Physical Cosmology and the Acceleration of Expansion
In this chapter we present a broad overview of the theoretical concepts crucial to our current understanding of cosmology. We begin by looking at the evolution of our concept of relativity, from Galilean to General Relativity. We then explore how General Relativity can give rise to a class of predictive models about our universe’s history, and how these models tie into the cosmological parameters. Afterwards, we present the so called ΛCDM model, the simplest model that can explain our current observations, in particular the relatively recent observation that the expansion of the universe is accelerating. Finally, we briefly explore the many other models that have been put forth to explain this acceleration.
A Historical Overview of Relativity
Which reference frames can be called inertial ? Answering this question has led to some of the most profound discoveries of physics, and fundamentally altered our very understanding of space and time. In this section, we take a historical approach to understanding why and how this question led to the formulation of General Relativity (GR). A principal motivating factor in defining such frames is understanding how the formulation of the laws of physics depends on the chosen reference frame. While such investigations have led to widely varying descriptions of space and time as the laws considered change, there is a single postulate that remains fundamentally the same in all theories of relativity :
Postulate 1 The postulate of relativity : The laws of physics are invariant in all inertial reference frames.
This postulate defines the very concept of inertial reference frames. To understand its im-plications, we begin with a simple thought experiment; one so fundamental most non physicists have already wondered about it. An observer is on a moving train. As the train begins to move, he has trouble telling if the train is moving forward or if the station is moving backwards. To answer this question, one must first consider what mechanical laws are being considered. To start, we begin by considering Newton’s laws, famously formulated in Newton (1760). These are :
Physical Cosmology and the Acceleration of Expansion
Newton’s Law 1 If no external forces are applied on a system, it remains at rest or continues moving at constant velocity.
Newton’s Law 2 The mass times acceleration of a system equals the amount of external force exerted on it : F = m ◊a
Newton’s Law 3 For every force one system applies on another, the latter system exerts a force on the former equal in force but in the opposite direction.
The galilean perspective on this issue is that both answers (either that the train is moving forwards or that the station is moving backwards) can be considered valid, so long as the relative motion between the two can be said to be rectilinear and uniform. The only thing that matters then, is that the proper transformations be applied when transferring coordinates from one reference frame to the other. We call the train station’s reference frame S, and that of the train SÕ. If the train tracks are aligned with the x-axis, and the train is moving in the positive x direction with speed v, these transformation are : yÕÕ = y ≠ ◊ Z (1.1)
This can be justified by the fact that Newton’s laws are invariant under this transformation. It is obvious that law 1 still holds, since equations 1.1 transform constant velocities into constant velocities. In addition, it is clear that the second derivative of xÕ is the same as that of x, provided v is constant in time. Hence, law 2 still holds. Finally, law 3 is left unaﬀected by these transformations. In other words, because only the second derivative in time of coordinates are thought to matter in the formulation of the laws of physics, adding a first order derivative (in time) to an inertial reference frame will lead to another inertial reference frame. This term will simply vanish after the second derivative is taken, and postulate 1 will hold for the laws of Newton. Our train passenger can now rest at ease in the knowledge that both of his answers are correct, and he can simply pick the frame which facilitates whatever particular physical problems he is trying to solve during his train ride.
Suppose, however, that the particular problem he is trying to solve happens to involve an electrically charged ball. Maxwell’s laws require that moving electrical charges create a magnetic field proportional to their speed. In other words, what happens when one introduces first order derivatives in the laws of physics ? Indeed, this is the case for Maxwell’s laws.
This poses a conundrum to our train passenger. The galilean tranformations described in equation 1.1 can reconcile the apparent motion of objects in reference frame S which are stationary in reference frame SÕ. It cannot, however, make magnetic forces appear out of thin air. It is clear then that the galilean transformations cannot satisfy the postulate of relativity if electromagnetic forces are involved.
To find a replacement for equation 1.1, we begin by reformulating the problem. First, we note that Maxwell’s equations can be shown to lead to a wave equation : 1 Ò2 ≠ c1 ˆ2 2 ˛ Zc = (1.2) Where ˛ and ˛ are the electrical and magnetic fields, and and are the permaebility E B µ ‘ and permittivity of vacuum. If the postulate of relativity is to hold, then the value of c must remain constant throughout any change of reference frame. Otherwise, that would require diﬀerent values for the permeability and permittivity of vacuum, violating the postulate. This electromagnetic wave equation actually describes light waves, leading to a second postulate that we will use in formulating our next relativistic theory, Special Relativity (SR) :
Postulate 2 The speed of light in a vacuum is the same for all observers.
The idea, then, is to change the transformations of equation 1.1 into transformations that will satisfy postulate 2. The only way to accomplish this is to allow the transformations to aﬀect the time coordinate as well. This leads to the Lorentz transformations : tÕ = “ 1t ≠ vx 2 Z
While these transformations had been known prior to the advent of special relativity, the contribution of Einstein (1905) was to derive these entirely from postulates 1 and 2, and to understand that they represented fundamental properties of the geometry of space and time, and not the eﬀects of motion on the size of rigid bodies (see the historical discussion in Brown (2003)). By allowing space and time to “mix” in these transformations, we have introduced the concept of spacetime. Before moving on to General Relativity, it is important to understand how norms are defined in spacetime. Given a 4-vector u with components uµ its norm squared is defined as :
u2 = ≠u02 + u12 + u22 + u33 (1.4)
It is worth noting that such a definition ensures that norms are invariant under the Lorentz transformations of equation 1.3, and is therefore a fixed quantity regardless of the choice of reference frame. To simplify explanations regarding general relativity, we introduce here the concept of the metric tensor gµ‹ . The metric tensor is defined such that for any 4-vector u, its norm is :
u2 = gµ‹ uµu‹ (1.5)
It is clear therefore, that in the case of special relativity, the metric tensor is always the same. This special case of the metric tensor is usually written as ÷µ‹ : Q ≠01 1 0 0 R
At this point, our train passenger still has one final question. He has derived transformations that will allow him to transform coordinates from one reference frame to another. These trans-formations will not alter the laws of physics provided the 2 frames are moving apart from each other at a constant speed v. Going back to our initial question, however, how then do we define inertial reference frames. These transformations allow us to say that if any given reference is an inertial one, then any other reference frame moving in a rectilinear and constant fashion relative to it is also an inertial reference frame. This defines a class of inertial reference frames up to an acceleration. How to tell then if our frame is an inertial one or an accelerating one?
Our observer might be tempted to draw upon his experiences aboard the train. When the train began to slowly edge forward, he could not tell if he was moving forward or if the train station was moving backward. On the other hand, when the train began to accelerate in order to reach its top speed, he felt pushed backwards against his seat, confirming that he was accelerating forward. We might then be tempted to use these virtual forces to define inertial reference frames : they are the class of frames that do not experience virtual forces. At this point, we are tempted to think that special relativity has completely solved the question of defining inertial reference frames.
However, much like Maxwell’s laws challenge Galilean relativity, so too does the law of gravitation present a challenge to special relativity. Recall that in a given gravity field g, the force applied on a body is simply m ◊g where m is the mass of the body. Applying Newton’s second law in this case becomes : m ◊g = m ◊a (1.7)
Because the force of gravity is proportional to the mass of the gravitating system, Newton’s second law implies that all gravitational forces are locally equivalent to an acceleration field. How then does one distinguish gravitating reference frames from accelerating ones ? The development of general relativity is rooted in the impossibility of this distinction. To understand this, we introduce our third and final postulate :
Postulate 3 The equivalence principle : The inertial mass of a body (right hand mass term in equation 1.7) is equal to its gravitational mass (left hand mass term in equation 1.7).
This postulate cements the indistinguishability of gravitating and accelerating reference frames. The solution of general relativity is to make the free falling frames the inertial frames. In this interpretation, the “force” of gravity as we experience it in everyday life is actually a virtual force, arising from our acceleration relative to local inertial frames. General relativity therefore sets out to write a set of equations relating the presence of energy to distortions in the spacetime metric. This derivation is constrained by the fact that these distortions must reproduce the observed eﬀects of gravity. These are known as the Einstein field equations (EFE), and were first derived in Einstein (1915). They are : Rµ‹ ≠ 1 = 8fi Tµ‹ (1.8)
Where Tµ‹ is the stress-energy tensor, R is the Ricci scalar, and Rµ‹ is the Ricci tensor. In this world view, Newton’s laws become replaced by the concept of geodesics which describe the path of inertial reference frames. The geodesic equations are also derived from the equivalence principle. We will not explore the diﬀerential geometry details of the EFE. Suﬃce it to say that our primary use for them as observational cosmologists lies in their ability to make predictions about the metric tensor gµ‹ . This tensor is the one most directly related to observables. In the next section, we will see how these equations give rise to the Hubble diagram. For now, we will conclude by saying that it is a most profound testament to the power of physics that merely asking the question “who is in motion and who is not” leads to a theory capable of elucidating our cosmic origins.
Theoretical Basis of Modern Cosmology
The Friedmann Equations
The simplest solutions to the EFE are actually not for the case of a spherical mass, as is often the case in physics. Though it is a subjective assessment, the EFE are most easily applied to the universe as a whole. In this case, a number of diﬀerent symmetries allow us to simplify the problem. Recall that the metric tensor is now a dynamical quantity. Rewriting equation 1.5 in diﬀerential form we find that the line element is given by : ds2 = gµ‹ dxµdx‹ (1.9)
We begin by making a simplifying assumption :
Assumption 1 The Cosmological Principle : The energy content of the universe is homoge-neous and isotropic in space.
Using this principle we can rewrite Tµ‹ as : Q 0 p 0 0 R
Tµ‹ = c fl 0 0 0 d
The homogeneity and isotropy required by the cosmological principle means that there exists a reference frame in which the oﬀ diagonal terms of gµ‹ are 0, and in which T11 = T22 = T33 = p. We use this to rewrite equation 1.9 in less general form using spherical coordinates. Taking r, ◊ and „ to be comoving coordinates (i.e. observers at rest in (r,◊,„) are in free fall), and defining the cosmic time t as the time measured by observers at rest in (r,◊,„) we arrive at the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric : ds2 = ≠c2dt2 + a2(t) C 1 drkr2 + r2 d◊2 + sin2◊d„2 D (1.11)
Where k is the curvature of space. The dynamical quantity a(t) is known as the scale factor. The simplifying symmetries constrain this to be the only dynamical quantity involved. The quantity multiplied by the scale factor is simply the line element for an infinitesimal displacement in 3D space, hence why we have rewritten it as dl(3)2. This is immediately apparent for the flat space case ( k = 0) where one finds the usual formula for the line element in spherical coordinates. In other words, before we even apply the EFE, the underlying symmetries allow us to describe the universe using a single dynamical quantity, which can easily be understood as the evolution over time of the relative distance between objects. Because it is found to strictly increase over time, this evolution is referred to as expansion.
Assuming that the metric takes the FLRW form described in equation 1.11, and that the stress energy tensor takes the form described by equation 1.10, we can use the EFE to relate the scale factor to the energy content of the universe. These relations are known as the Friedmann equations, and were first described in Friedmann (1922) : H2 © 3a˙ 4 2 = 8 3 3fl ≠ a2 4
H is defined as the Hubble parameter. Its value today, H0, is known as the Hubble constant. Assuming that the energy content of the universe is dominated by either matter or radiation, one can put general constraints on the values of fl and p :
Assumption 2 Energy density is positive.
Assumption 3 Pressure is positive.
This would imply that the right hand term of equation 1.12b is negative, leading to a net deceleration in the universe’s expansion. Here, we can already understand the implications of an observed acceleration. Namely that, in the context of the Friedmann equations, matter and radiation alone cannot explain the observations.
Before going further, it can be useful to rewrite equation 1.12a in a more wieldy form, better suited to compare it to various models of the universe. For a full description of the universe, we need to account for a multitude of possible contributions to its total energy density (traditionally thought to be matter and radiation). In what follows each such contribution is designated by the subscript i :
H2(a) = 8 3 Aÿi fli(a) ≠ a2 B
flc(a) Cÿi D = 8 3 Ωi(a) + Ωka≠2 (1.13) fiG
Where Ωi (a) is the density of fluid i relative to the critical density flc (a), and flc(a) is chosen such that the sum of all Ω(a), including Ωka≠2, is equal to 1 (i.e. flc(a) = 3H(a)2/8fiG). In this formalism, Ωka≠2 is referred to as the curvature of the universe.
We can simplify the first Friedmann equation even further by treating the universe’s energy content as an eﬀective fluid and introducing the concept of an equation of state. This equation relates the energy density of a fluid to its pressure : p = w ◊fl (1.14)
Where w is the equation of state parameter. For matter, whose energy density is dominated by its mass energy, w is eﬀectively 0. For radiation, whose energy density is dominated by its momentum, w is eﬀectively 1/3. Semi-relativistic matter can take any value in between. Using local conservation of energy (T;–—— = 0), the equation of state informs us about the rate of evolution of the energy density : Ωi(a) Ã a≠3(1+wi) (1.15)
Finally, plugging this rate of evolution into equation 1.13 and comparing the left hand side with the present day rate of expansion H0, we obtain : H2 = ÿi Ωia≠3(1+wi) + Ωka≠2 (1.16)
Note that here, the scale factor independent values of Ω represent their values today. Equation 1.16 is a particularly common form of the first Friedmann equation, since it contains many cosmological parameters of interest (the Hubble constant, the present day density of the various components that make up the universe, their equation of state, and curvature).
Table of contents :
1 Physical Cosmology and the Acceleration of Expansion
1.1 A Historical Overview of Relativity
1.1.1 Galilean Relativity
1.1.2 Special Relativity
1.1.3 General Relativity
1.2 Theoretical Basis of Modern Cosmology
1.2.1 The Friedmann Equations
1.2.2 Cosmological Redshift
1.2.3 The Hubble Diagram
1.3 Constructing the !CDM Model
1.3.1 On the Astrophysical Need for Dark Matter
1.3.2 The First Acceleration Observations From SN
1.3.3 Concordance With Other Probes
1.4 Theoretical Explanations of Observations
1.4.1 Corrections to General Relativity
1.4.2 The Impact of Inhomogeneities
1.4.3 Quintessence Models
2 Supernovae as Standard Candles
2.1 Empirical Properties of SNIa
2.1.1 Spectroscopic Properties
2.1.2 Photometric Properties
2.1.3 Peculiar SNIa
2.2 Proposed Physical Mechanisms
2.2.1 The Single Degenerate Model
2.2.2 The Double Degenerate Model
2.3 SNIa Modeling
2.3.1 Standardizing the Distance Modulus
2.3.2 On the Need for K-Corrections
2.3.3 Overview of Model Training
2.3.4 Accounting for Data Holes
2.4 Hints of New Standardization Parameters
2.4.1 Spectroscopic Correlations
2.4.2 Galaxy Dependence
3 Overview of the Supernova Legacy Survey
3.1 Overview of the Science Analysis
3.1.1 Photometry in Different Bands
3.1.2 The SALT2 Light Curve Fitter
3.1.3 The Cosmology Fit
3.2 The CFHT Legacy Survey
3.2.1 The “Very Wide” Survey
3.2.2 The “Wide” Survey
3.2.3 The “Deep” Survey
3.3 Observation Strategy
3.3.1 A Rolling Search
3.3.2 Spectroscopic Follow Up
3.4.1 The Upper End
3.4.2 Wide Field Corrector
3.4.3 Image Stabilizing Unit
3.4.4 Guiding and Focus
3.4.6 Around MegaCam
3.4.7 Modeling the Optical Path
3.5 Overview of the Data Flow
3.5.1 Preprocessing at CFHT
3.5.2 Local Processing
4 PSF Photometry of Dim Supernovae
4.1 Local Image Preprocessing
4.1.1 Sky Subtraction
4.1.2 Star Catalog
4.1.3 PSF fitting
4.2 Direct Simultaneous Photometry
4.2.2 Preserving Linearity
4.2.3 Effects of Refraction
4.3 Validations with simulations
4.3.1 Simulation goals
4.3.2 Simulation method
4.3.3 Expected biases
4.3.4 Simulation parameters
5 Photometric Calibration of the SNLS Supernova Sample
5.1 Calibrating Supernova Measurements
5.1.1 An Introduction to Photometric Calibration
5.1.2 The SNLS Magnitude System
5.2 Instrument Response Model
5.2.1 Transmission Model
5.2.2 Filter Measurements
5.3 Zero Point computation
5.3.1 Sky Pollution Bias
5.3.2 Chromatic PSF Bias
5.3.3 Results of Calibration uncertainty
6 Cosmology Analysis
6.1 Supernova Sample Selection
6.1.1 SALT2 Training Sample
6.1.2 For Cosmology
6.1.3 Flux Convention
6.2 Lightcurve Parameter Extraction
6.2.1 Results of the SALT2 Model
6.2.2 Lightcurve Fitting
6.2.3 Simulating the SALT2 Uncertainty
6.3.1 Host Galaxy Mass Corrections
6.3.2 Peculiar Velocity Corrections
6.3.3 Malmquist Bias Correction
6.3.4 Dust Correction
6.4 Fitting the Hubble Diagram
6.4.1 Correlated Calibration Systematics
6.4.2 Determining ‡coh
6.4.3 Constraints from Other Cosmological Probes
6.4.4 Overview of the Fit Method
6.5 Cosmological Results
6.5.1 A Blinded Analysis
6.5.2 Comparison with JLA Analysis
6.5.3 Impact of Corrections
6.5.4 Preliminary Analysis Results