ2 (1.7) 1 − kr2 where k denotes the spatial curvature of the Universe.

For the rest of this chapter I will let r0 denote the radial coordinate used in (1.5)

7 and r denote the radial coordinate in (1.7).

The rate at which the scale factor changes determines the speed at which the

Universe expands (or contracts). This is measured by the Hubble parameter H(t) deﬁned by a˙ H(t) = (1.8) a

Using the approximate distance redshift relationship (1.2), one could in principle measure the present value of the Hubble parameter, if the distances to galaxies were known accurately. However, the value of the Hubble parameter is still subject to some uncertainty characterized by the parameter h, and is usually expressed as follows:

h H = 100h kms−1Mpc−1 ' Mpc−1 (1.9) 0 3000

Observations suggest that h lies between 0.5 and 0.8, and the best present day

+.04 measurement is .73−.03 [3]. The inverse of the Hubble parameter sets a scale for the age of the Universe (the Hubble time) , as well as its spatial width (the Hub- ble distance). The present Hubble time and Hubble distance are 9.78h−1Gyr and

2998h−1Mpc respectively [9].

The second derivative of the scale factor measures the rate at which the Universe accelerates. This is usually characterized by the quantity:

aa¨ q = − (1.10) 0 a˙ 2 t=t0

Instead of the function S(r), curvature is now signiﬁed by the parameter k, which takes on values of (−1, 0, +1) for spaces of constant positive, zero or negative spatial

8 curvature. k also sets the curvature scale rcurv of the Universe which is deﬁned as:

−1/2 rcurv = a(t)|k| (1.11)

This is the physical distance at which effects of curvature start becoming signifi- cant. From (1.7), it is clear that when the physical distance rcurv, spacetime looks flat.

It is sometimes convenient to use conformal time (η) , which is deﬁned by dη = dt/a(t). Re-writing the FRW metric in terms of conformal time, it is easy to see that this metric is conformally ﬂat, that is, it has the same form as the Minkowski line element, up to a conformal factor2:

dr2 ds2 = a2(η) dη2 − + r2dθ2 + r2 sin2 θdφ2 (1.12) 1 − kr2

There are two sets of transformations [6] which leave this metric invariant. These

are:

1. The rigid rotations

0i i j x = Rjx (1.13)

where xi are Cartesian coordinates and R is a rotation matrix, and

2. The quasi-translations which translate the origin to the point deﬁned by the

vector a n h x.aio x0 = x + a 1 − kx21/2 − 1 − 1 − ka21/2 (1.14) a2

2The conformal ﬂatness of this metric can also be rigorously tested by computing its Weyl curvature, which turns out to be zero.

9 The second transformation implies that any point whose coordinates are ﬁxed with respect to the origin, can equally serve as the origin, and that the Universe will look the same from that point as it does from the origin. In other words, in an FRW spacetime, spatial homogeneity is guaranteed by isotropy at one location.

From the equation of motion for a particle in a gravitational ﬁeld [6],(where τ is the proper time)

d2xλ dxµ dxν + Γλ = 0 (1.15) dτ 2 µν dτ dτ

µ , and from the fact that the Christoffel symbol Γtt vanishes for the FRW metric, one can deduce that the trajectories of fixed x are geodesics. Typical galaxies follow geodesics, i.e., they are in free fall, and hence in this coordinate system they will have fixed coordinates. Hence this set of coordinates, aptly named comoving coordinates have the added benefit that typical galaxies have fixed spatial locations. Observers at rest in comoving co-ordinates are called comoving observers.

Only comoving observers ﬁnd the Universe isotropic. The fact that we see a dipole anisotropy in the CMB is because the motion of the earth is not comoving.

Borrowing an analogy from Weinberg [6], comoving co-ordinates can be thought of as a set of coordinates axes painted on the surface of a balloon. Regardless of whether the balloon expands or contracts, any given point on the balloon will have the same location with respect to the origin of these coordinates, since the axes expand or contract as well.

Any velocities with respect to comoving observers are called peculiar velocities and it can be shown that unless supported by interactions, peculiar velocities “red-

10 shift”, or decrease in proportion to the scale factor a(t). Furthermore, it is also easy to prove that the wavelength of light in an FRW Universe redshifts as well. Both of these points are described in detail in [1].

One can also understand comoving coordinates in the language of diﬀerential geometry as follows [15]. Homogeneity and isotropy require that our spacetime can be sliced into spacelike hypersurfaces which are maximally symmetric. This can be mathematically represented as R1 × Σ, where R1 is a one-dimensional Euclidean

manifold representing the time direction and Σ is a spherically symmetric three-

manifold. Let us deﬁne a set of coordinates (u1, u2, u3) and a maximally symmetric

three-metric γij(u) on Σ, such that the line element on Σ can be written as

2 i j dσ = γij(u)du du

If the spacetime metric can also be written in the form

ds2 = dt2 − a2(t)dσ2 (1.16)

which is free of cross-terms dtdui and the coeﬃcient of dt2 is independent of time,

then these coordinates (ui, t) are called comoving coordinates.

1.3 Distance Measures in an FRW Universe

The distance between two points in an expanding Universe is a subtle issue. An

idealized distance measure is the proper distance. However, as I explain below the

proper distance is not something one can actually measure.

11 Very short distances, such as distances to objects within the solar system, can be measured by bouncing radar signals of these objects. However, radar echoes become too weak to detect for distances on the order of 10 AU. Within our galaxy, distances can be determined using trigonometric parallax. The principle behind this that the apparent position of an astronomical object will shift if one changes the point of observation. Using this method the Hipparcos Satellite has determined distances to

2.5 million stars in our galaxy [16]. However, on cosmological scales (≈ 100) Mpc, trigonometric parallaxes become too small to measure.

On scales larger than our galaxy, two distance measures are available, called angular diameter distance or the luminosity distance. However, both of these can be shown to be approximately equal to the proper distance for small redshifts.

Below, I brieﬂy review these measures below.

1.3.1 Proper Distance

0 Consider an object located at the point E (re(or re), θ, φ) and an observer viewing

0 this object from the point O (r0(or r0), θ, φ). The proper distance of E measured from

O at time t0 can be deﬁned to be the length of a spatial geodesic between the two points when the scale factor is ﬁxed at the value a(t0). From the FRW metrics (1.5) and (1.7), one can see that this would be given by

Z r0 dr 0 0 dprop(t0) = a(t0) √ = a(t0)(r − r ) (1.17) 2 0 e re 1 − kr

The proper distance can also be intuitively understood as follows: imagine a line of comoving observers between E and O each separated by co-ordinate distance

12 0 δr(or δr ). Suppose at the present time t0, each observer measures the physical distance to the next observer. The distance δs(t) measured by any observer is given

a(t0) 0 δs(t0) = √ δr = a(t0)δr 1 − kr2

If one adds up these inﬁnitesimal sub-distances, it is clear that one gets the above

expressions for proper distance (1.17).

Now suppose E sends a light signal to O. The signal is emitted at t = te and

received at t = t0. Since light follows null geodesics, from (1.7) we can rewrite the expression for the proper distance as follows

Z t0 dt0 dprop(t0) = a(t0) 0 (1.18) te a(t )

The time diﬀerence (t0 − te) is called the lookback time. The expression (1.18) would have been practically useful if the exact functional

form of the scale factor were known, as well as the lookback time. In practice the

best we can do is approximately determine the scale factor in the form of a Taylor

expansion about the present time t0

1 a(t) = a(t ) + a0(t )(t − t ) + a00(t )(t − t )2 + ... 0 0 0 2 0 0

Plugging this into (1.18) and retaining the two lowest order terms in the lookback time, one gets H d (t ) ' (t − t ) + 0 (t − t )2 (1.19) prop 0 0 e 2 0 e

Using (1.3), one can express the lookback time in terms of the redshift, and hence

13 write an expression for the proper distance of an object at a redshift z:

1 1 + q0 2 dprop(t0, z) = z − z (1.20) H0 2

1.3.2 Luminosity Distance

The luminosity distance of a celestial object is a distance derived by comparing its observed luminosity to its absolute luminosity, which is assumed to be known.

Suppose we have a ”standard candle” (an object of known luminosity) with a measured ﬂux f. The luminosity distance is deﬁned as

L 1/2 d = (1.21) L 4πf

In a static Euclidean Universe, the above quantity would give the proper distance to the object. However, the fact that the Universe need not be Euclidean introduces

2 complications:

1. Spatial curvature: According to (1.5), the curvature of space changes the proper

0 0 0 area of a sphere centered at the object and with a radius of ρ = |re − r0| from

2 02 2 0 2 4πa(t0) ρ to 4πa(t0) S(ρ ) . For positive curvature the area of a sphere is greater than the ﬂat space case, and for negative curvature the area is lesser.

2. Cosmic expansion: The expansion of the Universe causes photons to redshift,

i.e., if a photon of wavelength λe emitted at time te is received now t0 and

14 measured to have a wavelength λ0, then we must have

a(t0) λ0 = λe = (1 + z)λe a(te)

which causes a corresponding shift in the energy of the detected photon:

E E = e (1.22) 0 1 + z

The expansion also causes photons emitted at time intervals δte to be received

at the detector at intervals δt0, where

a(t0) δt0 = δt0 = (1 + z)δte (1.23) a(te)

The net result of (1.22) and (1.23) is that the net ﬂux of power received at the detector is

L L f = 2 0 2 2 = 2 2 2 (1.24) 4πa (t0)S(ρ ) (1 + z) 4πa (t0)|re − r0| (1 + z)

From equations (1.21) and (1.24) we get the following expressions for the luminosity distance:

dL = a(t0)|re − r0|(1 + z) = dprop(1 + z)

Using the expression connecting the proper distance to the redshift (1.20), one can

15 express the luminosity distance in terms of the redshift:

1 (1 − q0) 2 dL ≈ z + z (1.25) H0 2

Hence in the limit that z → 0

z dL ≈ dprop ≈ (1.26) H0

1.3.3 Angular Diameter Distance

The angular diameter distance is a distance measure determined by computing the observed angular diameter of an object to its known diameter.

Suppose we have a ”standard yardstick” of known proper length l, subtending an angle δθ at O. The angular diameter distance dA is deﬁned by

l d = (1.27) A δθ

As in the previous case, this would give the proper distance to the object only if the

Universe is static and Euclidean.

To ﬁnd how this expression changes in curved and expanding space, consider a set of comoving coordinates whose origin is at O, according to which the two ends of the yardstick are located at (r0, θ, φ) and (r0, θ + δθ, φ).

From the FRW metric (1.5), the length of the object can be written as

0 l = δs = a(te)S(r )δθ = a(te)rδθ

16 which gives, according to (1.27)

a(t )S(r0) a(t )r d d = 0 = 0 = prop (1.28) A 1 + z 1 + z 1 + z

Again, using (1.20), the angular diameter distance for ﬂat space can be re-expressed as in terms of the redshift as

1 (3 + q0) 2 dA ≈ z − z (1.29) H0 2

Which implies, as for the luminosity distance, the angular diameter distance is practically identical to the proper distance for low redshifts.

z dA ≈ dprop ≈ (1.30) H0

For large redshifts, the angular diameter distance and the luminosity distance are generally different from each other and differ from the proper distance. The specifics of how they differ depends upon the particular model of matter-energy used.

The use of standard yardsticks and angular diameter distances in determining cosmological parameters is complicated by several practical difficulties. First of all, assigning an angle δθ to galaxies and clusters is difficult since these objects rarely have smooth luminosity distributions. Secondly neither galaxies nor clusters are static and isolated. Galaxies tend to grow with time as they merge with other galaxies. Clusters grow with time as well as more and more galaxies fall into them. Since angular diameter distances are defined such that they would have been proper distances in a static Euclidean Universe, they are useful in gravitational lensing calculations (see §

17 1.6).

Luminosity distances are somewhat more promising as cosmological probes and have been used extensively. Data from Type IA Supernovae, in combination from the

Cepheid data from the HST Key project, have played a key role in determining the

Hubble parameter.

Also, data from Supernovae and CMB anisotropies (see section 1.7) have allowed for the geometry of spacetime to be determined - they have shown that our Universe appears to be spatially ﬂat and dominated by vacuum energy.

1.4 The Friedmann Equation

The dynamical evolution of the FRW spacetime is governed by the Einstein equations, which follow directly from the action SEH + SM

1 Z √ S = − d4x −g (R + 2Λ) (1.31) EH κ Z X 4 √ SM = d x −gLﬁelds (1.32) all ﬁelds

SEH is the familiar Einstein-Hilbert action for general relativity and SM denotes the sum of the Lagrangian densities of all the ﬁelds. κ = 8πG, where G is Newton’s

2 √ constant and G = 1/mPl, the inverse Planck mass squared. −g is the determinant of the metric and Λ is a constant (which will later be identiﬁed as the cosmological constant).

Varying the action with respect to the metric gµν one obtains the Einstein equa-

18 tions 1 R − Rg = κT + Λg (1.33) µν 2 µν µν µν

Here Rµν and R stand for the Ricci tensor and scalar respectively, and Tµν stands for the energy-momentum tensor for all ﬁelds present. It is conventional to put the Λ

term on the right hand side, which allows it to be interpreted as a source of energy-

momentum of vacuum. The energy density associated with it is given by ρΛ = Λ/κ.

The simplest possible choice for Tµν which is in accordance with the symmetries of the FRW metric is that of a perfect ﬂuid with pressure p and density ρ.

Tµν = −pgµν + (p + ρ) uµuν (1.34)

Plugging (1.34) into the Einstein equations (1.33) one obtains the Friedman-

Lemaitre equations

a˙ 2 κρ k Λ H2 = = − + (1.35) a 3 a2 3 a¨ Λ κ = − (ρ + 3p) (1.36) a 3 6

(1.35) is called the Friedman equation and (1.36) is called the acceleration equation.

It is sometimes convenient to write the acceleration equation in terms of H as

follows: k H˙ = −4πG (p + ρ) + (1.37) a2

One more useful equation can be obtained, either by combining (1.35) and (1.36),

19 µν or from the energy conservation equation T;µ , or simply from the First Law of Ther- modynamics:

ρ˙ = −3H (p + ρ) (1.38)

The Friedman equation can be recast as

K = Ω − 1 (1.39) H2a2 tot

P where Ωtot = Ωi and Ωi = ρi/ρc.Ωi is the density parameter of the species i, i a dimensionless measure of density, expressed as a multiple of the critical density

2 −26 2 −3 ρc, defined as ρc = 3H /κ = 1.88 ∗ 10 h kg m [3]. Ωtot is the total density parameter, which is equal to the sum of the contributions of all the different fields P present in the model, as well as the vacuum: Ωtot = Ωi,. i

Combining (1.39) (at t = t0) and (1.35) we can get one more useful form of the Friedman equation: 2 H X Ωi,0 1 − Ωtot,0 = + (1.40) H2 a(t)3(1+wi) a(t)2 0 i An equation connecting the pressure p and density ρ of the ﬂuid is called the

equation of state:

p = wρ (1.41)

where w is the equation of state parameter. For pressureless matter w = 0, for

radiation w = 1/3 and for a cosmological constant w = −1/3.

20 Equation (1.39) makes it clear that Ωtot determines the curvature of the Universe:

k = +1 ⇒ Ω > 1

k = 0 ⇒ Ω = 1 (1.42)

k = −1 ⇒ Ω < 1

1.5 Model Universes

If the Universe has only one (spatially homogeneous) component, equations (1.35),(1.38) and (1.41) can be solved with appropriate boundary conditions to completely determine the pressure p(t), density ρ(t) and the scale factor a(t). Assuming a single component is obviously a oversimpliﬁcation since the Universe is clearly a mixture of several types of matter/energy. However, it is still useful to study such solutions as some of these components have individually dominated during diﬀerent phases of the Hot Big Bang evolution. For example, after nucleosynthesis the Universe passed through a radiation-dominated phase, followed by a matter-dominated phase and currently seems to be in a vacuum-energy dominated state.

In this section I will review a few simple one and two-component models (which can be analytically solved) and which help us understand diﬀerent epochs in the history of the Universe.

21 1.5.1 An Empty Universe

Let us start with the simplest solution - a Universe which has no matter or energy

in it (i.e. ρ = 0 and Λ = 0). From equation (1.35) we notice that there are two

possibilities:

1. A Universe with H(t) = k = 0, which is a static ﬂat spacetime.

2. A Universe with H(t) = t−1 and k = −1, an empty negatively curved universe

also called a Milne Universe.

A purely empty Universe is not very interesting from a physical point of view, but a Universe with a very small matter density can be approximated by it. However, observations clearly rule out this possibility for our Universe.

1.5.2 Generalized Flat One-Component Models

Now consider the general equation of state (1.41) with a constant w. From (1.38),

one obtains the relationship

ρ(t) ∝ a−3(1+w) (1.43)

Note that at early times, when a is small, if w > −1/3 the density term in the

Friedmann equation (1.35) will dominate the curvature term k/a2. This implies that

curvature can be neglected at early times, but can dominate at later times since it

redshifts slower than matter or radiation, unless of course the Universe is already

vacuum-dominated by then.

In the discussion below, we will neglect spatial curvature and deduce the properties

of a one-component Universe with a general equation of state parameter w.

22 For a spatially ﬂat Universe consisting of only a single component, one can integrate (1.35) to obtain (for w 6= −1)

2/3(1+w) a(t) = (t/t0) (1.44)

where the present time t0 (the age of the Universe) is given by

1 r 4 t0 = (1.45) (1 + w) 3κρ0

The energy density ρ scales as

t −2 ρ(t) = ρ0 (1.46) t0

The age of the Universe is related to the Hubble time by the relationship

2 t = H−1 (1.47) 0 3(1 + w) 0 implying that the Universe is younger or older than the Hubble time depending on whether w > −1/3 or w < 1/3.

The proper distance to an object from which light was emitted at a time t = te can be easily calculated from (1.18) to be (when w 6= −1/3)

" 1+3w # 3+3w 3(1 + w) te dprop = t0 1 − 1 + 3w t0 1 2 = 1 − (1 + z)−(1+3w)/2 (1.48) H0 (1 + 3w)

23 For a given point O in a spacetime, the particle horizon distance (or simply horizon distance for short) dh can be deﬁned as the farthest point in causal contact with O. An observer at O has no information about points more distant than the horizon distance because light has not had the time to reach O.

The distance to the horizon can be found from (1.48) in the limit z → ∞. Clearly, whether the horizon distance is finite or not depends on w. For w > −1/3 (which includes flat matter and radiation-dominated universes) the horizon distance is finite and is given by 1 2 dh = (1.49) H0 1 + 3w

On the other hand, if w < −1/3 (which includes a ﬂat cosmological constant dominated Universe), the horizon distance is inﬁnite.

1.5.3 A Cosmological Constant Dominated Universe

We now consider a Universe dominated by a cosmological constant, which is a species with w = −1.

If the cosmological constant Λ > 0, then from the Friedmann equation (1.35) we get

 q cosh Λ (t − t ) positive curvature (k = +1)  3 0  a(t)  q = exp Λ (t − t ) ﬂat space(k = 0) (1.50) a(t ) 3 0 0   q  Λ sinh 3 (t − t0) negative curvature (k = −1)

While apparently diﬀerent, these solutions all describe the same spacetime, de-

24 Sitter space, in diﬀerent coordinates.

If, on the other hand, the cosmological constant is negative, Λ < 0, then according

to the Friedmann equation the spatial curvature has to be negative as well. This is

called Anti-de Sitter space, and the scale factor behaves as

r ! a (t) Λ = sin − t (1.51) a(t0) 3

In the rest of this subsection we will study the properties of the ﬂat space solution in (1.50), which describes a Universe dominated by a cosmological constant.

First note that according to (1.38), the energy density of the w = −1 species, ρΛ is a constant. If Λ is interpreted as vacuum energy, then the constancy of the energy

density can be physically understood as the constant creation and annihilation of

virtual particle-antiparticle pairs.

The Friedmann equation (1.35) implies that

a˙ κΛ = ≡ H (1.52) a 3 0

which implies that the Hubble parameter is a constant (H = H0), and that space expands exponentially.

The proper distance at (t = t0) to an object with redshift z can be calculated from (1.18) to be

dprop(t0) = z/H0 (1.53)

The proper distance to the same object at (t = te) was obviously shorter by a factor

25 of (1 + z), i.e., 1 z dprop(t0) = (1.54) H0 1 + z

−1 In the limit z → ∞, dprop(t0) → ∞ but dprop(t0) → H0 . This implies that highly redshifted objects z >> 1 at actually located far beyond the Hubble distance at the time they are observed. This is simply a consequence of the existence of the MAS

(introduced in § 1.35): once an object moves outside the MAS (which coincides with the Hubble distance for an FRW cosmology) it can no longer be observed.

1.5.4 de Sitter space

The spacetime described above is called de Sitter space ([17, 18]). While the (r, θ, φ, t) coordinates together with the assumption of spatial ﬂatness make it easy to visualize the properties of this spacetime, in general, these coordinates do not span the entirety of the space, and ﬂatness is also not required.

A broader (albeit coordinate dependent) deﬁnition of de Sitter space is a maximally symmetric spacetime of positive curvature (k 6= 0). It might be visualized as a

4-dimensional hyperboloid

− u2 + w2 + x2 + y2 + z2 = α2 (1.55) which is embedded in a ﬁve-dimensional space having the metric

ds2 = −du2 + dw2 + dx2 + dy2 + dz2 (1.56)

Let a set of 4-dimensional coordinates (χ, θ, φ, t) be actually induced on the hyper-

26 boloid through the relations

u = α sinh(t/α)

w = α cosh(t/α) cos(r)

x = α cosh(t/α) sin(r) cos(θ) (1.57)

y = α cosh(t/α) sin(r) sin(θ) cos(φ)

z = α cosh(t/α) sin(r) sin(θ) sin(φ)

The induced metric on the hyperboloid therefore becomes

ds2 = −dt2 + α2 cosh2(t/α) dχ2 + sin2 (χ) dθ2 + sin2 (θ) dφ2 (1.58)

This metric clearly describes a spatial 3-sphere whose dimensions are governed by the t-dependent ”scale factor” cosh2(t/α). As t progresses from −∞ to +∞, the

sphere ﬁrst contracts, reaches a minimum radius at t = 0, and then expands. It

can be shown that this set of coordinates spans the entire space [19]. Clearly, the

coordinates in (1.50) are incomplete in the past.

We emphasize that this physical picture is purely an artifact of the choice of

coordinates, and there can be alternative descriptions which are equally valid.

1.5.5 Flat Matter Dominated Universe

The properties of such a Universe can be deduced directly from § 1.5.2 by setting

w = 0.

27 −3 2/3 The matter density and the scale factor evolve respectively as ρm ∝ a a(t) ∝ t . The proper distance to a galaxy of redshift z is given by

2 1 dprop(t0) = 1 − √ (1.59) H0 1 + z

implying a ﬁnite horizon size equal to 2/H0.

The age of the Universe is given by t0 = 2/(3H0).

1.5.6 Curved Matter Dominated Universe

A Universe ﬁlled with matter received considerable attention in the mid-twentieth

century. The fate of this Universe depends crucially on the total matter density Ωtot, as can be seen from the Friedmann equation (1.40):

2 H Ωtot,0 1 − Ωtot,0 2 = 3 + 2 (1.60) H0 a(t) a(t)

Since the Hubble term is squared, it is clear that the scale factor can have either a contracting (˙a < 0) or an expanding (˙a > 0) solution, and that the contracting

solution is the time reversal of the expanding solution.

First consider a positively curved Universe Ωtot > 1. From (1.60) it is clear that if the Universe starts out expanding from a small a (the ”Big Bang”), it will keep

expanding till it reaches a maximum scale factor aturn, and then start contracting till the scale factor vanishes and the matter density diverges (the ”Big Crunch”) . The

scale factor at turnaround (˙a = 0) can be easily deduced to be

28 Ωtot,0 aturn = (1.61) Ωtot,0 − 1

For a negatively curved Universe Ωtot > 1, (1.60) implies that if the Universe is

expanding at t = t0, it will continue to expand forever. Initially, the curvature term will be negligible and the Universe will behave like a ﬂat matter dominated Universe

(subsection 1.5.5). Later the curvature term will dominate and the Universe will

behave like the Milne Universe (subsection 1.5.1).

Interestingly, equation (1.60) does have analytical solutions for both the positive

and negative curvature cases, which allow for the above conclusions to be deduced

rigorously (see for example [1]).

For Ωtot > 1 the solution for the scale factor a(t) is given parametrically as

1 Ω a(θ) = tot,0 (1 − cos θ) 2 (Ωtot,0 − 1) 1 Ωtot,0 t (θ) = 3/2 (θ − sin θ) (1.62) 2H0 (Ωtot,0 − 1)

where θ runs from 0 to 2π. The above expression allows one to calculate the time duration τ between the ”Big Bang” and the ’Big Crunch”:

π Ωtot,0 τ = 3/2 (1.63) H0 (Ωtot,0 − 1)

For the negative curvature case (Ωtot < 1) the solution for the scale factor a(t) is given by

29 1 Ωtot,0 a (η) = 3/2 (cosh η − 1) 2 (Ωtot,0 − 1) 1 Ωtot,0 t (η) = 3/2 (sinh η − η) (1.64) 2 (Ωtot,0 − 1)

1.5.7 Flat Radiation Dominated Universe

The Friedmann equation (1.35) makes it clear that the early Universe was a hot and dense gas of relativistic particles. In addition, particles which make up matter now must have once been relativistic when the Universe was suﬃciently hot, and thereby contributed to radiation. The properties of such a Universe can be deduced by setting w = 1/3 in the equations in § 1.5.2.

The density of relativistic particles scales as ρ ∝ a−4 and the scale factor grows as a(t) ∝ t1/2. The extra factor of a diﬀerence from the matter dominated case can be physically understood as coming from the energy loss from redshift.

A galaxy at redshift z has a proper distance of

1 z dprop(t0) = (1.65) H0 (1 + z)

Taking the limit z → ∞ we note that there exists a ﬁnite horizon of size 1/H0.

Thermodynamics of a radiation-dominated Universe

During the radiation dominated phase photons and other relativistic species are in

thermal equilibrium at the same temperature with zero chemical potential. The

30 energies, number densities and pressures of all species can be readily computed from

thermodynamic formulae (see [1]). The total energy density can be written as

2 " 4 4# 2 π X Ti 7 X Ti π ρ = g + g T 4 ≡ g (T )T 4 (1.66) R 30 i T 8 i T 30 ∗ i=bosons i=fermions

Here only relativistic bosonic and fermionic species are being summed over, since

the contributions of non-relativistic species are exponentially smaller. We assume

that Ti ' T . The value of g∗ depends upon the particle physics model being used.

In the standard model of particle physics g∗ = 106.75 and can be as high as several hundred in more complicated models [9].

Using (1.66), and the time dependence of the energy density and scale factor as

described above, one can ﬁnd an expression linking the age of the Universe to its

temperature during the epoch of radiation domination:

1/2 90 −2 t = 2 T (1.67) 4π κg∗ (T )

Or, expressing t in seconds and T in MeV, one can rewrite the above as

t 2.4 T 1/2 = 1/2 (1.68) sec g∗ (T ) MeV

From the ﬁrst and second laws of thermodynamics, it is easy to show that (see [1]) under conditions of thermal equilibrium, the entropy per comoving volume S, given by

31 a3(p + ρ) S = (1.69) T

is a constant. Deﬁning the entropy density s as s = S/a3 = (p + ρ) /T . Since the entropy density is dominated by relativistic particles, it can also be expressed in a form similar to (1.66) as

2π2 s = g T 3 (1.70) 45 ∗S

T where g∗S = g∗ , but is numerically close to g∗ since all species are at the same Ti temperature.

3 3 The fact that g∗Sa T remains constant as the Universe expands shows that as long

1 as the number of degrees of relativistic species g∗S stays the same, T ∝ a . g∗S changes when entropy is produced, e.g.through a ﬁrst-order phase transition or the decoupling of a species, but even then the amount of entropy produced is small compared to the total entropy, and hence the inverse relationship between temperature and scale factor is a good approximation.

1.5.8 Matter Radiation Equality

The previous subsections have shown that density of radiation scales as a−4 and the density of matter scales as a−3. Initially radiation is dominant, but since it redshifts faster than matter, the latter comes to dominate at later times. This implies that the

Universe passes through a phase where matter and radiation have the same density.

This epoch in the history of the Universe is called the epoch of matter radiation

32 equality. In what follows, the subscript “eq” denotes the value of a quantity at the

time of equality.

The redshift at matter-radiation equality (zeq) is given by

Ωm,0 3 1 + zeq = ' 3 × 10 (1.71) Ωr,0

The temperature at matter-radiation equality is given by

3 Teq = T0 (1 + zeq) ' 9 × 10 K (1.72)

The Friedmann equation becomes

κρ a 3 a 4 H2 = eq eq + eq (1.73) 3 a a

This equation can be solved for the time behavior of the scale factor in terms of conformal time η [9]

a(η) √ η √ η 2 = (2 2 − 2) + 1 − 2 2 + 2 (1.74) aeq ηeq ηeq

where √ s (2 2 − 2) 3 ηeq = (1.75) aeq κρeq

For η ηeq the ﬁrst term in (1.74) dominates, giving the radiation-dominated

2 behavior a ∝ η . When η ηeq the second term dominates, giving the matter dominated behavior a ∝ η. Equation (1.74) shows that in the case of a Universe consisting

of both matter and radiation, the transition from radiation to matter domination is

33 Figure 1.2: Gravitational lensing geometry. The lens distorts the ”true” angles β (that would have been seen without any lensing eﬀects) into the angles θ. Source [15] smooth.

1.6 Gravitational Lensing

This is the phenomenon where the gravitational distortion of spacetime by a massive body like a galaxy causes the latter to act as a lens, creating a variety of different types of images of objects behind it. In mathematically treating gravitational lenses, one usually makes the assumption that they are thin-lens systems, where the bending of light occurs at a single distance, as shown in Fig. 1.2. The angles are thought of as 2-dimensional vectors on the sky.α ˆ is called the deflection angle, and ~α ≡ β~ − θ~ is called the reduced deflection angle.

Since lensing occurs in an expanding and possibly curved Universe, one needs to

34 choose distance measures carefully. It is clear that angular diameter distances are the most appropriate choice here, since the way these distances are deﬁned, ratios such as dLS/dS (where both are angular diameter distances) are not going to change either with expansion or curvature. All distances used in gravitational lensing are therefore angular diameter distances.

From the geometry of Fig. 1.2, one can write down the lens equation for a gravitational lens:

d β~ = θ~ − LS αˆ (1.76) dS

Consider the simplest case of a circularly symmetric lens of mass M. We know that the deﬂection angle for a photon traveling through a gravitational potential Φ is given by

Z ~ αˆ = 2 ∇⊥Φds (1.77)

It is easy to show that equation (1.76) reduces to

d 4GM β = θ − LS (1.78) dLdS θ

If the source is on-axis (β = 0), then the above equation has a unique solution:

r 4GMdLS θE = (1.79) dLdS

This implies that the images form a ring (the Einstein ring) around the source

35 with radius θE, which is also called the Einstein angle. The Einstein angle sets a scale for gravitational lensing and can be used for a rough estimate of the amount of

lensing by a given object. Measuring Einstein angles of lensing galaxies allows one

to determine masses of central parts of galaxies very accurately, and generally these

measurements are in good agreement with other independent observations.

If the source lies oﬀ-axis (β 6= 0), then one can solve (1.78) to get two image

locations, one inside and one outside the Einstein radius.

1 q θ = β ± β2 + 4θ2 (1.80) ± 2 E

In the case of lenses which cannot be treated as point masses, it is customary to

deﬁne a lensing potential by integrating the gravitational potential of the source over

past-directed geodesic paths emanating from the observer:

d Z ψ θ~ = 2 LS Φds (1.81) dLdS

The reduced lensing angle is now given by

~ ~α = ∇θψ (1.82)

Lensing can cause two eﬀects: convergence and shear, which can both be deﬁned from the lensing potential.

The convergence ζ, which measures the degree of focusing, is given by:

36 1 ζ = ∇2ψ (1.83) 2 θ

The shear γ which measures the shape distortion, and is given (in terms of carte-

sian components) by:

1 γ2 = (∂ ψ − ∂ ψ)2 + (∂ ψ)2 (1.84) 4 xx yy xy

The lens transforms an area element given by β~ into one described by θ~. This

transformation can be characterized by the magniﬁcation µ, which is the determinant

of the magniﬁcation tensor µij: ∂θi µ = (1.85) ij ∂βj

The magniﬁcation µ, can be expressed as a combination of the convergence and

shear as [15]

1 µ = (1.86) (1 − ζ)2 − γ2

Gravitational lensing eﬀects are classiﬁed into three categories:

Strong lensing

This is the strongest gravitational lensing eﬀect and occurs when the source lies within the Einstein radius of the lens. The lensing might cause an Einstein ring or (more likely) a series of tangential or radial arcs. From a given set of images, it might also be possible to reconstruct the lensing potential (a classic example of this is in [9801193]).

Also, if the source is time-varying, the multiple images will vary in time too, but

37 not exactly together, since light takes diﬀerent times to reach us in forming diﬀerent images. The time lags can be used, in principle, to measure the Hubble parameter.

However, the success of this method depends on the accuracy with which one can reconstruct the lensing potential.

Weak lensing

If the source is located more than an Einstein distance from the lens, the images will have very small amounts of shear and magniﬁcation. Here one uses a statistical approach: by studying a collection of galaxies (which might have been lensed), one looks for a statistical overall shear or magniﬁcation imposed on the distribution (which would have bee absent for an unlensed distribution of randomly oriented galaxies).

For instance one could search for correlated distortions in shapes. These statistical measures of the lensing eﬀect are then used to map the lensing potential. This tool has been very useful in measuring the dark matter haloes in which galaxies are embedded.

Micro lensing

This is the temporary one-time brightening of a source, possibly due to the passage of a MACHO (massive compact halo object) through the line of sight to the source.

The brightening is a consequence of the magniﬁcation caused by the lens. A microlensing event is easy to distinguish from a ﬂuctuation in the source itself, because the micro-lensing light curve can be precisely determined using general relativity, and should be identical in all frequency bands. The MACHO collaboration has found 450 microlensing events from studying 50.2 million lightcurves in the Large Magellanic

Cloud [20].

38 1.7 The Composition of the Universe

In § 1.4, I introduced the concept of the density parameter Ω, which measures the densities of diﬀerent components of the Universe. Of crucial importance to cosmology is the determination of the total density parameter Ωtot, and the density parameters of all the individual components. After several decade of measurements, these numbers have been determined.

It turns out that we live in a spatially ﬂat Universe consisting of three major components whose estimated density parameters are given below:

1. Ordinary (baryonic) matter (' 4%)

2. Dark Matter (' 26%)

3. Dark Energy (' 70%)

This is called the Λ-cold dark matter (ΛCDM) cosmology and seems to be supported by evidence from a variety of diﬀerent sources. In what follows I will brieﬂy review the observational evidence that we have for this model.

1.7.1 Measuring the Total Matter Content

Galaxy clusters, the largest collapsed structures known, prove to be an invaluable tool in placing constraints on ΩM . They are considered to have arisen from the collapse of initial perturbations about a tenth of the comoving horizon size. As a result their evolution is dominated by gravitational dynamics in the linear or weakly non-linear regime, where initial conditions have not yet been erased by gas dynamics.

39 The masses of clusters can be determined using three independent techniques, which agree with each other up to ∼ 1 Mpc (see [21] and references therein):

1. Velocity Dispersion: Under assumptions of hydrostatic equilibrium, the distri-

bution of velocities of galaxies in a cluster can be used to determine the cluster

mass.

2. ICM Temperature: The mass of the cluster can also be traced from the tem-

perature of the intercluster gas.

3. Lensing: If the cluster acts as a gravitational lens, the lensing eﬀects can be

used to determine its mass.

Clusters can be used to estimate ΩM in several diﬀerent ways, which I brieﬂy mention below.

Mass to Light ratio

The traditional method is to calculate the typical mass to light ratio (M/L) for rich clusters, and then integrate it over the entire observed luminosity density of the

Universe. It turns out that a median mass-to-light ratio of 300 ± 100h is observed for most clusters, almost independently of the individual characteristics of the clusters such as their luminosities and velocity dispersions [21]. Studies of this sort suggest that we live in a low density Universe with ΩM ∼ 0.2−0.3 [22]. The above method has been sharpened through the use of sophisticated techniques to measure the masses of clusters, such as gravitational lensing [23] and temperature proﬁles [24], but the conclusion remains the same.

40 Baryon Fraction

Another method is to use the baryon density instead of the luminosity density [25].

The idea here is to estimate the fraction of baryons fICM within the intercluster mediums of galaxy clusters, and then to put an upper limit to Ωb using the relation

ΩM = Ωb/fICM (1.87)

Here, of course, we are assuming that the entire Universe has the same ratio of baryons to total matter as do the clusters. This assumption is not unreasonable since on very large scales there is no reason to expect baryons to be signiﬁcantly segregated from dark matter. fICM has been measured by studying X-ray data from galaxies [26] as well as through the Sunyaev-Zeldovich eﬀect [27], and both indicate that the mass density of the Universe is low (Ω ' 0.2).

Cluster Abundance and Evolution

Constraints have also been placed on ΩM from observations of cluster abundances, combined with studies of evolution of rich massive clusters with redshift. The basic idea here is that the growth of high mass structures depends on cosmological parameters, in particular ΩM and σ8 (which is the root-mean-square mass fluctuation on 8h−1 Mpc scale and measures the bias in the distribution of mass vs light). It turns out that if ΩM is taken to be large (ΩM = 1) then fluctuations start growing late and produce a strong evolution in recent times z > 1. The opposite is expected to happen in lower density models where fluctuations freeze out much earlier. Observations seem to support the latter scenario we see very little evolution in recent times. The matter

41 density of the Universe is inferred to be ΩM ' 0.2 − 0.3 [28, 29, 30].

Gravitational Lensing Studies

Strong lensing provides constraints on ΩM and ΩΛ, and show that we live in a low- density Universe (Ω < 0.4) regardless of the value of ΩΛ. Weak lensing studies using several cluster lenses shows that Ωm ' 0.3 with a signiﬁcant contribution from Dark Matter [14].

Galaxy Correlation functions

Analyzing the bias parameter of galaxies b1 (which quantiﬁes the strength of clustering of the galaxies relative to the mass in the Universe) and the redshift correlation function β from the 2Df Galaxy Redshift Survey, one can get an independent mea-

0.6 surement of the mass density from the relationβ = Ωm /b1 which gives a value of

ΩM = 0.27 ± 0.06 [31]. This is completely independent of, yet in good agreement with, all the other measurements of ΩM described above.

1.7.2 Ordinary and Dark Matter

There is considerable evidence that there exist two types of matter - the ordinary matter we are familiar with, which is mostly made of baryons. The density parameter of baryonic matter is Ωb = 0.04 ± 0.02 [3]. This estimate comes from Big Bang nucleosynthesis, CMB power spectrum and direct counting of baryons.

Dark Matter is the other component is dark matter, which appears to be almost perfectly invisible except for its gravitational interaction. I now brieﬂy review some

42 of the observational evidence for the existence of a dark matter component.

Rotation curves of spiral galaxies provide strong evidence in favor of the existence

of dark matter (the ﬁrst such evidence came from the work of Zwicky in studying a

group of galaxies in the coma cluster [32]). For virialized galaxies, the mass up to a

given distance r, M(r) is related to the rotational velocity at that distance v by the relation

M (r) ∝ v2r/G (1.88)

The rotational velocities can be measured from the HI 21 cm lines. Away from the

luminous part of the galaxy, if there is no more matter, then equation (1.88) indicates

that the rotation velocities should fall oﬀ with r. However, observations show that

they are ﬂat, suggesting that the mass distribution goes as M(r) ∝ r beyond the

point where light ceases, in some unseen form [14].

Another piece of evidence comes from studying the distribution of X-ray emitting

hot gas in galaxies, calculating the amount of mass necessary to bind this gas, and

comparing it to the amount of visible matter in the galaxy. The latter has fallen short

of the required amount in several galaxies, indicating the presence of invisible forms

of matter. One example is the elliptical galaxy M-87 where the shortfall is ∼ 1% [14].

The latest evidence of Dark Matter comes from X-ray and weak lensing obser-

vations of the merging cluster system 1E0657-556. The collision shifted the X-ray

plasma (containing 90% of the baryons) in both the clusters away from the galaxies

(which only contain 10% of the baryons). This separation makes it possible to de-

termine which of these (i.e. plasma or galaxies) is predominantly responsible for the

43 lensing. Observations show that the lensing surface potential is more in agreement with galaxies and not with the X-ray plasma [33]. In [33], it is argued that these observations are a clear indicator of an invisible matter form regardless of the form of the gravitational force law. However, alternative explanations for these observations have also been put forward, based on modiﬁed gravity theories [34, 35].

Dark Matter is not just suggested by observations - it has at least one interesting theoretical motivation - dark matter seems to be crucial to the theory of structure formation. The early Universe was radiation dominated, a ﬂuid of baryons and photons coupled together. Gravitational clumping of the baryons was not possible at this time. As the ﬂuid cooled, the baryons and photons decoupled at 1 + z = 1100.

Assuming that perturbations are adiabatic, both the photon ﬂuid and the baryon

ﬂuid will inherit the curvature perturbations entering the horizon at that time, which also show up as anisotropies in the CMB background and have been measured (by

COBE and WMAP) to have an amplitude ∼ 10−5. With such a small initial amplitude we would expect the baryon overdensity to grow in proportion to the scale factor, i.e., in accordance with linear perturbation theory. As a result, by today these perturbations will have grown by a factor of 1100 , and hence we can expect typical structures today to have an overdensity of ∼ 0.1, which is far too low compared to what is observed (typical galaxies have over densities of 105. However, if we assume the existence of dark matter, then by the time of decoupling the dark matter will have formed potential wells into which the baryons can fall and hence immediately commence non-linear growth.

44 1.7.3 Supernovae and Cosmic Acceleration

Supernovae

Supernovae are spectacular stellar explosions, caused either by the collapse of a stellar core, or through runaway nuclear fusion in a white dwarf. Supernovae can release up to 1031 ergs of energy. Only a small fraction of which is released as visible light, but even that is typically suﬃcient to temporarily outshine the entire host galaxy. The explosion shoots out stellar material into the inter stellar medium (ISM) at speeds as high as a tenth of the speed of light, setting up a shock wave in the ISM.

The classiﬁcation scheme of supernovae is based on observation. The main categories are Type I and Type II depending on the absence or presence of Hydrogen lines in their spectra respectively. A detailed classiﬁcation scheme is shown in Fig. 1.3.

However from an astrophysical point of view, it is found that the Type II are similar to the Type Ib and Ic, in that they all originate from the collapse of the stellar core.

The Type II are believed to be formed when the cores of massive stars (> 8 solar masses) collapse to form black holes or white dwarfs. Type Ib and Ic are also formed from core-collapse after the hydrogen-rich (Ib) or helium-rich (Ic) outer mantles are blown away by stellar winds.

The Type Ia supernova have a diﬀerent astrophysical origin. They are formed from binary systems where one of the stars is a white dwarf. As the latter accretes more and more mass from the companion star, it eventually crosses the Chandrasekhar limit [36] and collapses until its density triggers nuclear fusion reactions throughout the star, causing the star to explode.

Since the Chandrasekhar limit is a universal phenomenon, one would expect the

45 Early Spectra: No Hydrogen / Hydrogen

SN I SN II Si/ No Si ~3 mos. spectra He dominant/H dominant

SN Ia He poor/He rich 1985A 1989B SN Ic SN Ib SN IIb “Normal” SNII 1983I 1983N 1993J 1983V 1984L 1987K Light Curve decay after maximum: Linear / Plateau Core collapse. Believed to originate Most (NOT all) from deflagration or H is removed during detonation of an evolution by tidal stripping. accreting white dwarf. SN IIL SN IIP Core Collapse. 1980K 1987A Outer Layers stripped 1979C 1988A by winds (Wolf-Rayet Stars) 1969L or binary interactions Ib: H mantle removed Core Collapse of Theory Ic: H & He removed a massive progenitor with plenty of H .

Figure 1.3: A ﬂow chart showing the classiﬁcations of Supernovae http://rsd-www. nrl.navy.mil/7212/montes/snetax.html.

46 luminosities of Type Ia’s to be identical, i.e., Type Ia’s are perfect candidates for standard candles. In practice, while a large majority of the true type Ia SNe have very similar light curve shapes, spectral time series and absolute magnitudes, there does exist a scatter in their peak luminosities (see [13] and references therein).

It was ﬁrst noted by Phillips [37] that a strong correlation existed between the rate at which a Type Ia SNe’s rate declines (measured by the parameter ∆M15 deﬁned as the reduction in brightness over 15 days following maximum light) and its absolute magnitude. Eliminating this correlation can reduce the dispersion in the maximum luminosities. Several techniques have been developed to do this.

Hamui [38] used this correlation between maximum luminosity and ∆M15 to reduce the scatter in the Hubble diagram for a sample of 30 SN1a from the CTSS search. A more sophisticated technique called the multi-color light curve shape method (MLCS) was developed by [39]. Other popular methods include the “stretch” method [40, 41] and the Color-Magnitude Intercept Calibration (CMAGIC) method [42]. All of these methods allow supernovae distances to be determined to a precision of 6% after correcting for photometric uncertainties and peculiar velocities [13] making them very useful standard candles.

Supernovae Magnitudes

The apparent magnitude m of a light source is deﬁned in relation to a reference

−8 −2 ﬂux fx = 2.53 × 10 watt m as

m ≡ −2.5 log10 (f/fx) (1.89)

47 The absolute magnitude M of a light source is similarly deﬁned in relation to

a reference luminosity of Lx = 78.7 solar luminosities

M ≡ −2.5 log10 (L/Lx) (1.90)

(Lx is the luminosity of an object that produces a ﬂux of fx when it is placed at a luminosity distance dL = 10 parsec. The physical interpretation of the absolute magnitude is therefore clear: it is the apparent magnitude a source placed at a luminosity distance of dL.) The distance modulus, deﬁned as m − M, is a convenient measure of the luminosity distance in terms of the apparent and absolute magnitudes. From equations

(1.89) and (1.90)

d m − M = 5 log L + 25 (1.91) 10 1 Mpc

From equation (1.25), we can connect the distance modulus to the redshift and the acceleration parameter:

H m − M ≈ 43.17 − 5 log 0 + 5 log z + 1.086 (1 − q ) z (1.92) 10 70km s−1 Mpc−1 10 0

Hence it is clear that if the apparent ﬂux and the absolute ﬂux (or equivalently, m and M) are known for a population of standard candles, the parameters H0 and

q0 can be determined from a plot of the distance modulus m − M vs z.

48 Supernovae results

Fig. 1.4 shows such a Hubble plot incorporating data from both the Supernova

Cosmology Project [41] and the High-z Supernova search team [43]. The data are

compared to three model Universes as shown. Both sets of datapoints show that

distant supernovae are fainter than one would expect from the Hubble expansion

alone, and hence the Universe is accelerating. The upper panel clearly reveals the

validity of the Hubble law for small redshifts (z 1). The lower panel subtracts out

a negatively curved matter-only Universe from the data and clearly reveals that at

high redshift (z ≥ .5) the supernova are about ∼ .25 mag fainter than what one can

expect from a pure Hubble ﬂow in a matter-only Universe.

The supernovae data allow for a wide range of both possibilities for ΩM and ΩΛ (see Fig.1.5), but in general are better ﬁt by cosmological constant dominated models. If we assume that we know either one of ΩM or ΩΛ, we can establish tight constraints on the other one. For instance, choosing ΩM = 0.3 based on, for instance, the large scale structure evidence described in § 1.7.1, the supernovae results imply that ΩΛ = .7

1.7.4 Cosmic Microwave Background Anisotropies

In § 1.1.3, I had introduced the CMB. I now proceed to discuss the wealth of information that lies in its anisotropies.

The CMB has a dipole anisotropy (see Fig. 1.6), meaning that in one half of the sky the CMB blackbody spectrum is blue-shifted and in the other half it is redshifted.

This is a consequence of the earth not being a comoving frame, and having a relative velocity in a frame in which the CMB is isotropic. The COBE satellite is orbiting the

49 44 High-redshift (z > 0.15) SNe: High-Z SN Search Team 42 Supernova Cosmology Project

40 Low-redshift (z < 0.15) SNe: CfA & other SN follow-up 38 Calan/Tololo SN Search ΩΜ=0.3, ΩΛ=0.7 36 ΩΜ=0.3, ΩΛ=0.0 Distance Modulus (m-M) 34 ΩΜ=1.0, ΩΛ=0.0

1.0 =0.0 Λ Ω 0.5 =0.3, Μ

Ω 0.0

-0.5 (m-M) - 0.01 0.10 1.00 z

Figure 1.4: top panel: A Hubble diagram made from data from both the Supernova Cosmology Project and the High-z Supernova Search Team taken from [13].bottom panel The residual of the distances relative to a ΩM = 0.3, ΩΛ = 0.7 Universe.

50 3

No Big Bang SCP

2 HZSNS

1 accelerating decelerating

expands forever 0

(cosmological constant) lapses eventually vacuum energy density recol

closed

flat -1 open

0 1 2 3 mass density

Figure 1.5: Best ﬁt regions in the (ΩM ,ΩΛ) plane for data from both the Supernova Cosmology Project and the High-z Supernova Search Team. The agreement of the two experiments is remarkable. Source [13]

51 Figure 1.6: The Dipole Anisotropy in the CMB as seen by COBE http://map.gsfc. nasa.gov/m_uni/uni_101Flucts.html. earth at ∼ 8 km s−1, the earth is orbiting the sun at 30 km s−1, the sun is orbiting the galactic center at ∼ 220 km s−1, and our galaxy is orbiting the center of mass of the Local Group at ∼ 80 km s−1. Finally, the Local Group is accelerated towards the

Virgo cluster, which in turn is accelerated towards the Hydra-Centaurus supercluster, as a result of which the Local group is headed in the direction of Hydra at a speed of

.2% the speed of light [5].

After the dipole is subtracted (see Fig. 1.7), the remaining temperature ﬂuctua- tions are extremely small. In fact in Fig. 1.7 the temperature of the hot regions (red) exceeds the temperature of the cold regions (blue) by .0002K.

Let the temperature at any point in the sky be denoted by the function T (θ, φ), where θ and φ are spherical polar angles on the sky. The dimensionless temperature

ﬂuctuation at a given point in the sky is deﬁned as

52 Figure 1.7: The CMB spectrum once the dipole is subtracted. http://map.gsfc. nasa.gov/m_uni/uni_101Flucts.html.

δT T (θ, φ) − hT i (θ, φ) = (1.93) T hT i

Results from COBE, WMAP and a variety of other experiments shows that

−5 (δT/T )rms ∼ 10 . Since these ﬂuctuations are deﬁned on the surface of a sphere centered on the observer, it is convenient to expand the function (δT/T )(θ, φ) in the

spherical harmonics:

δT X (θ, φ) = a Y (θ, φ) (1.94) T l,m l,m l,m and

∗ halmal0m0 i = δl,l0 δm,m0 Cl (1.95)

where the Kronecker delta’s spring from the assumption that the mechanism gener-

53 ating the anisotropies was isotropic. The information regarding the anisotropies is

encoded in the correlation function C (θ) between two points on the last scattering surface in directionsn ˆ andn ˆ0 separated by an angle θ:

δT δT C (θ) = (ˆn) (ˆn0) (1.96) T T

Using the properties of spherical harmonics, the correlation function can be writ-

ten as an expansion in the Legendre polynomials as follows:

1 X C (θ) = (2l + 1) C P (cos θ) (1.97) 4π l l l

The Cl’s can be interpreted as a measure of temperature ﬂuctuations on angular scales 180◦/l. On small sections of the sky where spatial curvature can be neglected, the Spherical Harmonic expansion becomes identical to a Fourier expansion in two dimensions, with l as the Fourier wavenumber [8]. For small angular separations

R 2 2 l 1, the correlation function in Fourier space is d lCl/ (2π ), using which, the power spectrum of temperature ﬂuctuations is conventionally written as

l(l + 1) ∆2 ≡ C T 2 (1.98) T 2π l

The right hand side is now clearly the power per logarithmic interval in l for l 1.

(Another reason for adopting this convention is because the right hand side of (1.98)

is a constant for perturbations generated via the Sachs Wolfe Eﬀect).

How accurately the Cl’s can be measured is limited by cosmic variance. This is the notion that our sky is only one realization of an ensemble, and hence there are

54 only 2l + 1 m-samples of the power in each multipole moment [8] causing an error of:

r 2 ∆C = C (1.99) l 2l + 1 l

Cosmological models generally predict the behavior of Cl vs l. Figure 1.8 shows such a plot, together with data from recent experiments measuring the CMB power spectrum.

The diﬀerent sources of temperature ﬂuctuations in the CMB are the following:

1. Dipole: These are caused by the earth’s peculiar velocity (already discussed

above).

2. Sachs Wolf: These are caused by ﬂuctuations in the gravitational potential

on the last scattering surface, which in turn are created when superhorizon

metric perturbations enter the horizon. These ﬂuctuations dominate on scales

comparable to the Horizon size at last scattering [45].

3. Intrinsic: These are caused by ﬂuctuations in the photon-baryon plasma.

4. Doppler: These are caused by the non-zero velocity of the plasma at recom-

bination (the last scattering surface) leading to Doppler shifts in the frequency

(and hence the temperature) of the CMB.

The ﬁrst peak in Fig. 1.8, located at an angle of 1◦ is crucially important. This peak corresponds to the horizon size at last scattering and can be understood as follows. Prior to recombination, the baryons and photons are tightly coupled and photon pressure can provide a restoring force to baryon clustering. As a result coherent oscillations are set up in the baryon-photon plasma. The rise in temperature associated

55 Angular Scale 90° 2° 0.5° 0.2° 6000

WMAP Acbar 5000 Boomerang CBI VSA 4000 ] 2 K + [ /

/2 3000 l +1)C l ( l 2000

1000

0 10 100 5001000 1500 Multipole moment l

Figure 1.8: The WMAP three-year power spectrum (in black) together with data from other recent experiments measuring the CMB angular power spectrum. Taken from [44]

56 with the compression of the photons acts as damping force (the Silk damping, [46]), and hence the system as a whole can be modelled as a damped harmonic oscillator

[47]. Now, heuristically, we can expect that oscillation modes which are larger than the horizon will not have had enough time to evolve appreciably till last scattering.

For subhorizon modes, the ones which are “caught at the maxima or minima of their oscillation at recombination” [47] will show up as peaks in the Cl vs l plot. Owing to the eﬀect of photon damping, we can expect the modes much smaller than the horizon, which have had considerable time to equilibriate, to make smaller peaks than the Horizon sized mode. We therefore expect the largest peak in the Cl vs l plot to correspond to the angular size of the Horizon at the time of last scattering.

Since we know the physical size of the Horizon at last scattering, we can deduce the location of this peak for diﬀerent spatial geometries. For a ﬂat matter-dominated

Universe, the peak should lie at l ' 220, which is exactly what is observed experimentally.

A careful analysis of the entire CMB spectrum allows us to constrain almost all cosmological parameters. However, there are degeneracies in the parameters which allow constraints to be set only when priors are assumed. For example, if one assumes a ﬂat vacuum energy dominated Universe, the WMAP data constrains Ωtot and h. If one assumes curvature, there is a degeneracy between Ωm, h and the curvature, and one of these has to be assumed in order to constrain the others [48].

The combined data from large scale structure, supernovae and the CMB seem to consistently bear out the ﬂat ΛCDM cosmology.

57 1.8 Shortcomings of the Standard Cosmology

For all its successes, the Standard Hot Big Bang model suﬀers from one major ﬂaw:

several observational features of the Universe, though not logically inconsistent with

the Hot Big Bang, still require remarkable initial conditions. I now brieﬂy review

some of these instances in this section, and in the next section show how the theory

of inﬂation comes to the rescue.

1.8.1 The Flatness Problem

The Friedmann equation, in the form (1.39) indicates that the diﬀerence between

2 Ωtot and 1 increases with time. It increases in proportion to a for a ﬂat radiation- dominated Universe (see § 1.5.7) and in proportion to a for a ﬂat matter-dominated

Universe (see § 1.5.5). If the Universe starts out flat, it stays flat forever. However, if it does not start out flat, then in order to be spatially flat today, then |Ωtot − 1| has to be extraordinarily close to, but not exactly, 0 in the past, a very unseemly initial condition requiring severe fine-tuning.

Peebles and Dicke [49] have pointed out that at t = 1 second, at the beginning of

−15 nucleosynthesis, Ωtot must have equalled 1to an accuracy of 10 . If we assume that General Relativity is valid all the way up to the Planck scale, then from (1.39) (and

−64 assuming radiation domination) one can estimate that |Ωtot − 1|tPl ≈ O [10 ] Another way of looking at the ﬂatness problem is to ask how the Universe got to be so old. Only the highly specialized initial conditions described in the previous paragraph would cause the Universe to survive in its present state for this long. Any other initial conditions would have lead to a curved Universe that recollapses very

58 quickly, or an open Universe that cools to below 3K within one second [9].

1.8.2 The Entropy Problem

This problem concerns the constancy of the entropy per comoving volume S described in § 1.5.7. Using equations (1.39), (1.69) and (1.70) one can show that at the Planck

Time, the entropy within our causal horizon is ≈ O [1060].

Such an incredibly large number demands an explanation.

1.8.3 The Horizon Problem

The crux of this problem lies in the remarkable isotropy of the CMB, even among regions of the sky that have never been in causal contact. In other words, diﬀerent parts of the sky, which cannot have had any microphysical interaction, seem to have the same temperature. The following simple calculation serves to illustrate this point very clearly:

The size of our visible patch today is roughly equal to the inverse Hubble length

−1 H0 . Let LLS be the radius of this volume at last scattering. Clearly,

−1 a(tLS) −1 T0 LLS = H0 = H0 (1.100) a(t0) TLS

(the subscripts 0 and LS stand for today and last scattering respectively). On the other hand, the Hubble length at last scattering (which deﬁnes the causal volume at last scattering), is given by (assuming matter domination since last scattering)

59 −3/2 −1 −1 TLS HLS = H0 (1.101) T0

Taking the ratio of the two volumes (our visible patch and the causal volume) we

ﬁnd that 3 LLS T0 6 −3 = ≈ 10 (1.102) HLS TLS

This implies that the volume occupied by our visible patch at the time of last scattering contained ∼ 106 causally disconnected regions. The horizon problem is nicely illustrated Fig. 1.9, which shows that for an FRW cosmology, physical scales entering the horizon at a certain time have never been inside the horizon before, simply because the Horizon evolves faster than the scales.

Further, not only do causally disconnected regions in the sky have the same temperature, they also seem to have the same degree of temperature anisotropy ∼ 10−5

This leads to the question of whether the large-scale inhomogeneity that is observed

must also be attributed to initial conditions.

1.8.4 The Monopole Problem

All Grand Uniﬁed Theories (GUT’s) predict the existence of magnetic monopoles, i.e.,

massive particles carrying a net magnetic charge. In fact Preskill [50] argues that if

the GUT phase transition were second order (or weakly ﬁrst order), the production of

monopoles would be so copious that they would come to dominate the energy density

of the Universe before the time of Helium synthesis. The standard cosmology pro-

vides no explanation for the observed absence of magnetic monopoles in the Universe

60 Figure 1.9: A plot of length scale vs (logarithmic) scale factor in an FRW cosmology. The blue line shows the evolution of the Hubble scale. The red lines show the evolution of physical scales. Since the Hubble length evolves faster than the physical scales, sub-Horizon modes have never been in causal contact prior to Horizon entry.

61 (especially given the age of the Universe determined from other sources).

1.9 Inﬂation

1.9.1 Inﬂation and the Problems of the Standard Cosmology

The theory of inflation [51] dramatically resolves all these problems, simply by propos- ing that for a very brief period in its history, the Universe underwent a period of rapid expansion. Formally, an inflationary phase is one during which the scale factor accelerates (ä > 0) [9]. In most models of inflation the scale factor grows exponentially.

In order to not disturb the successes of the Hot Big Bang, inflation must occur before nucleosynthesis. After inflation ends, all the energy responsible for inflation is released as thermal energy through a non-adiabatic process which increases the entropy of the Universe by a very large factor. This process is dubbed reheating.

Equation (1.36) implies that an inﬂationary phase can only occur if the overall pressure of the Universe is negative to the extent that p < −ρ/3. In the next subsection I describe how this is made possible from the particle-physics perspective.

It is not hard to see how inﬂation resolves the problems mentioned in the previous section (see, for example, [1] for a detailed discussion). The resolution of the horizon problem can be seen from Fig. 1.10 which plots the Hubble scale vs time and shows the evolution of modes. In this scenario, it is clear that superhorizon modes entering the Horizon now were once inside the Horizon.

The fact that modes which were superhorizon at the time of recombination were once inside the horizon provides a causal mechanism for the generation of large scale

62 −1 H length scale FRW

k’

−1 H inf

t tt t iH0

Figure 1.10: Physical scales entering the horizon at the time of last scattering have been in causal contact before. structure. Metric perturbations in the inflationary spacetime (probably generated from quantum fluctuations in the inflaton) are stretched to superhorizon scales during inflation. When these modes re-enter the Horizon at the time of last scattering, they leave an imprint on the matter content of the Universe at that time. However,

Trodden and Vachaspati [52] have argued that if inflation was preceded by a pre- inflationary stage subject to some reasonable characteristics (i.e., General Relativity being valid, the null energy condition being satisfied, and spacetime topology being trivial), then homogeneity on super-Hubble scales is required as an initial condition.

Models in which inﬂation originates at the Planck epoch may not be subject to this

63 condition.

The ﬂatness problem is resolved since the inﬂationary phase drives Ωtot very close to 1 rather than away from it as can be seen from (1.39), especially if the growth of the scale factor is exponential.

Monopoles and other relics can be removed simply by assuming that inﬂation occurs before (or during) their production. The exponential expansion of spacetime makes their density suﬃciently low in order for them to be undetectable later.

The entropy problem is resolved by assuming that the reheating process is non- adiabatic, leading to the creation of a large amount of entropy.

1.9.2 The Dynamics of Inﬂation

In the previous subsection I show that a brief period of exponential expansion in the early history of the Universe is extremely beneﬁcial in resolving the problems of the

Standard Cosmology. However, the question remains as to how such an inﬂationary epoch can come about.

As I have shown in § 1.5.3, the Friedmann equation admits an inﬂationary solution if the equation of state parameter w = −1, implying p = −ρ. A substance with negative pressure is hard to imagine in our everyday existence, scalar ﬁelds (which have never been observed experimentally yet) can easily have negative pressure, as I show below.

A scalar ﬁeld φ is typically described by a Lagrangian:

µ L = ∂ (φ)∂µ(φ) − V (φ) (1.103)

64 and a stress tensor given by:

Tµν = ∂µ(φ)∂ν(φ) − gµνL (1.104) gµν represent the components of the metric, and V (φ) represents the potential. If the field is a quantum field, then φ is actually a quantum operator. The scalar field responsible for inflation is usually called the inflaton. The equation of motion of the

ﬁeld is the Klein-Gordon equation:

1 √ ∂V √ ∂ −ggµν∂ φ + = 0 (1.105) −g µ ν ∂φ

The energy density and pressure of the ﬁeld are given by

1 1 ρ (φ) = T 0 = φ˙2 + (∇φ)2 + V (φ) (1.106) 0 2 2 1 1 p (φ) = −T i = φ˙2 − (∇φ)2 − V (φ) (1.107) 0 2 6

In inflationary cosmology, it is customary to split the inflaton field into a homogenous background and a time and space dependent perturbation.

φ (~x, t) = φ0 (t) + δφ (x, t) (1.108)

φ0 is the expectation value of the inflaton field and can be treated as a classical field, and δφ represents quantum fluctuations about the homogenous background. In what follows, I will suppress the subscript 0 in denoting the homogenous part of the field.

65 The equation of motion of φ (t) and δφ (x, t) can be derived from (1.105):

φ¨ + 3Hφ˙ + V 0 (φ) = 0 (1.109)

δφ00 + 3Hδφ0 − ∇2δφ0 + V 00δφ − 4φ0Φ0 + 2V 0Φ = 0 (1.110)

To first order, the equations (1.106) and (1.107) now indicate that if the potential energy of the field dominates its kinetic energy, e.g., V (φ) φ˙2, then p ' −ρ will hold, and the Universe will inflate exponentially. In the next subsection, we will investigate this condition in greater detail.

1.9.3 The Slow Roll Parameters

In order to ensure an inﬂationary phase, the usual assumption is that the ﬁeld is rolling slowly down its potential. To make this possible, one uses the approximations:

1. φ˙2 V (φ), and

2. φ¨ 3Hφ˙

For these approximations to be valid, it is necessary for two conditions to hold

|η| 1 (1.111)

66 These are called the slow-roll conditions. The slow roll parameters are deﬁned as

[9]

H˙ φ˙2 1 V 0 2 ≡ − = 4πG = (1.112) H2 H2 16πG V 1 V 00 η ≡ (1.113) 8πG V

quantifies the change in the Hubble rate during inflation. Inflation ends when ≥ 1.

1.9.4 Models of Inﬂation

An enormous number of models have been proposed to satisfy this condition in in-

genious ways, using one or more scalar ﬁelds (see [53] for a comprehensive review of

inﬂationary models). The models usually specify the potentials of the scalar ﬁeld(s)

responsible for inflation. A typical potential for single-field inflation can be charac-

terized [10] by two parameters A and B and a function f as follows:

φ V (φ) = A4f (1.114) B

where A measures the vacuum energy density during inﬂation and B measures the

change in the field value during inflation. The function f specifies the model.

Based on the above characterization, inﬂationary models can be (very broadly)

classiﬁed into three categories [54]:

1. Large Field Models

2. Small Field Models

67 3. Hybrid Models

Large Field Models

These are models in which the scalar ﬁeld is displaced very far away from the minimum

of its potential, typically to values of several times the Planck mass. The ﬁeld is

assumed to be in this state as a result of having emerged from a quantum-gravity

state where the energy density was on the order of the Planck energy. The potential

energy of the ﬁeld is therefore very high V (φ) ∼ mPl, leading to a high value of the Hubble parameter, and the consequent heavy Hubble damping causes the ﬁeld to roll

slowly. Inﬂation ends when the ﬁeld starts rolling fast, and this can be shown to

occur when the ﬁeld is of the order of the Planck mass. This kind of inﬂation is called

chaotic inﬂation [55]. Examples of large ﬁeld models are polynomial potentials

V (φ) = A4 (φ/B)p and exponential potentials V (φ) = A4 exp (φ/B).

Small Field Models

This is a class of models in which the field is initially near the origin and is evolving away from an unstable equilibrium at the origin toward a nonzero vacuum expectation value of hφi = 0. Such potentials arise naturally from spontaneous symmetry breaking, such as the class of models referred to as “new” inflation ([56, 57]) and “natural” inflation [58]. Inflation ends when the field reaches the minimum and starts oscil- lating. A generic form of potentials of this kind are V (φ) = A4 [1 − (φ/B)p], which

can be regarded as a lowest order Taylor expansion (since the ﬁeld is small) of an

arbitrary potential about the origin.

68 Hybrid Models

These models typically involve more than one scalar ﬁeld. One ﬁeld responsible for

inﬂation rolls towards a non-zero vacuum expectation value. An instability in the

other field ends the inflationary stage. A typical potential for the field responsible

for inﬂation is V (φ) = A4 [1 + (φ/B)p]. The presence of the second ﬁeld introduces another free parameter in the model. Examples of hybrid models can be found in

[59, 60, 61].

Other Models

While the above classification is useful in comparing the predictions of different models with CMB observations, it is certainly not exhaustive. There are plenty of models which do not neatly fit into the above scheme, such as models involving brane inflation

[62], which does not require a scalar ﬁeld at all. Other examples are the logarithmic potentials V (φ) = ln (φ) typically used in supersymmetric inﬂation (see [63] for a review), or the potentials of the type V (φ) = φ−p used, for instance, in intermediate

inﬂation [64].

1.9.5 Issues with Inﬂation

While inﬂation is a very elegant solution to a number of problems of the standard

cosmology, it has some problems of its own, some of which I list below. An extensive

discussion can be found in [65].

1. First of all, being a class of models rather than a model, the inﬂationary

paradigm allows for considerable freedom in choosing the inﬂaton potential,

69 the parameters of which can be tuned to match any observations. This leaves

unresolved several questions such as the start of inﬂation itself - why the inﬂaton

started out at the top of the hill.

2. The “solution” to the ﬂatness problem described above is not entirely satisfac-

tory. Open inﬂationary models [66] are designed to make the Universe end up

positively curved.

3. As I show in Chapter 3, the power spectrum generated from inﬂation depends

on the slow roll parameters, and is hence easily tunable. By varying these

parameters, one can make inﬂationary models agree with scale-invariant, or

either slightly red or blue-shifted spectra. Also, for scalar perturbations, there

is no minimum predicted amplitude - their amplitude depends entirely on the

model.

4. Trodden and Vachaspati [52] have shown that inﬂationary models based on cer-

tain mild conditions (the classical Einstein equations, the null energy conditions,

and trivial topology), require the initial inﬂationary patch to be homogenous.

Hence inﬂation does not quite resolve the homogeneity problem by itself.

5. Finally, there is no evidence of the existence of a ﬁeld suitable to be an inﬂaton.

As a result of all the above, it is useful to investigate cosmological models which do not fall within the inﬂationary paradigm. In the next two chapters I describe a cosmological model which relies on a diﬀerent hypothesis, and examine its viability.

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75 Chapter 2

Island Cosmology

2.1 Introduction

As I discussed in the previous chapter, the theory of inflation, together with the standard Big Bang Cosmology, is the most widely accepted description of the origin of our cosmos. At the core of the inflationary paradigm lies the hypothesis that an inflaton exists in Nature.

In this chapter, I describe an alternative cosmological model which I have proposed together with Tanmay Vachaspati called “Island Cosmology” [1]. Island Cosmology relies on a very diﬀerent hypothesis than inﬂation.

In a nutshell, Island Cosmology works as follows: the universe is initially assumed to be ﬁlled with cosmological constant of the currently observed value but is otherwise empty. In this eternal or semi-eternal de Sitter spacetime, local quantum ﬂuctuations violate the null energy condition (NEC) and create islands of matter, one of which is our Universe. With further cosmic evolution the island disappears and the local

76 spacetime returns to its initial cosmological-constant dominated state.

2.2 The Model

I begin with a short overview of Island Cosmology, describing each of its diﬀerent stages.

1. In the beginning the universe is inﬂating due to the observed dark energy that

1 −1 we assume is a cosmological constant (Λ) . The de Sitter horizon size (HΛ ) is

−1 comparable to our present horizon, (H0 ). We call this spacetime the Λ-sea.

2. A quantum fluctuation of some field (e.g. scalar field, photon) in a horizon-size

volume (which we label as an I-region) in the expanding phase of de Sitter

spacetime drives the Hubble constant to a large value. Even as the Hubble

length scale is decreasing the universe continues to expand. From the accel-

eration equation (1.37), we see that this can occur only when (p + ρ) < 0, or

µν NµNνT < 0, or in other words, the Null Energy Condition (NEC) is violated. Later I will show, following Refs. [2, 3, 4], that quantum ﬁeld theoretic

ﬂuctuations allow for this possibility.

3. After the NEC violating ﬂuctuation (which is assumed to be of a very short

temporal duration - see § 2.4) is over, the Hubble constant within the I-region is

large, and the I-region gets ﬁlled with classical radiation, as the NEC violating

ﬁeld decays into relativistic particles. Thereafter, the I-region evolves as a

1We make no attempt to address why Λ is so small compared to the Planck scale.

77 Figure 2.1: Sketch of the behavior of the Hubble length scale with conformal time, η, in the Island model, and the evolution of fluctuation modes. At early times, inflation is driven by the presently observed dark energy, assumed to be a cosmological constant. As the cosmological constant is very small, the Hubble length scale is very large – of order the present horizon size. Exponential inflation in some horizon volume ends not due to the decay of the vacuum energy as in inflationary scenarios but due to a quantum fluctuation in the time interval (ηi, ηf ) that violates the null energy condition (NEC). The NEC violating quantum fluctuation causes the Hubble length scale to decrease. After the fluctuation is over, the universe enters radiation dominated FRW expansion, and the Hubble length scale grows with time. The physical wavelength −1 of a quantum fluctuation mode starts out less than HΛ at some early time ηi. The mode exits the cosmological horizon during the NEC violating fluctuation (ηexit) and then re-enters the horizon at some later epoch (ηentry) during the FRW epoch (The modes are drawn as straight lines for illustrative purposes only, they actually grow in proportion to the scale factor).

78 radiation-dominated FRW Universe, as described in § 1.5.7, and eventually

forms our observed Universe. We call a Universe created in this manner an

Island Universe.

4. With further evolution, the Island Universe dilutes and eventually the I-region

is again dominated by the cosmological constant and the spacetime returns to

its normal inﬂating state.

Island Cosmology does include elements of earlier work such as eternal inﬂation models, steady state models and ekpyrotic models. While eternal inﬂation models [5,

6], especially Garriga and Vilenkin’s “recycling universe” [7], (see also the discussion in [8]) use NEC violating quantum fluctuations in the inflaton field to drive the Hubble length scale to smaller values, in Island cosmology, these quantum fluctuations can occur in any quantum field and have to be large. In both inflationary cosmology and our case, the quantum fluctuation needs to violate the NEC. Furthermore, in both cases the back-reaction of the fluctuation is assumed to lead to a faster rate of cosmological expansion. In the language of [9], the evolution we are considering is one of the “miraculous” trajectories that go directly from a dead de Sitter region of spacetime to a region that is “macroscopically indistinguishable from our universe”

(MIFOU). Eventually the trajectory leaves the MIFOU region and returns to the dead de Sitter region. In Steady State Cosmology [10, 11], matter is sporadically produced by “minibangs” in a hypothetical C-ﬁeld. The explosive events in Island

Cosmology, on the other hand, are quantum ﬁeld theoretic in origin and seed the matter content of an entire Universe. The decreasing Hubble scale is also a feature of the ekpyrotic cosmological model [12]. However, in that model, the motivation

79 for the decrease lies in extra-dimensional brane-world physics and results in a period

of contraction of our three dimensional universe. Island Cosmology does not involve

any brane-world physics, and has no contracting phase, as the Universe continues to

expand even while the Hubble scale drops.

Having summarized the main idea, I now discuss each step of this model in greater

detail.

2.3 NEC violations in de Sitter space

In the ﬁrst stage of the model, we assume that the Universe is ﬁlled with cosmological

constant. As I have explained in the previous chapter, observations are consistent

with some form of dark energy, the simplest explanation of which is a cosmological

constant. The model does not necessarily have to start out with a singularity, and

nor does spacetime need to be created out of nothing as in quantum cosmology. All

that we need is an expanding de Sitter background, and this could be the expanding

phase of a classical de Sitter spacetime with no beginning and no end. The scale

factor of the universe at this stage is given by:

1 HΛt a(t) = a0e ≡ − (2.1) HΛη

where η ∈ (−∞, 0) is the conformal time.

In de Sitter spacetime, as well as any other spacetime, there are ﬂuctuations of

the energy-momentum tensor, Tµν, of quantum ﬁelds. This is simply a consequence of the fact that the vacuum, |0i, is an eigenstate of the Hamiltonian but not of the

80 ˆ energy-momentum density operator, Tµν. In short-hand notation:

ˆ X † † † Tµν|0i = [(... )alak + (... )al ak] |0i X = [(... )|0i + (... )|2; k, li] (2.2)

where, the ellipses within parenthesis denote various combinations of mode functions

† and their derivatives; ak, al are creation and annihilation operators and |2; k, li is a two particle state. The ﬁnal expression is not proportional to |0i, implying that ˆ the vacuum is not an eigenstate of Tµν and there will be ﬂuctuations of the energy- momentum tensor in de Sitter space.

It has been shown [2, 3, 4] that quantum ﬁeld theory of a light scalar ﬁeld in the

Bunch-Davies vacuum [13] in de Sitter space leads to violations of the NEC. I now

brieﬂy summarize the general arguments behind this conclusion.

The ﬁrst step is to construct a “smeared NEC operator”

Z √ ˆren 4 µ ν ˆren OW ≡ d x −g W (x; R,T ) N N Tµν (2.3)

where W (x; R,T ) is a smearing function on a length scale R and time scale T . The vector N µ is chosen to be null, and the superscript ren denotes that the operator has been suitably renormalized. As shown in [2] the smeared operator will be not be proportional to the vacuum state either, and will ﬂuctuate. The scale of the

ﬂuctuations can be estimated on dimensional grounds:

2 ˆren 2 8 Orms ≡ h0|(OW ) |0i ∼ HΛ (2.4)

81 −1 ˆ in the special case when R = T = HΛ . Since, in de Sittter space, h0|Tµν|0i ∝ gµν, we also have:

ˆren h0|OW |0i = 0 (2.5)

ˆren Therefore the ﬂuctuations of OW are both positive and negative. Assuming a symmetric distribution, we come to the conclusion that quantum ﬂuctuations of a scalar

field violate the NEC with 50% probability. Exactly the same arguments can be applied to quantum fluctuations of a massless gauge field such as the photon.

Note that the above calculation does not give us the probability distribution of the violation amplitude, for which we would have to calculate the actual probabil-

ˆren ity distribution for the operator OW . However, by continuity we can expect that large amplitude NEC violations will also occur with some diminished but non-zero probability.

2.4 Extent and duration of NEC violation

What is the spatial and temporal extent of these NEC-violating ﬂuctuations? Such

ﬂuctuations can occur on all spatial and temporal scales, but based on causality and predictability, I now argue that only ﬂuctuations of a large (horizon-sized) spatial extent and small temporal duration are relevant to creating islands of matter. Smaller

ﬂuctuations are irrelevant because the spacetime is likely to respond only locally before returning to its original state.

Consider the spacetime diagram of Fig. 2.2. In that diagram we show an initial de Sitter space that later has a patch in which the space is again de Sitter though

82 + η

η b b η Q Q ? ? η ? ? ? P P η a a

Figure 2.2: We show a classical de Sitter spacetime for conformal time η < ηP , that transitions to a faster expanding classical de Sitter spacetime for η > ηQ. The inverse Hubble size is shown by the white region. A bundle of ingoing null rays originating at point a is convergent initially but becomes divergent in the superhorizon region at point b. This can only occur if the NEC is violated in the region η ∈ (ηP , ηQ). In the quantum domain, a classical picture of spacetime may not be valid and this is made explicit by the question marks.

83 −1 with a larger expansion rate. Hence the initial Hubble length scale Hi is larger than

−1 the ﬁnal Hubble length scale Hf . Therefore there are ingoing null rays that are within the horizon initially that propagate and are eventually outside the horizon.

An example of such a null ray is the line from a to b. At point a a bundle of such rays will be converging whereas at point b the bundle will be diverging. It can be demonstrated from the Raychaudhuri equation (provided some mild conditions are satisﬁed, such as general relativity being valid, and spacetime topology being trivial) that the transition from convergence to divergence of a bundle of null rays can only occur if there is a NEC violation somewhere along the null ray (see [14]).

−1 Now if the NEC violation only occurred on a scale smaller than Hi , one could imagine a null ray that would never enter the NEC violating region and yet go from being converging to diverging (see Fig. 2.3). This would clearly be inconsistent with the Raychaudhuri equation.

Furthermore, after the energy condition violations are over, the faster expanding region would have to either instantly revert to the ambient expansion rate, or some spacetime feature, such as a singularity, would have to occur to prevent a null ray from entering the faster-expanding region from the slower-expanding region. Addi- tional boundary conditions would have to be imposed on the singularity to restore predictability. An example of such a process can be found in Ref. [15] in connection with topological inﬂation [16, 17].

Another way of understanding the loss of predictability is the following. Whenever a faster expanding universe is created, it must be connected by a wormhole to the ambient slower expanding region. The wormhole can be kept open if the energy

84 η

η b b Q η Q ? ? η ? P P η a a

Figure 2.3: A spacetime diagram similar to that in Fig. 2.2 but one in which the NEC violation occurs over a sub-horizon region (shaded region in the diagram). Now the null ray bundle from a to b goes from being converging (within the horizon) to diverging (outside the horizon). However, it does not encounter any NEC violation along its path, and this is not possible as can be seen from the Raychaudhuri equation. Since the ingoing null rays are convergent as far out as the point P , the size of the quantum domain has to extend out to at least the inverse Hubble size of the initial de Sitter space. Therefore the NEC violating patch has to extend beyond the initial horizon.

85 conditions are violated [18]. But, if the wormhole neck is small, as soon as the energy condition violations are over, it must collapse and pinch oﬀ into a singularity. Signals from the singularity can propagate into the faster expanding universe destroying predictability. However, if the neck of the wormhole is larger than the horizon size of the ambient universe, the ambient expansion can hold up the wormhole and the neck does not collapse even after the NEC violation is over.

Our argument that NEC violations on scales larger than the horizon are needed to produce a faster expanding universe is consistent with earlier work [19] showing that it is not possible to produce a universe in a laboratory without an initial singularity

(also see [20]). Subsequent discussion of this problem in the quantum context [21,

22, 23], however, showed that a universe may tunnel from nothing without an initial singularity, just as in quantum cosmology [24, 25]. Such a tunneling event, however, is irrelevant to Island Cosmology, as the newly created universe is causally disconnected from the ambient Λ-sea. Without an inﬂaton, the process would therefore produce only a second Λ-sea.

Based on the above arguments, and on the results of the earlier investigations cited, we conclude that to get a faster expanding region that lasts beyond the duration of the quantum ﬂuctuation and remains predictable, the spatial extent of the NEC

−1 violating ﬂuctuation must be larger than Hi :

−1 R > Hi (2.6) where R is the spatial extent of the ﬂuctuation and shows up as the spatial smearing ˆ scale in the calculation of Orms.

86 We also argue that the temporal scale of the ﬂuctuation has to be small. This is ˆ because, an explicit evaluation [2] shows that Orms is proportional to inverse powers of the temporal smearing scale and diverges as the smearing time scale T → 0. Hence the briefer the ﬂuctuation, the stronger it can be, as we might also expect from an application of the Heisenberg time-energy uncertainty relation. Therefore we take the time scale of the NEC violation to be vanishingly small:

T → 0 (2.7)

2.5 Likelihood – the role of the observer

In the preceding sections I have described the nature of the ﬂuctuation. I now turn to the question of how likely are ﬂuctuations of the kind described.

In Sec. 2.4 I have pointed out that the NEC-violating fluctuations need to have two requirements to be cosmologically relevant - they need to have a superhorizon spatial extent and must be of vanishningly small duration. There is one more require- ment that is absolutely crucial - the fluctuations must have the correct amplitude in order to have sufficient energy density to lead to our observed Universe. Clearly, only if the temperature produced is high enough and the end point of the NEC violating fluctuation is a thermal state with all the different forms of matter in thermal equilibrium, further evolution of the island will simply follow the standard big bang cosmology.

Admittedly the three requirements of large spatial extent, small temporal extent and large amplitude make these ﬂuctuations rare. However, since spacetime is eternal

87 in this model we can wait indeﬁnitely for such a ﬂuctuation to occur.

The probability of ﬂuctuations in the Λ-sea that can lead to an inﬂating cosmology versus those that produce an FRW universe have been considered by several researchers [9, 26]. In particular, Dyson et al. [9] estimate probabilities based on a

“causal patch” picture, which assumes that the physics beyond the de Sitter horizon is irrelevant to the physics within, and that the latter should be regarded as the com- plete physics of the Universe. Based on this picture, the authors of [9] conclude that it is much more probable to directly create a universe like ours than to arrive at our present state via inflation. Albrecht and Sorbo [26] have argued that the conclusion rests crucially on the causal patch picture, and provide a different calculation leading to the conclusion that inflationary cosmology is favored. Both the above calculations assume the existence of fields that are suitable for inflation. However, Island Cos- mology does not rely on the hypothesis of the inflaton, and so the comparison of the likelihood of inflation versus no inflation is moot.

The monopole overabundance problem can be resolved in Island Cosmology in a manner similar to that proposed in Ref. [27], by assuming that the temperature required for magnetic monopoles production is higher than that required for matter- genesis. If the temperature at the beginning of the FRW phase is below that needed for monopole formation but above the matter-genesis temperature then there will be no cosmological magnetic monopole problem.

Another important question is where we are located on the island. Are we close to the edge of the island (“beach”)? In that case we would observe anisotropies in the

CMB since in some directions we would see the Λ-sea while in others we would see

88 inland. However, the island is very large (by a factor a0/af ) compared to our present

−1 horizon, HΛ . If we assume a uniform probability for our location on the island, our

−1 distance from the Λ-sea will be an O(1) fraction of HΛ a0/af . Since a0/af is of order

Tmg/T0 – the ratio of the matter-genesis temperature to the present temperature – we are most likely to be suﬃciently inland so as not to observe any anisotropy in the

CMB.

Whereas inﬂationary models crucially rely on the existence of a suitable scalar

field (inflaton), I have so far not specified the quantum field that causes the NEC violating fluctuation. I now turn to this issue.

2.6 The NEC violating ﬁeld

The phantom energy that is assumed to describe the effects of the NEC violating quantum fluctuation, by definition, satisfies ρ + p < 0. In addition, the assumption that the backreaction is given by Eq. (3.38), requires ρ > 0. Hence we need a quantum field that can give NEC violating fluctuations while still having positive energy density. In other words, the energy density should be positive but the pressure should be sufficiently negative so that the NEC is violated.

First consider a scalar ﬁeld, φ, with potential V (φ). The energy density and pressure are:

1 1 ρˆ = φ˙2 + (∇φ)2 + V (φ) 2 2 1 1 pˆ = φ˙2 − (∇φ)2 − V (φ) (2.8) 2 6

89 where the hats on ρ and p emphasize that these are quantum operators Therefore:

(∇φ)2 ρˆ +p ˆ = φ˙2 + (2.9) 3

The operatorsρ ˆ andρ ˆ +p ˆ are not proportional to each other and fluctuations in one do not have to be correlated with fluctuations of the other. The energy density in a region can be positive while the NEC is violated. Therefore a scalar field, even if V (φ) = 0, can provide suitable NEC violating fluctuations.

The particle physics in the very early stages of the model is described by low energy particle physics that we know so well. At present we do not have any experimental evidence for a scalar field. One field that we know of today is the electromagnetic field.

Could the electromagnetic ﬁeld give rise to a suitable NEC violating ﬂuctuation?

For the electromagnetic ﬁeld we have:

1 ρˆ = (E2 + B2) 2 1 1 pˆ = (E2 + B2) = ρˆ (2.10) 6 3

So nowρ ˆ andp ˆ are not independent operators and

4 ρˆ +p ˆ = ρˆ (2.11) 3

From this relationship between the operators, it is clear that the only electromagnetic

ﬂuctuation that can violate the NEC also has negative energy density. This means that even though the electromagnetic ﬁeld can violate the NEC, it does not satisfy

90 the positive energy density condition needed in the working hypothesis to ﬁnd the backreaction. (Our working hypothesis for the backreaction is described in § 3.3.2).

It may be possible that the electromagnetic ﬁeld will still be found to be suitable once we know better how to handle the backreaction problem. Then perhaps we will not need to rely on the working hypothesis that requires positive energy density.

There is a possible loophole in our discussion of the electromagnetic ﬁeld. The equation of statep ˆ =ρ/ ˆ 3 follows from the conformal invariance of the electromagnetic

ˆµ ﬁeld Tµ = 0. However, we know that quantum eﬀects in curved spacetime give rise to ˆµ a conformal anomaly and the trace hTµ i is not precisely zero. So we can expect that the equation of statep ˆ =ρ/ ˆ 3 is also anomalous. Whether this anomaly can allow for

NEC violations with positive energy density is not clear to us.

Note that it is not necessary for the NEC violation to originate from a fluctuation of a massless or light field. The arguments of Sec. 2.3 are very general and apply to ˆ massive fields as well. Though, for a massive field, Orms will be further suppressed by exponential factors whose exponent depends on powers of HΛ/m. While the likelihood of a suitable NEC violating fluctuation from a very massive field is much smaller compared to that of a light or massless field, the massive field fluctuations are clearly more important if the light field doesn’t even exist! The discussion in the previous section of the likelihood still applies.

2.7 Assumptions

Island cosmology involves several assumptions that I have pointed out above but now summarize and discuss.

91 Our ﬁrst assumption is that the dark energy is a cosmological constant. This is consistent with observations and moreover is the simplest explanation of the Hubble acceleration. We assume that the cosmological constant provides us with a background de Sitter spacetime that is eternal2. As de Sitter spacetime also has a contracting phase, the singularity theorems of Ref. [29] are evaded.

The second assumption is that there is a scalar field in the model responsible for the NEC violation. It would have been more satisfactory if the electromagnetic field could have played this role but we have shown (up to the loophole of the conformal anomaly) that the conformal invariance of the electromagnetic field prevents NEC violations with positive energy density. It is possible that with a better understanding of the backreaction of quantum energy-momentum fluctuations on the spacetime, the electromagnetic field might still provide suitable NEC violations (see Sec. 3.3.2).

The basic formalism of quantum ﬁeld theory in curved spacetime clearly leads to

NEC violations and this is not an assumption. (Though one could reasonably question the applicability of quantum field theory on systems with horizons.) Then there seems little doubt that there should exist large amplitude NEC violations, though occurring much more infrequently than the small amplitude violations. The idea that NEC violating fluctuations could have played an important cosmological role is also to be found in the “eternal inflation” scenario [30]. Indeed, the current scenario may also be viewed as an eternal inflation scenario – since the universe is eternally inflating due to a cosmological constant! While we may not be able to test the idea of cosmological

NEC violating ﬂuctuations, we can certainly test quantum ﬂuctuations with and

2For a discussion of the timescale on which the spacetime can remain de Sitter, see Ref. [28].

92 without horizons in laboratory experiments [31, 32, 33, 34].

The third assumption we have made has to do with NEC violations in regions of small spatial extent. Based on work done on the possibility of creating a universe in a laboratory, topological inﬂation, and wormholes, we have argued for the conjecture that small scale violations of NEC can only give rise to universes that are aﬀected by signals originating at a singularity. Hence predictability is lost in such universes.

Our assumption is that even if we did know how to handle the spacetime singularities aﬀecting these universes, they would turn out to be unsuitable for matter genesis.

Without this assumption, we should also be considering such universes as possible homes.

The fourth assumption is that the ﬁnal state of the ﬂuctuation is a thermal state.

All the diﬀerent energy components are also assumed to be in thermal equilibrium.

We have then assumed that the critical temperature needed for observers to exist is the temperature at which matter-genesis occurs. One could relax this assumption but one would need an adequate characterization of the most likely state to be able to calculate cosmological observables (e.g.spectrum of density ﬂuctuations).

This brings us to the part of the model where we argue that even if the large amplitude fluctuations are infrequent, they are the only ones that are relevant for observational cosmology. This is quite similar to the arguments given in the context of eternal inflationary cosmology where thermalized regions are relatively rare but these are the only habitable ones. It also occurs in chaotic inflation [35], where closed universes of all sizes and shapes are produced but only a few are large and homogeneous enough to develop into the present universe. So this part of Island

93 Cosmology is no weaker (and harder to quantify) than other cosmological models.

2.8 Conclusions

To conclude, we have investigated a new cosmological model, which we call “island cosmology”, where large NEC violating quantum ﬂuctuations (“upheavals”) in a cosmological constant de Sitter universe create islands of matter. In island cosmology, spacetime may be non-singular and eternal3. and an inﬂationary stage is not necessary.

Island cosmology is attractive because it is a minimalistic model. It uses currently observed features of the Universe as its ingredients and combines them with well- established results from quantum ﬁeld theory to account for the Universe that we live in now. However, the crucial test for any cosmological model comes from the spectrum of density perturbations that the model predicts. The next chapter is devoted to exploring this question in detail.

3The essential point is to have an expanding de Sitter phase; whether this is part of an eternal de Sitter spacetime or originates at a big bang makes no diﬀerence.

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97 Chapter 3

Perturbation Spectra

3.1 Introduction

This chapter explores the issue of determining the nature of perturbations generated by Island Cosmology. I ﬁrst review the theory of cosmological perturbations, and demonstrate how the theory predicts a scale invariant spectrum for inﬂation. I then describe the challenges involved in computing the spectrum generated in Island Cos- mology. Finally I review my research [1] in calculating this spectrum based on some classical assumptions.

3.2 The Theory of Cosmological Perturbations

The goal of this theory is to ﬁnd the relationship between matter and metric perturbations by expanding the Einstein equations to linear order about the background metric. The following discussion is based on the books and reviews ([2, 3, 4, 5, 6]). In

98 what follows, Greek letters denote spacetime indices, Roman numerals denote spatial

indices, and primes denote derivatives with respect to conformal time (except when

noted otherwise).

3.2.1 The metric perturbations

The most general linear perturbation in the spatially ﬂat FRW metric (see § 1.2 for

an introduction to the FRW metric) can be written as

2 2 2 i i j ds = a (1 + 2A) dη − 2Bidx dη − (δij + hij) dx dx (3.1)

where i and j stand for spatial indices. The space and time perturbations shown above can be regarded as three-dimensional tensors whose indices can be raised or lowered by the spatial metric δij. Since general vector and tensor ﬁelds are often a mixture of “pure” scalar, vector and tensor modes (based on their transformations

under spatial rotations), it is instructive to decompose the perturbations in (3.1)

accordingly.

A spatial vector ﬁeld Bi can be decomposed into a longitudinal part and a

transverse part, where the longitudinal part is curl-free and the transverse part is

divergence-free:

¯i ¯i Bi = ∇iB + B , ∇iB = 0 (3.2)

The longitudinal part is written as the divergence of a scalar B, and the transverse part B¯i has two vector degrees of freedom (the index i can take on three values, but one degree of freedom is lost from the second relation in (3.2).

99 Variable Scalar Vector Tensor

AA ¯ Bi B Bi (2) ¯ ¯ij hij C, E Ei (2) E (2)

Total 4 4 2 Table 3.1: Scalar-Vector-Tensor decomposition of metric perturbations

Similarly, any symmetric tensor can be built from two scalar, two vector and two

transverse and traceless tensors as follows:

¯ ¯ hij = 2Cδij + 2∇i∇jE + 2∇(iEj) + Eij (3.3)

¯ ¯ij ¯ij where the Eij are transverse ∇iE = 0 and traceless δijE tensors. The perturbations in (3.1) therefore break up into 4 scalar, 4 vector, and 2 tensor modes, as summarized in Table 3.2.1.

At linear order, the scalar, vector and tensor modes can be treated independently.

The tensor ﬂuctuations do not couple (at linear order) to matter perturbations, and vector perturbations redshift away in an expanding background. As a result it is only the scalar perturbations that are most important in understanding the formation of structure.

100 3.2.2 Gauge Issues in Cosmology

One must be particularly cautious about gauge artefacts (spurious gauge modes

which do not correspond to physical degrees of freedom) in the theory of cosmological

perturbations because of the way perturbations are deﬁned. A perturbation in a given

physical quantity (e.g. the Ricci scalar) at a given point is the diﬀerence between the value of the quantity in the real physical spacetime, and the value it assumes at the

same point in an unperturbed spacetime (see, for example [2]). Clearly, one needs a

map to transform between “corresponding” points on the two diﬀerent spacetimes.

A particular choice of such a map is called a gauge choice. Fixing a gauge amounts to threading the spacetime into lines of ﬁxed x, and slicing the spacetime into hypersurfaces of ﬁxed t.

It is not hard to see that some of the metric perturbation degrees of freedom in table 3.2.1 are spurious gauge modes. This can be seen by studying the eﬀect of coordinate transformations on the metric. Consider the coordinate transformation

xµ → x˜µ = xµ + ξµ (3.4)

Decomposing the spatial part of ξµ as in (3.2) into the gradient of a scalar ξ and a transverse piece ξi, it is clear that there are four physical degrees of freedom (2 scalar and 2 vector) associated with this transformation. Clearly, the tensor modes are not aﬀected by this transformation, and are automatically gauge-invariant.

One can now apply this transformation to the metric (3.1), and (keeping in mind that the line-element is left invariant by this transformation), one can derive trans-

101 formation rules for the scalar perturbation variables

a0 A˜ = A − ξ0 − ξ00 a B˜ = B + ξ0 − ξ0 a0 1 C˜ = C + ξ0 − ∇2ξ a 3 E˜ = E − ξ (3.5)

Also from (3.4) it is easy to deduce that a perturbation in a scalar quantity f

(such as a ﬁeld or an energy density) transforms as

δf˜ = δf − f 0ξ0 (3.6)

There are two standard approaches to dealing with the ambiguity of having more gauge modes than physical modes. One of them is to work with gauge invariant variables, which are combinations of perturbation variables, and which correspond to physical degrees of freedom and are hence unaﬀected by gauge transformations.

One set of gauge invariant variables introduced by Bardeen [7] is the following:

1 Φ = A + [(B − E0) a]0 (3.7) a a0 Ψ = C − (B − E0) (3.8) a

Φ and Ψ are commonly known as the Bardeen potentials.

A diﬀerent approach is to ﬁx a gauge, for which there is a variety of gauge choices

available [3]. A very common choice is the longitudinal or conformal Newtonian

102 gauge, in which B = E = 0. An advantage of longitudinal gauge is that the Bardeen potentials Φ and Ψ are numerically equal to the metric perturbations A and C in this gauge. The slicing and the threading are clearly orthogonal.

Another commonly used gauge is the synchronous gauge, which corresponds to the choice A = B = 0, so that the only non-zero perturbations are D and E. The advantage of this gauge is that the threading consists of geodesics, making this gauge a natural choice if one wishes to work in comoving co-ordinates. Synchronous gauge is a popular choice for numerical computations and is used in the CMBFAST1 code.

[8] provides a detailed review and comparison of both of these gauges.

3.2.3 The Comoving Curvature Perturbation

The curvature perturbation ψ is deﬁned in terms of R(3), the intrinsic spatial curvature on hypersurfaces of constant conformal time η, for a ﬂat Universe, through the relationship [3] 4 4 1 R(3) = ∇2ψ = ∇2 D + E (3.9) a2 a2 3

From the connection with the metric perturbation variables, it is clear that ψ is gauge dependent.

In a generic gauge, let the perturbation in some scalar quantity φ be δφ.

Now suppose we transform to a synchronous gauge, where the threading consists of geodesics and the slicing is comoving. Observers in these comoving coordinates will ﬁnd the φ to be isotropic, i.e., δφcom = 0.

1http://cfa-www.harvard.edu/~mzaldarr/CMBFAST/cmbfast.html

103 But the value of the perturbation in the two gauges is linked via (3.6):

0 δφcom = δφ − φ δη (3.10) implying that the time step δη necessary to switch from a generic gauge to a comoving gauge is δη = δφ/φ0.

From the gauge transformation laws 3.5, we conclude that the change in ψ is as follows: δφ R ≡ ψ ≡ ψ + Hδη = ψ + H (3.11) com φ0

The above quantity, the curvature perturbation on comoving hypersurfaces R, or the comoving curvature perturbation is a crucial cosmological observable. It is particularly useful in inﬂationary cosmology as it is gauge invariant (by construction) and roughly constant on superhorizon scales [9], and can therefore be used to relate scales leaving the horizon during inﬂation to scales re-entering the horizon at last scattering [2].

In the classical treatment of Island Cosmology that follows (§ 3.6.2), the relationship between R and the Mukhanov variable (explained in § 3.2.6) is used to compute the latter.

3.2.4 The Power Spectrum

Consider a quantum ﬁeld g(x, t) expanded in Fourier modes gk(t) as follows:

Z d3k g (x, t) = eik·xg (t) (3.12) (2π)3/2 k

104 The power spectrum of the ﬁeld g(x, t), Pg is deﬁned through the relationship

2π2 h0| g∗ g |0i = δ3 (k − k ) P (3.13) k1 k2 1 2 k3 g

where |0i denotes the vacuum state.

The variance of the ﬁeld g(x, t) from equations (3.12) and (3.13) works out to

Z dk h0| g2(x, t) |0i = P (k) (3.14) k g

However, using Parseval’s theorem, the variance of g(x, t) can be written as

Z 3 2 d k 2 h0| g (x, t) |0i = |gk (t)| (3.15) (2π)3

Comparing (3.14) and (3.15), we get another very useful deﬁnition of the power

spectrum k3 P (k) = |g (t)|2 (3.16) g 2π2 k

The power spectrum measures the variance of the ﬁeld or “power” per unit logarithmic

wavenumber interval. A scale-invariant spectrum Pg (k) = const is one in which the power per unit logarithmic wavenumber interval is independent of the wavenumber.

The departure of a spectrum from scale-invariance (or the spectral tilt) is measured by the power spectral index ns which is usually deﬁned by the relation

d ln P n − 1 = g (3.17) s d ln k

105 3.2.5 The Equations of Motion

To derive the linearized equations describing the evolution of the perturbations, one

can either work with gauge-invariant perturbation variables, or one can ﬁx a gauge.

For simplicity, we choose the latter option and work in the longitudinal gauge (see

i § 3.2.2). We also assume that there is no anisotropic stress, e.g., δTj = 0, which at once implies, from the constraint equation, that the two perturbation variables are

equal: A = C ≡ Φ. The assumption of no anisotropic stress is appropriate for perfect

fluids and scalar field matter. The field is linearized as in (1.108).

We therefore have two ﬂuctuating variables: the metric perturbations represented

by Φ and the ﬁeld perturbations by δφ. The equations governing the evolution of the perturbations are the following: (the subscript 0 in the homogenous part of the ﬁeld

0 ∂V is dropped, and V (φ) ≡ ∂φ ):

∇2Φ − 3HΦ0 − H0 + 2H2 Φ = 4πG φ0δφ0 + V 0a2δφ

Φ0 + HΦ = 4πGφ0δφ (3.18)

Φ00 + 3HΦ0 + H0 + 2H2 Φ = 4πG φ0δφ0 + V 0a2δφ

Combining these equations, and eliminating δφ one gets a second order equation governing the metric perturbation Φ:

φ00 φ00 Φ00 + 2 H − − ∇2Φ + 2 H0 − H Φ = 0 (3.19) φ0 φ0

The fact that the perturbations are expressible in terms of a single ﬂuctuating variable

indicates that the system has one true dynamical degree of freedom.

106 The form of (3.19) provides some insight into the physics of the evolution of the perturbations. The last term represents the gravitational force causing the instability, the Laplacian term represents the restoring force due to pressure, and the second term represents Hubble friction. For superhorizon modes k < H there is no pressure force, and the perturbations freeze out, while the growth of their amplitude is governed by the other two forces. For subhorizon modes, the pressure force leads to damped oscillations.

(3.19) can actually be solved asymptotically in the short and long wavelength approximations [2]. Introducing the new variable

a u = Φ (3.20) φ0 one can reduce (3.19) to the form

θ00 u00 − ∇2u − u = 0, θ = H/ (aφ0) (3.21) θ

Solving this in the asymptotic limits, one can obtain explicit expressions for Φ and

δφ (see [2]) for details).

A diﬀerent way of manipulating (3.19) is as follows. Deﬁning a variable ζ

2 H−1φ˙ + φ ζ ≡ Φ + (3.22) 3 1 + w where w = p/ρ, then (3.19) can be expressed in a very simple form

ζH˙ (1 + w) = O ∇2Φ (3.23)

107 Since the right side of (3.23) is negligible for superhorizon modes, (3.23) implies conservation of ζ on large scales. The variable ζ happens to be a gauge-invariant quantity which can be physically interpreted as the curvature perturbation on slices of constant energy density. It is approximately equal to the comoving curvature perturbation R (see § 3.2.3) on large scales [3].

The constancy of ζ on large scales allows one to use (3.22) to relate initial and

ﬁnal values of the metric perturbation. For example, if the amplitude of Φ is known at the time of horizon exit during inﬂation, one can use (3.22) to deduce the amplitude of Φ at Horizon entry during last scattering.

Finally, note that the constraint equations link Φ to the amplitude of density perturbations, and hence the amplitude of large angle temperature anisotropies in the CMB via the Sachs Wolfe eﬀect [10]:

δT (e) 1 = Φ(x ) (3.24) T 3 ls

−1 where e signiﬁes the direction of observation, and xls = 2H0 e.

3.2.6 Quantum Theory of Cosmological Perturbations

The inhomogeneities in the inflaton are the result of quantum fluctuations, so in order to fully understand their origin and evolution, a quantum mechanical treatment is essential. The smallness of the fluctuations allows one to sidestep the problem of metric backreactions by using the semiclassical approximation, namely, that the spacetime responds only to the expectation value of the energy-momentum tensor. Given this hypothesis (which is well-motivated by the smallness of the CMB anisotropies), and

108 given that the system has only one true dynamical degree of freedom (as demonstrated in the previous subsection), the quantum mechanical theory of cosmological perturbations reduces to the quantum theory of a single free scalar ﬁeld with a time dependent mass. I now provide a very brief summary of the steps in this analysis.

1. The starting point is the Einstein-Hilbert action (1.31), and a single scalar ﬁeld.

2. The metric is perturbed using the longitudinal gauge prescription, and the ﬁeld

is split as in (1.108).

3. The action is then expanded up to second order in the perturbative variables.

4. A gauge-invariant variable v, which is a combination of metric and matter

perturbations, is identiﬁed, such that the action takes on a canonical form in v:

1 Z z00 S(2) = d4x v02 − (∇φ)2 + v2 (3.25) 2 z

The variable v, called the Mukhanov variable plays a crucial role in the theory of cosmological perturbations. v and z are deﬁned as follows:

φ v = a δφ + Φ (3.26) H aφ0 z = (3.27) H

From equation (3.11), we see that v R = (3.28) z

109 The action (3.25) immediately dictates the following equation of motion for the

Mukhanov variable (in Fourier space):

z00 v00 + k2 − v = 0 (3.29) k z k

The next step is to solve this equation. If we assume that the slow roll conditions

hold (see § 1.9.3), then φ0 and H evolve much slower than the scale factor, and (3.27) implies that z00 a00 ≈ (3.30) z a

If we also assume that the spacetime evolves as de Sitter (with the scale factor given

by (2.1)) i.e., we take → 0, then equation (3.29) turns into the familiar equation of

a massless scalar ﬁeld in de Sitter space:

a00 v00 + k2 − v = 0 (3.31) k a k

The exact solution of Eq. (3.31) with the boundary condition that small wave-

length modes go over into Minkowski space modes is:

e−ikη i vk = √ 1 − (3.32) 2k kη

∗ The other independent mode is vk. These are the mode functions for the Bunch- Davies vacuum [11]. A derivation of these mode functions using inverse scattering

technology can be found in [12]. The Bunch Davies modes make for a good choice of

initial conditions also because they are known to be a local attractor in the space of

110 initial conditions in an expanding background [13].

Now using (3.16), and in the limit of superhorizon modes k |η| 1, the power spectrum of the comoving curvature perturbation (due to the modes exiting the Hori- zon during inﬂation) turns out to be

1 H4 PR = (3.33) 2 ˙2 4π φ k=aH

This is the well-known scale-invariant, or Harrison-Zeldovich spectrum, which is one of the most successful predictions of inﬂation, and seems to be bourne out by

CMB observations. Note that our simplified analysis (where we effectively took the slow-roll parameters to be zero) hides the fact that H and φ˙ slowly evolve during inflation, and so a small scale-dependence is implicit in the ratio H2/φ˙. In fact it

2 ˙ 2 ˙ is better to replace H /φ by Hk /φk, where the subscript k signiﬁes the value of a quantity at the time the mode k exists the horizon.

In terms of the ﬁrst slow roll parameter (deﬁned in (1.112)) the power spectrum can be written as

1 V PR = 2 4 (3.34) 24π mPl

This equation shows how CMB observations of the amplitude of the perturbations can be used to constrain the scale of the inﬂation potential.

We take the scale-dependence into account in calculating the spectral tilt from

(3.17)

111 d ln P d ln H4 d ln φ˙2 n − 1 = R = k − k = 2η − 6 (3.35) s d ln k d ln k d ln k where we have used the manipulation d ln k = d ln (aH) ' d ln a. The form of the spectral index in terms of the slow roll parameters (3.35) agrees with the standard result from inﬂation.

3.3 Diﬃculties in computing perturbations from

Island Cosmology

3.3.1 Classical vs Quantum Fields

Unlike most models of inﬂation, however, computing the perturbation spectrum in

Island Cosmology is extremely difficult. First of all, in Island Cosmology, both the background field φ0 and the perturbation δφ are quantum operators. To calculate density fluctuations due to δφ, one needs a suitable model for the evolution of φ0 during the NEC violating fluctuation. This evolution is quantum and not described as a solution to some classical equation of motion. The closest related problems that have been addressed in the literature are the production of particles during the quantum creation of the universe and the fluctuations of a vacuum bubble that has itself been produced in a tunneling event [14, 15, 16]. These analyses rely on the existence of an instanton describing the tunneling event. In our case, the NEC violation is not described by an instanton; instead it is described by the most probable fluctuation leading to matter-genesis. Hence the existing techniques do not apply directly and

112 new techniques are needed.

3.3.2 Backreaction on spacetime

Another major diﬃculty in computing the perturbation spectrum is the lack of an adequate characterization of the backreaction of the metric. Hence the best one can do is assume some hypothesis for the backreaction. I explain this in detail below and also discuss our working hypothesis for the backreaction on the spacetime.

In this model, we have quantum ﬁelds in a classical spacetime. On the surface, this seems to be an ideally suited for the machinery developed in the theory of semiclassical relativity:

ˆren Gµν = 8πGhTµν i (3.36) where hi denotes the expectation value in some speciﬁed quantum state. In other words, semiclassical relativity assumes that the spacetime responds only to the expectation value of the energy-momentum tensor - ﬂuctuations in the energy-momentum tensor about the mean are unimportant. However, in studying Island Cosmology,

ˆren we need to determine the effects of fluctuations of Tµν . So it is essential that we go beyond semiclassical relativity to be able to treat the backreaction of the NEC violating fluctuations on the spacetime. For small fluctuations, one could envisage expanding the metric around a fixed background and quantizing the metric fluctuations. Such an attempt has been made in Refs. [17] though not in the context of NEC violations. The perturbative scheme can not however hope to capture the physics of large fluctuations of the kind we are interested in.

Since a rigorous treatment of the backreaction is not possible, we shall adopt a

113 “working hypothesis” in which the NEC violating ﬂuctuation behaves like “phantom

energy” i.e. a classical perfect ﬂuid with equation of state w ≡ p/ρ < −1 where

ρ > 0. Furthermore, in the sudden approximation discussed in the previous section,

the phantom energy exists only for a vanishingly small time period. Hence the energy

content of the universe has the following time dependence:

ρ = Λ , w = −1 , η < ηf 1 ρ = ρ , w = + , η > η (3.37) FRW 3 f

where ηf denotes the instant at which the NEC violating ﬂuctuation occurs and

ρFRW denotes the energy density after the NEC violating ﬂuctuation is over and this is assumed to be dominantly in the form of radiation. The initial condition for the

FRW phase is: ρFRW(ηf ) = ρmg, the radiation density required for matter-genesis. With this working hypothesis for the energy content of the local universe, the backreaction on the spacetime is given by:

8πG H2 = ρ (3.38) 3

where, as usual, H =a/a ˙ and a(t) is the local scale factor. For η < ηf , ρ is a constant and H = HΛ which is constant. In a vanishingly small interval around

ηf , ρ increases rapidly due to the NEC violating fluctuation and this means that H also increases correspondingly. This implies that the scale factor grows faster than exponentially during the NEC violating fluctuation, yielding a vanishingly short period of “super-inflation”. After the NEC violating fluctuation is over, the region is

114 ﬁlled with radiation energy density and the FRW epoch starts. We summarize the

behavior of the Hubble scale, H, as follows:

H = HΛ , η < ηf

H = HFRW , η > ηf (3.39)

with the initial condition HFRW(ηf ) = Hmg where Hmg is the Hubble constant at the epoch of matter-genesis.

In writing Eq. (3.38), we are assuming that spacelike surfaces of constant ρ are ﬂat.

This is a consequence of our choice of the Bunch Davies modes as initial conditions

which assume a spatially ﬂat spacetime slicing. That is why we have not included

the spatial curvature term, k/a2. This also consistent with our working hypothesis

since the initial state (η < ηf ) is ﬂat and the universe expands even faster during the phantom energy stage which is entirely classical.

In the oft studied example where a scalar ﬁeld tunnels through to a diﬀerent

value, it is known that the surfaces of constant ﬁeld have negative spatial curvature.

The energy density in the ﬁeld is purely due to potential energy and so the surfaces

of constant ﬁeld are also surfaces of constant energy density. Hence the tunneling

event produces an open universe with negative spatial curvature [18]. The scenario

in this model is diﬀerent from the tunneling scenario because there is no instanton

that describes the NEC violating ﬂuctuation. It can be shown explicitly that the

tunneling process preserves de Sitter invariance [15] (though see the caveat mentioned

in Footnote 33 in [19]) and this symmetry implies hyperbolic spatial slicings (i.e. open universe slicings) of the spacetime. In our case we know that NEC violations only

115 occur if de Sitter invariance is broken. This can be seen by considering

ˆren ˆren hTµν Tλσ i (3.40)

If we demand that this be a tensor respecting the de Sitter symmetry, then it must be expressible in terms of the metric tensor since this is the only tensor available to us.

µ ν ˆren 2 However, then, when we contract with null vectors to get h(N N Tµν ) i, the result

α β will be zero since gαβN N = 0, and there will be no NEC violating ﬂuctuations. This “working hypothesis” is probably the weakest assumption in our analysis.

However, we cannot do any better at the moment because the backreaction of quantum ﬂuctuations on the spacetime requires that we consider a quantum theory of gravity as well. The backreaction problem also occurs in eternal inﬂationary cosmology where a similar working hypothesis is used.

A rigorous calculation perturbation spectrum from Island Cosmology, particularly addressing the backreaction issue using quantum cosmology, string theory, or loop gravity is an open question for future work. In the rest of this chapter I describe some efforts to compute the perturbations using assumptions which alleviate these difficulties. The next section calculates the spectrum of perturbations generated in a field uncoupled to the NEC violating field, assuming that the NEC violation is practically instantaneous. In the section after that I investigate the scenario of a phantom field mimicking the NEC violation.

116 3.4 Perturbations in a Spectator Field

In this section, we consider a light field other than the NEC-violating field φ, and not interacting directly with φ (a “spectator” field), and find its power spectrum based on the assumption that the NEC violating fluctuation is instantaneous at the conformal time ηf .

Let us denote such a ﬁeld generically by χ, its eigenmodes by χk(η) exp(ik · x). From the discussion on cosmological perturbations earlier in this chapter, one can conclude that this ﬁeld will have a spectrum of perturbations given by:

k3 v 2 P (k, η) = k (3.41) χ 2π2 a

After the quantum ﬂuctuation is over, During the radiation dominated FRW epoch, i.e. for η > ηf , we have s t a(t) = af (3.42) tf where tf is the cosmic time at which the NEC violating ﬂuctuation occurs, and af ≡ a(tf ). (Note that the Hubble parameter is discontinuous at ηf in the sudden approximation but the scale factor is continuous.) In terms of the conformal time one

ﬁnds:

2 a(η) = af + af Hf (η − ηf ) (3.43)

Clearly a00 = 0. Therefore, (from (3.31)):

−ikτ +ikτ vk = αke + βke , τ ≡ η − ηf > 0 (3.44)

117 Next we need to solve Eq. (3.31) at η = ηf . This step is non-trivial since a is

0 00 continuous at ηf but a is discontinuous. Hence a has a delta function contribution. Using Eqs. (2.1) and (3.43) we ﬁnd:

a00 2 = Θ(η − η) + a ∆Hδ(η − η ) (3.45) a η2 f f f

where Θ(·) is the Heaviside function and

∆H ≡ Hf − HΛ ≈ Hf (3.46)

Integrating Eq. (3.31) in an inﬁnitesimal interval around ηf , we ﬁnd the junction conditions:

vk(ηf +) = vk(ηf −)

0 0 vk(ηf +) = vk(ηf −) + af ∆Hvk(ηf ) (3.47) where the last term is due to the δ−function piece in a00/a. We can now ﬁnd the coeﬃcients αk and βk by inserting the de Sitter and FRW mode functions and their derivatives at η = ηf in the junction conditions. This gives:

1 i α = v + v0 + a H v k 2 kf− k kf− f f kf− 1 i β = v − v0 + a H v (3.48) k 2 kf− k kf− f f kf−

0 where vkf− ≡ vk(ηf −) and similarly for the (conformal) time derivative vk.

118 We are interested in the long wavelength ﬂuctuations for which k|ηf | → 0. Then

0 the dominant contributions come from the vkf− and af Hf vkf− terms in Eq. (3.48)

2 and are of order 1/(kηf ) . However, the af Hf vkf− term is much larger than the vkf− term because Hf >> HΛ. (Recall from Eq. (2.1) that af ηf = −1/HΛ.) Therefore

1 1 H √ f αk ≈ + 2 2 2k (kηf ) HΛ 1 1 H √ f βk ≈ − 2 (3.49) 2 2k (kηf ) HΛ

Therefore −i 1 H √ f vk(η) ≈ 2 sin(kη) (3.50) 2k (kηf ) HΛ

Using Eqs. (3.43) and (3.50) in (3.41), together with η >> ηf gives:

1 1 sin(kη)2 Pχ(k, ηk) ≈ 2 4 2 4 (3.51) 4π af HΛηf kη

Making use of Eq. (2.1), af ηf = −1/HΛ, and taking the limit kη → 0, we ﬁnally get:

H2 P (k, η ) ≈ Λ (3.52) χ k 4π2

Since the result does not depend on k, the spectrum of χ ﬂuctuations is scale invariant, as in the inﬂationary case [20], with amplitude set by the cosmological constant.

(Corrections to scale-invariance, of course, can be expected from the ﬂuctuation not being exactly instantaneous).

As discussed earlier in this section, the result in Eq. (3.52) applies to all very light or massless ﬁelds other than, and not interacting directly with, the NEC violating

119 ﬁeld. In particular, in the context of the gravitational wave power spectrum, the

perturbation of the metric is equivalent to χ/mP where mP is the Planck mass.

2 Hence the power in gravitational waves is proportional to (HΛ/mP ) and is very tiny.

3.5 A Classical Treatment of Island Cosmology

3.5.1 Introduction

In this section we present a classical computation of the perturbations generated

in Island Cosmology by assuming that the NEC-violating ﬁeld behaves as a classical

phantom ﬁeld for the duration of the ﬂuctuation. Using an exactly-solvable potential,

we show that the model generates a scale-invariant spectrum of scalar perturbations,

as well as a scale-invariant spectrum of gravitational waves. The scalar perturbations

can have suﬃcient amplitude to seed cosmological structure, while the gravitational

waves have a vastly diminished amplitude.

We set our notation as follows: let us choose our cosmic time coordinate t such that

the phantom phase (described in Section 3.5.3) begins at t = ti = 0. In this paper we ﬁnd it more convenient to work in conformal time (η) where dt = a(η) dη and η ranges between (−∞, 0) as t goes from −∞ to +∞. We set our conformal time coordinates such that the period of phantom cosmology lasts between η = ηi = −1/HΛ and

η = ηf . The above choices of ti and ηi are arbitrary and for convenience. Primes denote derivatives with respect to conformal time, and dots denote derivatives with

respect to cosmic time. We adopt the convention that the suﬃx i denotes the value

of a quantity at η = ηi and the suﬃx f denotes the value at η = ηf .

120 I now brieﬂy recapitulate the diﬀerent stages of Island Cosmology, with special emphasis on the assumptions relevant to the present analysis.

3.5.2 The de Sitter phase

This phase represents the initial state of the Universe, before the onset of the phantom behavior. In this phase, we assume that the Universe is de Sitter space inﬂating due to the observed dark energy, which we assume is a cosmological constant. The Hubble parameter (H) has the same value that it has today, which we call HΛ . As discussed in the previous chapter, this expanding de Sitter background can be part of a classical de Sitter spacetime with no beginning and no end, with early contraction and then expansion. We will only consider the expanding phase of the de Sitter spacetime in the following discussion.

We assume that the matter content of the Universe is a classical scalar ﬁeld (φ) having a Lagrangian L given by:

µ L = λ∂ (φ)∂µ(φ) − V (φ) (3.53) and a stress tensor given by:

Tµν = λ∂µ(φ)∂ν(φ) − gµνL (3.54) gµν represent the components of the metric, and V (φ) represents the potential. For the sake of generality, we have inserted the constant λ which determines the sign of the kinetic term. Obviously, λ = +1 for an ordinary classical scalar ﬁeld, and λ = −1

121 for a phantom ﬁeld.

The equation of motion of the ﬁeld is the Klien Gordon equation:

1 √ ∂V λ√ ∂ −ggµν∂ φ + = 0 (3.55) −g µ ν ∂φ

The scale factor a(t) and Hubble value H(t) during this period can be written as follows:

For (−∞ < t ≤ ti = 0):

a(t) = eHΛt (3.56) a˙(t) H(t) = = H (3.57) a(t) Λ

In terms of conformal time, for (−∞ < η ≤ ηi):

1 a(η) = − (3.58) ηHΛ a0(η) 1 H(η) = = − (3.59) a(η) η

3.5.3 The Phantom Phase

In this phase, lasting between the times ηi and ηf , the Universe undergoes an NEC- violating quantum fluctuation. We model this phase by assuming that the matter content of the Universe behaves like a phantom field φp for the duration of the fluctuation.

122 We assume that this hypothetical phantom ﬁeld φp is classical, i.e., its Lagrangian and stress tensor are given by Eq. (3.53) and Eq. (3.54), and it satisﬁes the Klien

Gordon equation Eq. (3.55).

Using the above equations, one can readily determine the pressure p and energy

density ρ of the ﬁeld. These work out to:

φ02 p = 1 P T = λ p − a2(η)V (φ ) (3.60) 3 ii 2 p φ02 ρ = T = λ p + a2(η)V (φ ) (3.61) 00 2 p

2 Clearly, p + ρ = λφp indicating, as one would expect, that the NEC is violated during this period if our matter ﬁeld is phantom (λ = −1).

−1 −1 Also during this phase, the Hubble horizon size drops from HΛ to Hf . We assume that this drop is linear in cosmic time, ending at time tf . The validity of this assumption, as well as its implications on the matter content of the Universe are discussed later. Thus,

−1 −1 H (t) = HΛ − αt (for 0 ≤ t ≤ tf ) (3.62)

Here α is a dimensionless parameter measuring the rate at which the horizon size changes during the phantom phase. We assume that the quantum ﬂuctuation is very abrupt, and hence α is very large.

Using the deﬁnition of conformal time, it is easy to deduce that during this phase

123 (ηi ≤ η ≤ ηf ),

− 1 a(η) = [−α − HΛ(1 + α)η] 1+α (3.63) a0(η) H H(η) = = Λ (3.64) a(η) [−α − HΛ(1 + α)η] H(η) H(η) = = H [a(η)]α (3.65) a(η) Λ

We also need to address the nature of the back-reaction of matter on geometry.

Let us make the working hypothesis that the back-reaction is fully described by the

Friedmann equation.

The actual time duration of this phase can be calculated by demanding continuity of the Hubble value at t = tf (or equivalently, η = ηf ). Thus we require (see Eq. (3.65)),

H(ηf ) = Hf (3.66)

Solving the above equation for ηf , we obtain

" 1 # 1+ α 1 HΛ ηf = − − α (3.67) HΛ(1 + α) Hf

We assume that Hf HΛ and since α → ∞, the ﬁrst term in the square brackets can be ignored leaving us

α 1 ηf ' − − O (3.68) HΛ(1 + α) (1 + α)Hf

From this we can compute the duration of the phantom phase in conformal time (∆η),

124 as follows:

∆η = ηf − ηi 1 1 = (3.69) HΛ α + 1

Again, since α → ∞, this is a vanishingly small interval.

3.5.4 The Radiation Dominated FRW Phase

−1 In this epoch we have a volume of space of Hubble length Hf ﬁlled with classical radiation. Rapid interactions thermalize the radiation, after which this volume follows a standard FRW evolution.

The scale factor in this radiation-dominated epoch can therefore be written as

s t a(t) = af (3.70) tf

In terms of the conformal time, this reduces to

2 a(η) = af + af Hf (η − ηf ) (3.71)

3.5.5 Calculational Strategy

We make two key assumptions to facilitate our calculation, which we discuss below.

These are:

1. During the NEC-violating explosive event the energy content of the Universe

125 behaves as a phantom ﬁeld.

2. During the NEC violation, the drop in the Hubble scale is linear in cosmic time.

Assumption (1) is an attempt to model the behavior of the matter field during the NEC-violating event. To calculate density fluctuations due to fluctuations in the

NEC-violating field, one needs a suitable model for the evolution of the field itself during the NEC-violating fluctuation. This evolution is quantum and not described as a solution to some classical equation of motion. For the purpose of this calculation, we have made the simplifying assumption that the matter field behaves in the same manner as a classical object that would also violate the NEC and produce the same effect on the spacetime. Of course this purely classical treatment cannot substitute for a rigorous quantum mechanical treatment of the NEC-violation, but we hope that it captures the essential elements of the physics involved.

Assumption (2) can be justiﬁed considering that the drop in the Hubble length need not be linear throughout the explosive event, but only during the window of time δt that it takes for the scales observed today to leave the horizon. Since the

−1 ﬂuctuation itself is very short lived, by expanding H (t) in a Taylor series about ti, the drop in H−1(t) over δt can be well approximated to be linear.

We now turn our attention to computing the spectrum of perturbations that would be generated in this cosmological model. Our plan of action is the following:

1. Working in k (momentum or wavenumber) space, we ﬁrst ﬁnd expressions for a

gauge-invariant variable vk(η) , representing the true degrees of freedom of the system in all the three stages of the model.

126 2. The unknown coeﬃcients that arise in the above expressions are then deter-

0 mined by demanding continuity of vk(η) and its time derivative vk(η) at tran-

sition times ηi and ηf .

3. As discussed earlier, the adiabatic density perturbation responsible for structure

in the Universe is conveniently characterized by the curvature perturbation R

seen by comoving observers. Once we fully determine vk(η) in the radiation dominated phase, we obtain the co-moving curvature perturbation spectrum

from the relations (3.28) and (3.16).

3.6 Spectrum from a Classical Treatment

We next ﬁnd expressions for vk in the three stages of the model. The unknown coeﬃcients that appear in these expressions will be determined through a matching

process in § 3.6.3, and then the scalar and tensor power spectra are computed in §

3.6.4 and § 3.6.5 respectively.

3.6.1 vk in the de Sitter and FRW phases

For a scalar ﬁeld in de Sitter space, the variable vk(η) satisﬁes (3.31) with the solution (3.32)

During the FRW phase after the NEC-violating ﬂuctuation, from the discussion in §

3.4 we know that vk satisﬁes

ikτ −ikτ vk(τ) = αke + βke (3.72)

127 where αk and βk are constants of integration and τ = η − ηf > 0

3.6.2 vk in the Phantom Phase

In this case we will calculate vk starting from ﬁrst principles.

Matter and metric perturbations

The ﬁrst step is to perturb the matter and metric. Working in longitudinal gauge

and assuming no anisotropic stress, the scalar metric perturbations are written as:

  1 + 2Φ 0 0 0      0 −1 + 2Φ 0 0  2   gµν = a   (3.73)    0 0 −1 + 2Φ 0    0 0 0 −1 + 2Φ

Given our choice of gauge, the metric perturbation Φ(η, x) coincides with the gauge- invariant Bardeen potential (see, for example [2]).

The phantom matter ﬁeld φp(η, x) is perturbed as follows:

φp(η, x) = φ(η) + δφ(x, η) (3.74)

128 Evolution of the perturbations

To ﬁnd time evolution of the perturbations, we use the perturbed Einstein equations up to ﬁrst order. The i-i component of the zero-th order equations reads:

a0 2 a00 φ02 − Λa2 + − 2 = 8πG λ − a2V (φ) (3.75) a a 2 while the 0-0 component reads:

a0 2 φ02 Λa2 + 3 = 8πG λ + a2V (φ) (3.76) a 2

Adding these equations, one obtains the familiar relationship:

H2 − H0 = 4πGλφ02 (3.77)

The i-i, 0-0 and 0-i components of the ﬁrst order equations are respectively:

a0 2 a00 a0 Λa2Φ − 2Φ + 4Φ + 3 Φ0 + Φ00 = a a a 1 1 8πG a2V (φ)Φ − a2V δφ + λφ0δφ0 − λΦφ02 (3.78) 2 φ 2 a0 Λa2Φ − 3 Φ0 + ∇2Φ = a 1 1 8πG a2V (φ)Φ + a2V δφ + λφ0δφ0 (3.79) 2 φ 2 a0 Φ0 + Φ = 4πGλφ0δφ (3.80) a

(where Vφ = ∂V/∂φ).

129 Perturbing the Klein-Gordon equation Eq. (1.105) using Eq. (3.74) yields, at zero-

th order:

00 0 2 λ [φ + 2Hφ ] = −a Vφ (3.81)

Using Eq. (3.77), Eq. (3.78), Eq. (3.79), Eq. (3.80) and Eq. (3.81), one obtains the

equation of motion of Φ(η, x):

φ00 φ00 Φ00 − ∇2Φ + 2 H − Φ0 + 2 H0 − H Φ = 0 (3.82) φ0 φ0

Applying the Fourier transform:

Z d3k Φ(x, η) = Φ (η) eik.x (3.83) (2π)3/2 k

we obtain: φ00 φ00 Φ00 + 2 H − Φ0 + k2 + 2 H0 − H Φ = 0 (3.84) k φ0 k φ0 k

Note that Eq. (3.84) is independent of λ, indicating that it has the same form for a phantom ﬁeld as it would for a normal ﬁeld. This is a surprising result, since all the equations used to derive Eq. (3.84) are λ dependent. Physically, this result implies that the evolution of the metric perturbation is insensitive to whether the matter content of the Universe is normal or phantom.

To solve Eq. (3.84), we need to determine the dynamics of the phantom ﬁeld φ(η), which is in turn determined by the potential V (φ). We choose a particular form of the potential which allows for a solution in closed form:

130 (3 + α) 2αH V (φ ) = K2 exp Λ φ (3.85) p 2α K p

Here K is a constant that has the dimensions of mass squared, and sets the scale of the potential.

For this potential, it is easy to verify that the exact form of φ(η) which satisﬁes

Eq. (3.81) (with of course λ = −1 as is the case in the phantom phase) is:

K φ(η) = − ln [−α − HΛ(1 + α)η] (3.86) HΛ(1 + α)

Further, to facilitate the back-reaction as discussed in Section 3.5.3, and satisfy our ansatz given by Eq. (3.62) we must require that our field satisfies the Friedmann equation Eq. (3.76). The result of this is to fix the value of K:

r 3α K = H (3.87) Λ 4πG

The matter ﬁeld φ(η) has the interesting property that

φ00(η) = (1 + α) H(η) (3.88) φ0(η)

Using Eq. (3.88), Eq. (3.84) reduces to

00 0 2 2 Φk − 2HαΦk + k − 2H α Φk = 0 (3.89)

131 This is a familiar second order diﬀerential equation of the form:

00 0 Φk + P (η)Φk + Q(η)Φk = 0 (3.90) with

P (η) = −2H(η)α

Q(η) = k2 − 2H2(η)α which has the solution (see e.g.[21])

− 1 R P (η) dη α Φk(η) = e 2 χ(η) = a χ(η) (3.91) where χ(η) satisﬁes the diﬀerential equation

1 1 χ00(η) + Q − P 0 − P 2 χ(η) = 0 (3.92) 2 4

In our case, this reduces to

2 00 2 αHΛ χ (η) + k + 2 χ(η) = 0 (3.93) [α + HΛ(1 + α)η]

At this point it is convenient to temporarily switch to a new time variable x deﬁned by

x = α + HΛ(1 + α)η (3.94)

Note that x = −1 when η = ηi and from Eq. (3.68), x = O [HΛ/Hf ] or x ' 0 when

132 η = ηf . With x as the time variable, Eq. (3.93) takes the form

d2χ(x) 1 1 + m2 − p2 − χ(x) = 0 (3.95) dx2 4 x2

where m and p are deﬁned by

k m = (3.96) HΛ(1 + α) 1 α 1/2 p = − (3.97) 4 (α + 1)2

The solution to Eq. (3.95) can be written in terms of Bessel functions as

√ χ(x) = mx [AmJp(mx) + BmYp(mx)] (3.98)

where Am, Bm are constants and Jp and Yp denote Bessel functions of the ﬁrst and second kind of order p respectively.

Putting together Eq. (3.98) and Eq. (3.91), we conclude that the time evolution of Φk is fully described by the equation

α√ Φk(η) = a mx [AmJp(mx) + BmYp(mx)] (3.99)

where m, x and p are deﬁned by Eq. (3.96), Eq. (3.94) and Eq. (3.97) respectively,

and α is deﬁned in Eq. (3.62).

133 Calculating the Mukhanov Variable

We are ﬁrst going to compute Rk and z and then use Eq. (3.28) to compute vk.

Calculation of Rk: Noticing that for this space, we have

H0 − 1 = α (3.100) H2 and using equations Eq. (3.80) and Eq. (3.77), we can eliminate the δφ in Eq. (3.11),

0 giving us an expression for Rk involving Φk and Φk as the only ﬁrst order variables:

1 R = Φ − [Φ0 + HΦ ] (3.101) k k αH k k

(Again, the absence of λ indicates that the expression for Rk is insensitive to whether the ﬁeld is real or phantom.)

Calculation of z: In the deﬁnition of z Eq. (3.27) we eliminate φ0 using Eq. (3.77), and introduce α using Eq. (3.100) to get

r α z = a −4πGλ r 2α = am (3.102) Pl −λ

p where mPl is the reduced Planck mass deﬁned by mPl = 1/8πG, where G is New- ton’s gravitational constant.

134 Final expression for vk: Combining Eq. (3.28), Eq. (3.101) and Eq. (3.102), we

0 can express vk entirely in terms of Φk and Φk as follows:

r α 1 v = a Φ − (Φ0 + HΦ ) (3.103) k −4πGλ k αH k k

3.6.3 Calculation of Unknown Constants

The above calculations produced four unknown constants Am, Bm, in Eq. (3.98) and

αk and βk, in Eq. (3.72). These constants can be determined by demanding continuity

of vk and its time derivative at the two transition times ηi and ηf .

To determine Am and Bm, we perform the above matching process at η = ηi =

1 − , or in terms of x, (from Eq. (3.94)), at x = xi = −1 . In other words, we need HΛ to simultaneously solve the equations

vk(de Sitter)|(η=−1/HΛ) = vk(phantom)|(x=−1)

0 0 vk(de Sitter)|(η=−1/HΛ) = vk(phantom)|(x=−1) (3.104)

to ﬁnd the unknowns Am and Bm.

The expression obtained for vk in the phantom phase (by substituting the values of the above constants) is fairly complicated. However, since we are only interested in

the super-horizon modes, we can make the approximation that k (or m) → 0, and use

the appropriate asymptotic forms of the Bessel functions Jp(mx) and Yp(mx). Also,

1 since α is large, from Eq. (3.97), p ' 2 . With these simpliﬁcations the expression for

135 vk (in the phantom phase) reduces to:

ik/HΛ (1+α) i e a HΛ(−1 + xα) vk(x) ' − √ 2k3/2(1 + α) +O k−1/2 (3.105) i eik/HΛ a(1+α)H2 v0 (x) ' − √ Λ + O k−1/2 (3.106) k 2k3/2x

Now having fully determined the form of vk(η) during the phantom phase, we can

determine the coeﬃcients αk and βk in Eq. (3.72) to determine vk(η) in the ﬁnal FRW phase. In particular, we need to solve simultaneously the equations:

vk(phantom)|(x'0) = vk(FRW)|(η=ηf )

0 0 vk(phantom)|(x'0) = vk(FRW)|(η=ηf ) (3.107)

For brevity, let us call the leading term in k in the expression for vk (Eq. (3.105)) at

0 η = ηf (or x ' 0) as l1 and the leading term in k in the expression for vk (Eq. (3.106))

at x ' 0 as l2. Thus we have

ik/HΛ (1+α) i e a HΛ l1 = √ (3.108) 2k3/2(1 + α) and,

ik/HΛ (1+α) 2 i e a HΛ l2 = − √ 3/2 2k xf (2+2α) i ei/kHΛ a H2 = √ f Λ (3.109) 2k3/2

136 −(1+α) Where in the last manipulation we have used the result that −xf = af , which follows from the form of the scale factor during the phantom phase (Eq. (3.63)) and

the deﬁnition of x (Eq. (3.94)). Eq. (3.107) now implies that

1 l2 αk = 2 l1 + ik

1 l2 βk = 2 l1 − ik (3.110)

Using Eq. (3.65) at (η = ηf ) to ﬁnd

α af HΛ = Hf (3.111) we note from equations Eq. (3.109) that,

l1 1 1 = 1+α = 1 (3.112) l2 (1 + α)af HΛ af Hf (1 + α)

since both α and Hf are large. Eq. (3.110) now reduces to

1 l α ' 2 k 2 ik 1 l β ' − 2 (3.113) k 2 ik

Hence the form of vk(η) in the phantom phase becomes:

l l v (η) ' 2 eikτ − 2 e−ikτ k 2ik 2ik sin(kτ) = l τ (3.114) 2 kτ

137 3.6.4 Determination of the Scalar Power Spectrum

Now we are in a position to determine the power spectrum of the co-moving curvature perturbation in the FRW space using Eq. (3.16) which gives

3 2 k sin(kτ) τ PR = 2 l2 (3.115) 2π kτ zFRW(η)

2 Making the approximation aFRW(η) = af Hf τ from Eq. (3.71) since at the time a mode re-enters the horizon, η ηf , using the relation Eq. (3.111), and taking the limit k → 0, we ﬁnally obtain

3 −ik/H 2 k i e Λ PR ' √ Hf (3.116) 2 3/2 2π 2 2k mPl H 2 = f (3.117) 4πmPl

Hence we ﬁnd that our model produces a (nearly) scale invariant spectrum of cosmo-

14 logical perturbations, with amplitude set by Hf /mPl. If we assume Hf ∼ 10 GeV (approximately GUT scale), then the power spectrum matches the COBE DMR observations [22] of CMB temperature ﬂuctuations of order 10−5. In other words, the perturbation spectrum can have an amplitude suﬃciently large to seed the cosmological structure that we see today.

3.6.5 Determination of the Tensor Power Spectrum

We know from the theory of cosmological perturbations (see, for example, [5]) that gravitational waves are essentially equivalent to two massless scalar ﬁelds (for each

138 polarization) up to a renormalization factor of 2/mPl. Hence we can write the tensor power spectrum PRT as 4 PRT = 2 2 PRψ (3.118) mPl where the ﬁrst factor on the right comes from the two polarization states, the second represents the renormalization mentioned above, and PRψ is the spectrum of perturbations of a massless scalar ﬁeld ψ other than, and not interacting with, the

NEC-violating field. The ψ-field perturbations can be computed using essentially the same machinery as above, with a small difference in the final step:

1. We solve Eq. (3.84) with ψ representing the matter ﬁeld.

2. We ﬁnd expressions for vk in the three stages of the model (up to constants of integration).

0 3. We demand the continuity of vk and vk at ηi and ηf to fully specify the expression

for vk in the FRW region (that is, evaluate the undetermined constants obtained

in the previous step), and from this, compute the perturbation spectrum PRψ using the relation 3 2 k vk(η) PRψ(η) = 2 (3.119) 2π a(η) FRW

The ﬁnal expression for the power spectrum of tensor perturbations turns out to be !2 2 HΛ HΛ PRT = 8 3 ' 8 (3.120) af πmPl πmPl

The last step follows because it is easy to show (by Taylor expanding a(η) about the point η = ηi) that the scale factor hardly changes from its initial value of 1 during

139 the NEC-violating event. The result agrees (up to O(1) numerical factors) with the corresponding result obtained in (3.52).

This calculation is independent of the nature of the NEC-violating phantom ﬁeld.

It is valid as long as the spacetime responds to the NEC-violation with a sharp drop in the Hubble parameter.

2 Eq. (3.120) indicates that power in gravitational waves is proportional to (HΛ/mPl) = 10−122. This clearly precludes any possible detection of these gravitational waves, either directly, or in the CMB polarization.

A similar scenario was investigated by Y.S Piao [23], and while he obtains a scale invariant spectrum of scalar perturbations, the tensor perturbation spectrum in his calculation turns out to be blue-shifted. The source of the discrepancy could lie in his assumption that the canonical relationship satisfied by vk in the case of the tensor spectrum has a time-dependent mass given by a00(η)/a(η) (Eq. (12) in [23]) during the phantom phase. In our approach we derive vk from first principles and find that the time dependent mass can have a more complicated form.

3.7 Conclusion

In this chapter, I have brieﬂy reviewed the theory of cosmological perturbations and demonstrated how inﬂationary models lead to an approximately scale invariant spectrum.

I have then discussed the subtleties and issues inherent in attempting to compute the perturbation spectrum in Island Cosmology. The spectrum of density perturbations due to the NEC violating ﬁeld itself is still an open problem. Determining this

140 spectrum will be crucial to determining if the model agrees with observations. For

2 example, if the scale of fluctuations in this field is still set by (HΛ/mP ) then the fluctuations are too small to seed the structure that we know and the island will be

2 a desert. On the other hand, if the scale is set by (Hf /mP ) then there is a chance that island cosmology can be a viable model. In that case, quantum NEC violations provide a definite mechanism by which regions that are “macroscopically indistinguishable from our universe” can be produced from the dead de Sitter sea. Assuming that the fluctuation is instantaneous, I show that fields other than the NEC violating

ﬁeld are likely to have a scale invariant spectrum of perturbations.

The third part of this chapter reviews a classical treatment of Island Cosmology, which attempts to alleviate some of the diﬃculties mentioned earlier through classical assumptions regarding the behaviour of matter and spacetime during the quantum

ﬂuctuation. These calculations yield an adiabatic spectrum of scale-invariant perturbations, whose amplitude is determined by the value of the Hubble constant at the end of the NEC-violating ﬂuctuation. If we assume the latter to be approximately

GUT scale, the perturbation spectrum turns out to have an amplitude suﬃcient to seed the cosmological structure seen today. We also obtain a scale-invariant spectrum of gravitational waves of amplitude set by (HΛ/mPl).

141 Works Cited

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[4] Robert H. Brandenberger. Lectures on the theory of cosmological perturbations. Lect. Notes Phys., 646:127–167, 2004. hep-th/0306071.

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142 [15] Tanmay Vachaspati and Alexander Vilenkin. Quantum state of a nucleating bubble. Phys. Rev., D43:3846–3855, 1991.

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[17] N. C. Tsamis and R. P. Woodard. The quantum gravitational back-reaction on inﬂation. Annals Phys., 253:1–54, 1997. hep-ph/9602316.

[18] Martin Bucher, Alfred S. Goldhaber, and Neil Turok. An open universe from inﬂation. Phys. Rev., D52:3314–3337, 1995. hep-ph/9411206.

[19] Alexander Vilenkin. Approaches to quantum cosmology. Phys. Rev., D50:2581– 2594, 1994. gr-qc/9403010.

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143 Chapter 4

Dark Energy Voids

4.1 Introduction

The question of why the Universe accelerates has puzzled cosmologists for over a decade. In the framework of Einstein’s General Relativity, there are two possible explanations: a cosmological constant, or a dynamical dark energy component (DDE).

Determining which of these is correct (assuming, of course, that General Relativity can be trusted on cosmological scales) is one of the most crucial problems in physics, with far reaching implications particularly in particle physics.

In this chapter, I report on my work attempting to find ways of distinguishing a cosmological constant from a dynamical component via gravitational clustering. A cosmological constant, by definition, will remain unaffected in the vicinity of gravitationally collapsing matter. The behavior of a field, on the other hand, is not so clear.

As I describe below, we ﬁnd that a scalar ﬁeld tends to form voids in the presence of collapsing matter, a behavior that could have interesting observational implications.

144 4.2 What is causing the acceleration?

As I have described in Chapter 1 there is plenty of evidence that the Universe is

accelerating [1, 2, 3, 4, 5]. Nonetheless, the source of the accelerated expansion is as

elusive as ever.

If one assumes that Einstein’s general relativity can be trusted on cosmological

scales, then the only way one can have an accelerating Universe is by hypothesizing the

existence of some form of Dark Energy, which could either be a cosmological constant

or a dynamical ﬁeld. While the cosmological constant has a ﬁxed ratio of pressure to

energy density, w = p/ρ = −1, dark energy due to a dynamical component such as a scalar ﬁeld will in general have a varying equation of state (EOS), w(z). Observing a

deviation from −1 or a time-evolution in the EOS will be decisive evidence in favor of

the existence of DDE. However, there are known degeneracies [6] which make this task

extremely diﬃcult, unless the deviation from a cosmological constant is strong. The

current observational limit on the EOS of dark energy is roughly w ≈ −1 ± 0.1 at the

1σ level [1, 2], which is consistent with a cosmological constant. Future experiments hold out the possibility of narrowing this limit by maybe a factor of 10. For recent reviews, see [7, 8].

The eﬀect of the cosmological constant and DDE on the expansion rate can be identical, and there is a need for probes that go beyond the background to distinguish between the two. An example of such a probe is structure formation. There are numerous works exploring the formation of structure in the presence of homogeneous dark energy [9, 10, 11, 12, 13, 14, 15, 16, 17]. An exciting and somewhat controversial possible diﬀerence between the cosmological constant and DDE is their clustering

145 behavior. While the cosmological constant is exactly homogeneous on all scales, DDE is expected to be not perfectly homogeneous [18], and the implications of this on the

CMB are well known [19]. However, it is usually assumed that the clustering of DDE is negligible on scales less than 100 Mpc. Whether small perturbations in DDE can be neglected is debatable, and a deeper understanding of the DDE inhomogeneous dynamics is clearly needed.

Several recent works have explored the consequences of DDE clustering on scales shorter than 100 Mpc. Some have adopted a phenomenological approach, param- eterizing the clustering degree of DDE [20, 21, 22, 23, 24]. These works point out potential observables of DDE clustering, justifying further investigation. Works which attempt a more fundamental treatment are mostly in the context of coupled dark energy [25, 26, 27, 28], or other non-trivial models of DDE [29, 30, 31, 32, 33, 34], as clustering is most probable in such theories. However, less attention has been given to the clustering in simpler models of DDE.

In this chapter, I describe my research (with Irit Maor) into exploring the inhomogeneous behavior of DDE. Our approach is straightforward: starting with a gravitational action which includes matter and DDE, we numerically follow the linear evolution of spherical perturbations of matter and the DDE response to these perturbations. For the sake of simplicity, our model for the DDE is a light scalar

ﬁeld, which is not explicitly coupled to the matter density. As the only coupling between the DDE and the matter is gravitational, our results are conservative in the sense that any model more complicated can be expected to show stronger DDE perturbations than shown here, simply because of the additional coupling beyond

146 gravity.

The striking feature that emerges from our calculation is that in the vicinity of collapsing matter, the DDE develops a spatial profile and tends to form voids. The mechanism that allows the void to form is that although initially the field’s evolution is friction dominated due to the cosmic expansion, the collapse of matter slows down the local expansion. This allows the field to locally roll down and lose energy, creating the void. The presence of the matter perturbation is necessary to trigger this mechanism.

The plan of this chapter is as follows: in § 4.3 I describe our model in detail.

In § 4.4 I present our results. Discussions and conclusions are in § 4.5 and § 4.6 respectively.

4.3 The model

We are interested in spherical perturbations around a ﬂat FRW universe. The most general line element in comoving coordinates is then

ds2 = dt2 − U(t, r)dr2 − V(t, r) dθ2 + sin2 θdϕ2 , (4.1) where U(t, r) and V(t, r) are general functions [35].

We take a cosmic mix of non-relativistic matter and a DDE component as the energy source. The matter component is described by a perfect and pressureless

ﬂuid, with an energy-momentum tensor given by

Tµν(m) = diag (ρ, 0, 0, 0) , (4.2)

147 where ρ is the energy density of matter.

We model the DDE with a classical scalar ﬁeld φ with a Lagrangian L given by

1 L = (∂ φ)2 − V (φ) , (4.3) 2 µ

and an energy-momentum tensor given by

Tµν(φ) = ∂µφ∂νφ − gµνL . (4.4)

The EOS of the DDE w is deﬁned as

p w = φ , (4.5) ρφ

with the energy density ρφ and the pressure pφ are read oﬀ the energy momentum

ij tensor, T00(φ) and −g Tij(φ)/3 respectively.

It is convenient to rewrite Einstein’s equations in the following way,

1 R = K T − g T α . (4.6) µν µν 2 µν α

where Rµν is the Reimann tensor, and K = 8πG. As there is no explicit interaction between the matter and the DDE, energy conservation applies to each separately,

µν µν ∇µT (m) = 0 , ∇µT (φ) = 0 . (4.7)

148 The time-evolution of the system is given by the following equations (where dots denote time-derivatives and primes denote derivatives with respect to the radial co-

0 dV ordinate, except for V (φ) ≡ dφ ):

1 U¨ 1 U˙ V˙ 1 U˙ 2 1 1 U 0 V0 1 V02 V00 + − + + − 2 U 2 U V 4 U 2 U 2 U V 2 V2 V 1 1 −K ρ + V (φ) + φ02 = 0 (4.8) 2 U 1 V¨ 1 U˙ V˙ 1 1 1 U 0 V0 V00 + + + − 2 V 4 U V V 2U 2 U V V 1 −K ρ + V (φ) = 0 (4.9) 2 ! V˙ 1 U˙ ρ˙ + + ρ = 0 (4.10) V 2 U ! V˙ 1 U˙ φ¨ + + φ˙ + V 0(φ) V 2 U 1 V0 1 1 U 0 − − φ00 − φ0 = 0 . (4.11) U V U 2 U

These are subject to the following constraint equations:

1 1 U˙ V˙ 1 V˙ 2 1 1 U 0 V0 1 V02 V00 + + + + − V 2 U V 4 V2 U 2 U V 4 V2 V 1 1 −K ρ + V (φ) + φ˙2 + φ02 = 0 (4.12) 2 2U 1 U˙ V0 1 V˙ V0 V˙ 0 + − − Kφφ˙ 0 = 0 . (4.13) 2 U V 2 V V V

149 4.3.1 Linearization

We now proceed to separate our variables to a homogeneous background and a time and space-dependent perturbation, which we will then linearize. Working in the synchronous gauge [36], we redeﬁne the metric functions U and V as follows:

U(t, r) = a(t)2e2ζ(t,r)

V(t, r) = r2a(t)2e2ψ(t,r) .

(4.14)

Here a(t) is the scale factor of the spatially homogenous and ﬂat background, and

ζ(t, r) and ψ(t, r) are the deviations. We introduce a perturbation around a homogeneous background also in the matter and the DDE,

ρ(t, r) = ρ(t) + δρ(t, r)

φ(t, r) = φ(t) + δφ(t, r)

V (φ + δφ) = V (φ) + δV (φ, δφ) .

The zeroth order of equations (4.8) - (4.13) gives

1 3H2 − K ρ + V + φ˙2 = 0 (4.15) 2 1 H˙ + 3H2 − K ρ + V = 0 (4.16) 2 ρ˙ + 3Hρ = 0 (4.17)

φ¨ + 3Hφ˙ + V 0 = 0 , (4.18)

150 where H =a/a ˙ is the Hubble function.

To linear order, the evolution equations (4.8)-(4.11) give

2 ζ0 2ψ0 ζ¨ + 4Hζ˙ + 2Hψ˙ + − − ψ00 a2 r r 1 −K δρ + δV = 0 (4.19) 2 1 2ζ 2ψ ζ0 4ψ0 ψ¨ + 5Hψ˙ + Hζ˙ + − + − − ψ00 a2 r2 r2 r r 1 −K δρ + δV = 0 (4.20) 2 δρ˙ + 3Hδρ + ρ ζ˙ + 2ψ˙ = 0 (4.21)

δφ¨ + 3Hδφ˙ + δV 0 1 2 + ζ˙ + 2ψ˙ φ˙ − δφ00 + δφ0 = 0 , (4.22) a2 r and the constraint equations (4.12)-(4.13) reduce to

2 ζ ψ 2ζ0 6ψ0 2Hζ˙ + 4Hψ˙ + − + − − ψ00 a2 r2 r2 r r −K δρ + δV + φδ˙ φ˙ = 0 (4.23) 2 ψ˙ + ζ˙ − Kφδφ˙ 0 + 2ψ˙ 0 = 0 . (4.24) r

Combining equations (4.19), (4.20) and (4.23) gives

ζ¨ + 2ψ¨ + 2H ζ˙ + 2ψ˙ +K δρ − δV + 2φδ˙ φ˙ = 0 . (4.25)

151 The only combination which is relevant to the equations of motion (4.21) and

(4.22) is χ ≡ ζ˙ + 2ψ˙. Comparing (4.17) and (4.21) it is clear that χ can be thought of

as 3δH, and therefore characterizes the spatial proﬁle of the Hubble function. At the

cost of losing some information about the metric, we can reduce the number of our

variables and equations from 4 to 3 by solving for χ instead of for ζ and ψ. Equations

(4.21), (4.22) and (4.25) yield

δρ˙ + 3Hδρ + ρχ = 0 (4.26) 1 δφ¨ + 3Hδφ˙ + δV 0 + χφ˙ − ∇2 (δφ) = 0 (4.27) a2 1 χ˙ + 2Hχ + K δρ − δV + 2φδ˙ φ˙ = 0 . (4.28) 2

Finally, by Fourier-transforming δρ(t, r), δφ(t, r) and χ(t, r) into δρk(t, k), δφk(t, k)

and χk(t, k) respectively, equations (4.26)-(4.28) can be written as a set of ordinary diﬀerential equations:

δρ˙k + 3Hδρk + ρχk = 0 (4.29) k2 δφ¨ + 3Hδφ˙ + V 00 + δφ + φχ˙ = 0 (4.30) k k a2 k k 1 χ˙ + 2Hχ + K δρ − V 0δφ + 2φ˙ δφ˙ = 0 , (4.31) k k 2 k k k k where we have used the fact that to linear order, δV = V 0δφ and δV 0 = V 00δφ.

152 4.3.2 Potential

Observationally distinguishing between various potentials of DDE is a formidable task

[37, 38, 39], and a careful analysis of the growth of structure in various potentials might prove a useful tool. Our present goal though is to trace generic properties of

DDE. Accordingly, we choose to work with a simple mass potential,

1 V (φ) = m2φ2 . (4.32) 2

We take the mass scale comparable to the present Hubble scale, mφ/H0 ∼ 1. The light mass assures a slow roll behavior, which will provide accelerated cosmic expansion.

Unless noted otherwise, the ﬁgures presented here refer to this mass potential.

In order to verify the generality of our results, we repeated the analysis for a more complicated potential - the double exponential [40],

√ √ Kαφ Kβφ V (φ) = V0 e + e , (4.33) with α = 20.1 and β = 0.5. As we later show, the resulting behavior for the two potentials is qualitatively the same.

4.3.3 Initial conditions

We want to study how the DDE reacts to the clustering of matter. Thus our initial conditions are of perturbed matter and homogeneous DDE. The matter perturbation

153 is taken as a spherical Gaussian,

2 2 δm(ti, r) ≡ δρm(ti, r)/ρm(ti) = A exp −r /σ , (4.34) and the DDE is taken to be initially homogeneous, δφ = 0. A non-homogeneous evolution for the dark energy is nonetheless allowed. To ensure that the matter perturbation has no peculiar velocity, we choose the initial condition for the metric variable as χ = 0. From equation (4.26), we ﬁnd that this amounts to the statement that initially, δρ ∝ a−3, i.e. the matter particles making up the perturbation are being simply carried along with the Hubble expansion (or in other words, they are at rest in a comoving frame).

Note that the equations of motion give us the freedom to specify the initial value of χ. A little bit of algebra shows that once χ is specified, one can always find a corresponding ζ and ψ. However, we have verified that choosing different initial values of χ does not qualitatively affect our results.

The numerical calculation begins at a redshift of z = 35 with a perturbation amplitude of A = 0.1, and is run to the present, z = 0. We focus on relatively short- scale perturbations, σHi ∼ 0.01. With this choice of redshift span, we aim to explore the full linear range of the matter perturbation, while also allowing for a period of

DDE dominance.

The initial values of the background variables ρi, φi and Hi are chosen such that their present values (denoted with a 0 subscript) converge to Ωm,0 = 0.3, Ωφ,0 = 0.7, and a normalized Hubble value, H0 = 1. This is done with the help of a root-ﬁnding ˙ algorithm. We take φi = 0, which means that the initial state of the scalar ﬁeld is

154 homogeneous and with equation of state w = −1, similar to the cosmological constant.

We now have the layout to numerically evolve equations (4.29)-(4.31). The solutions for δρk, δφk and χk are then Fourier transformed back to real space.

4.4 Results

4.4.1 Density contrasts

The results shown in this section are for the mass potential, equation (4.32), unless speciﬁcally noted otherwise.

The numerical run begins at a redshift of z = 35, when the DDE is subdominant.

The DDE remains subdominant through most of the growth time of the perturbation as well, and accordingly, we expect the matter density contrast δm ≡ δρm/ρm to grow as the scale factor a. This is indeed conﬁrmed by our results.

The most striking result that emerges from our calculation is that in the vicinity of collapsing matter, the DDE tends to form voids, or growing regions of underdensity. The anti-correlation between the perturbations of the matter and the DDE was noted in [19] and in [41] for particular cases. Figure 4.1 plots the DDE density contrast δφ ≡ δρφ/ρφ at the center of the matter perturbation, r = 0, against the redshift (1 + z). We ﬁnd that the amplitude of the perturbation grows sharply at late times.

Figure 4.2 shows the growth of the absolute value of the DDE perturbation with redshift on a logarithmic scale. This plot reveals another interesting eﬀect: the initial response of the DDE to the gravitationally collapsing matter is a very weak tendency

155 0.00 m/H =1 0 σH =.01 i φ

δ −0.01

−0.02

0 10 20 30 1+z

Figure 4.1: The evolution of the DDE overdensity δφ at the center of the matter perturbation, r = 0, with redshift (1+z).The scale of the perturbation is σHi = 0.01, and the mass is m/H0 = 1. Initially homogenous, the DDE develops an underdensity at late times in response to the matter perturbation.

−2 m/H =1 0 σH =.01 | i φ −4 δ | 10

log −6

−8

−10 0 5 10 15 20 25 30 1+z

Figure 4.2: Same as Figure 4.1, with the y-axis on a logarithmic scale. The DDE tends to cluster initially, but eventually forms a void. The kink in the plot signiﬁes the change-over from positive to negative perturbation.

156 to collapse. The collapsing phase, however, is extremely short-lived (an O[10−1] fraction of the total time of the run). The kink in the ﬁgure is the cross-over from a positive to negative perturbation.

We next look at the spatial profiles of the perturbations. Figure 4.3 shows the profiles of both the δm and δφ at an early stage of the run (z = 27.5), at an intermediate redshift (z = 0.8) and at the final stage (z = 0). The x-axis shows the physical scale as a fraction of the horizon size. The amplitude of the perturbations is shown on a logarithmic scale, and it is worth noting the change in gap between the two scales.

The growth rate of δφ is signiﬁcantly faster than that of δm, so that the two amplitudes are almost comparable at late times. This suggests that a calculation of the non-linear regime might reveal interesting behavior of the DDE.

Figures 4.4 and 4.5 show how the growth of the DDE perturbation is affected by the initial width of the matter perturbation. The figure confirms the sensitivity of

DDE perturbations to the scale, and as expected, shorter scales exhibit a suppressed behavior. Nonetheless, a possibly signiﬁcant amplitude can be found on relevant

−1 scales. For instance, our local supercluster has diameter of order 0.01H0 and an overdensity of about δm ≈ 1.2. Our runs of σHi = 0.01 end with an overdensity of

δm ≈ 1.6, but roughly they can be taken as a measure of what we should expect on these scales. Also, one must remember that this analysis only captures the physics of the linear regime. Given the sharp increase in the void amplitude at later times, it is not unreasonable to expect interesting eﬀects in the strongly non-linear regime.

Figures 4.6 and 4.7 examine the sensitivity to the ﬁeld’s mass. As expected, increasing the mass of the scalar ﬁeld causes δφ to grow stronger and at increasingly

157 Figure 4.3: Logarithmic profiles of the matter density contrast log10 |δm| and the dark energy density contrast log10 |δφ|, at three different redshifts. Solid lines denote the DDE profiles, and dotted lines denote the matter profiles.

158

m/H =1 0 0.000

−0.005

φ −0.010 δ

σ H =.1 −0.015 i σ H =.01 i −0.020 σ H =.001 i

−0.025 1.0 1.5 2.0 2.5 3.0 1+z

Figure 4.4: DDE density contrast δφ at the center of the matter perturbation r = 0 as a function of the redshift (1 + z) for fixed mass m/H0 = 1 and different initial matter perturbations’ widths. The larger the initial matter perturbation, the stronger is the void. The curves of σHi = 0.01 (dashed) and 0.1 (solid) almost overlap. The figure zooms on late times, z < 3.

0 σ H =.1 i m/H =1 0 σ H =.01 −2 i σ H =.001 i |

φ −4 δ | 10

log −6

−8

−10 0 10 20 30 1+z

Figure 4.5: Same as Figure 4.4 with the y-axis on a logarithmic scale. The shorter scales start evolving at later times than the longer scales, but their evolution is faster. The curves of σHi = 0.01 (dashed) and 0.1 (solid) almost overlap. 159 earlier redshifts. One can also see that if the mass is heavy enough (an order of magnitude larger than the Hubble mass), the ﬁeld will have had enough time to enter a period of rapid oscillations, and will eﬀectively behave as regular matter.

4.4.2 Equation of state

We would now like to focus on the local behavior of the equation of state (EOS) of the scalar field. Let us define w0 as the background homogeneous EOS, δw as the leading order correction to w0 and let the first-order corrected w be w1 = w0 + δw:

1 ˙2 p0 2 φ − V w0 = = (4.35) ρ 1 ˙2 0 2 φ + V 1 δw = δp − w0δρ (4.36) ρ0

w1 = w0 + δw , (4.37) where to ﬁrst order δρ = φδ˙ φ˙ + δV and δp = φδ˙ φ˙ − δV , and we have suppressed the

φ subscripts.

Figure 4.8 shows how w1 increases with time and becomes non-homogenous. The EOS at the perturbation is less negative than the background, but for this choice of mass it is still negative enough to behave as dark energy.

To quantify the extent of the inhomogeneity in w we deﬁne ∆w to characterize the % deviation of the local w from the background value,

δw 1 ∆w = 100 = 100 δp − w0δρ . (4.38) w0 p0

160

m/H =10 0 σ/H =.01 i m/H =1 0.3 0 m/H =0.1 0 0.2 φ δ 0.1

−0.1 1 2 3 4 5 6 1+z

Figure 4.6: The DDE density contrast δφ at the center of the matter perturbation, r = 0, against redshift (1 + z) for σHi = .01 and three diﬀerent masses. The ﬁgure zooms on late times, z < 7.

0 m/H =10 σ H =.01 0 i m/H =1 −5 0 m/H =0.1 0 |

φ −10 δ | 10

log −15

−20

−25 0 10 20 30 1+z

Figure 4.7: Same as Figure 4.6 with the y-axis on a logarithmic scale. The perturbation’s is extremely sensitive to the mass scale.

161 −0.75 z=35 m/H =1 0 z=.8 −0.80 σ H =.01 i z=0

−0.85 w δ + 0

w −0.90

−0.95

−1.00 0 0.01 0.03 0.05 r/r horizon

Figure 4.8: Plot of w1 vs r for three diﬀerent redshifts.

8 m/H =1 z=35 0 z=0.8 σ H = 0.01 i 6 z=0

w 4 Δ

0 0 0.02 0.04 0.06 r/r horizon

Figure 4.9: Plot of % deviation in w vs r at three diﬀerent redshifts.

162 5

m/H =1 −5 0 σ H =.01 −10 i

H/H (%) −15 δ −20 z=27.5 z=1.2 −25 z=0.0

−30 0 0.01 0.02 0.03 0.04 0.05 0.06 r/r horizon

Figure 4.10: Percentage variation of the local Hubble parameter at three diﬀerent redshifts

The evolution of the ∆w spatial proﬁle is shown in Figure 4.9.

4.5 Discussion

4.5.1 Void formation

We now aim to intuitively explain why the DDE perturbation initially increases and

then sharply drops in the presence of the matter perturbation. Initially, both the

scalar ﬁeld and the Hubble function are homogenous. The matter perturbation intro-

duces an inhomogeneity in the Hubble function, as regions with matter overdensity

expand slower. As a result, H acquires a spatial proﬁle which evolves in time along with the matter overdensity, due to the ρχ term in equation (4.26). This is illustrated

163 0.02

0.00

m/H =1 −0.02 0 σH =.01 i ¨ φ

δ −0.04

−0.06 z=27.5 z=0.8 −0.08 z=0

−0.10 0 0.02 0.04 0.06 r/r horizon

Figure 4.11: Proﬁle of δφ¨ at three diﬀerent redshifts

in Figure 4.10, which plots the spatial proﬁle of the % deviation of H from the homo-

geneous background value, at three diﬀerent redshifts. Regions which have a lower

local value of H oﬀer less Hubble damping to the scalar ﬁeld, as the φχ˙ contribution

in equation (4.27) is negative. Therefore, in these regions the scalar ﬁeld accelerates

down its potential slightly faster, the local φ¨ + δφ¨ has a bigger absolute value than the acceleration of the background, φ¨. Thus the matter perturbation imparts a local

“downhill kick” to the ﬁeld, and the strength of the kick depends on the magnitude of the matter perturbation at that point. The presence of matter is essential to trigger this mechanism. A similar conclusion was reached in [42].

Since the matter perturbation is Gaussian, we can expect the proﬁle of δφ¨ to be

Gaussian as well, which is confirmed in figure 4.11. The acceleration of the field

perturbation δφ¨ is zero initially (z = 35), but it quickly takes on a Gaussian proﬁle,

164 which initially grows, and then shrinks at later times. The important point to note

in ﬁgure 4.11 is that the acceleration weakens, but doesn’t change its sign throughout

the evolution.

The acceleration proﬁle leads to spatial variations in the energy density of the

DDE. For the mass potential, the linear order of the density perturbation is δρφ = φδ˙ φ˙+m2φδφ, where the ﬁrst term represents the local variation in kinetic energy (KE) and the second term represents the local variation in potential energy (PE). Whether the initial value of the scalar ﬁeld was shifted to the right or left of its minimum, φ˙

and δφ˙ will be of the same sign, assuring a positive correction to the KE. On the

other hand, φ and δφ will be of opposite signs, assuring the PE correction will be

negative. The total correction to the energy density then will be positive when the KE

dominates, creating an overdensity, and negative when the PE correction dominates,

creating a void. The fact that δφ¨ weakens but doesn’t change its sign assures that

δφ˙ approaches a constant value, but δφ keeps growing. The PE contribution m2φδφ

soon becomes dominant over the KE correction, φδ˙ φ˙. As soon this happens, a void

is created.

Similar reasoning should apply to other slow-roll potentials.

4.5.2 Generality

The results we have presented so far were for the mass potential, equation (4.32).

This is not a very attractive model from the theoretical point of view, as the choice

of the mass scale and initial conditions are ﬁne tuned.

While the general problem of ﬁne tuning has not yet been resolved, the issue of the

165 5 V(φ)=V [exp(K1/2αφ)+exp(K1/2βφ)] 0 σ H =.01 i 0 z=0 | at r=0

m z=2.3 δ | −5 10 z=22.9

−10 | and log φ δ |

10 −15

log z=0 z=2.3 z=22.9

−20 0 0.02 0.04 0.06 r/r horizon

Figure 4.12: The equivalent of ﬁgure 4.3, with the double exponential potential, equation (4.33). initial conditions is alleviated in tracking potentials such as equation (4.33). To verify that our results are not unique to the mass potential, we have repeated the analysis for the double exponential potential. We present here the equivalent of ﬁgures 4.3 and

4.8 to show that the results are essentially similar: as a reaction to the perturbation in the matter ﬂuid, the DDE quickly forms a void. Starting very low, the rate of growth of its amplitude is signiﬁcantly faster than that of the matter perturbation, motivating a further investigation into the non linear regime.

4.6 Conclusions

In this work we have investigated the clustering properties of DDE. We modeled the DDE as a scalar ﬁeld with a light mass, and have shown that in the vicinity of

166 −0.92 V(φ)=V [exp(K1/2αφ)+exp(K1/2βφ)] 0 σ H =.01 i −0.94 z=35 z=0.8 z=0 w δ

+ −0.96 0 w

−0.98

−1.00 0 0.02 0.04 0.06 r/r horizon

Figure 4.13: The equivalent of ﬁgure 4.8, with the double exponential potential, equation (4.33). gravitationally collapsing matter, the DDE develops inhomogeneities and forms voids.

Our results show a high sensitivity to the mass scale of the field. For a mass much larger or smaller than the Hubble scale, the field imitates the behavior of dust or the cosmological constant, respectively. The interesting dynamics is most prominent in the window where the mass and Hubble scales are comparable. This window is within the relevant mass range for dark energy models. As the Hubble function was larger in the past, heavier fields would have had comparable mass and Hubble scales at some point in the past.

Our results should apply to any model of DDE which achieves the present acceleration through a slow roll phase, as the slow roll assures that only a small patch of the potential is probed. We have shown this explicitly for the double exponential potential. Whether our results apply to an even wider class of models which achieve

167 accelerated expansion not through slow roll should be further investigated.

One thing which is clear from our results, is that DDE has potentially non-trivial behavior during the growth of inhomogeneities, though full non-linear analysis is needed to conﬁrm whether the amplitude of the DDE inhomogeneities is relevant to observations. As inhomogeneities of dark energy are a clear signature diﬀerentiat- ing between the cosmological constant and DDE, such possibilities should be fully explored and exhausted.

A full treatment of the observational consequences of our results is beyond the scope of this work, but we would like to mention a few possibilities. Some obvious places to look for DDE inhomogeneities include lensing, the ISW eﬀect in the CMB, number counts, and mass functions. Some of these directions are being pursued

[20, 21, 22, 23, 24, 25, 26, 27, 28].

Our results show that both the energy density and the EOS of DDE develop a spatial dependence. Thus any observation constraining either of the above which can separately be measured locally and globally is valuable.

Another possibility which we would like to point out, is that it would be useful to quantify the eﬀect of a statistical distribution of DDE voids on the CMB, or, following reference [41], on the directional distribution of supernovae. For example, detecting an angular inhomogeneity in H which is not in accordance with the matter distribution might suggest the presence of a DDE void.

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