Astr 5460 “Longair Cosmology”

Lecture slides based on Longair, Ch. 5-8

- More detail than Combes et al.

- May explore after Thanksgiving

Unless noted, all figs and eqs from Longair.

Theoretical Framework

Section 5.1, the “Cosmological Principle”

Isotropic, homogenous, uniform expansion

Can write relativistic equations in different forms (famous names cited here…)

Weyl’s postulate: “The particles of the substratum (representing the nebulae) lie in space-time on a bundle of geodesics diverging from a porint in the (finite or infinite) past.”

Theoretical Framework

Section 5.1, the “Cosmological Principle”

Geodesics are “world-lines” of galaxies and do not intersect except at a singular point in the past. Weyl’s idea predates Hubble’s law.

Fundamental observers on each world line, each with standard clock measuring cosmic time from that singular point.

“We are not located at any special location in the universe.”

Theoretical Framework

Sections 5.1-5.4 cover underpinnings of GR (curved spaces, space-time metrics) and in particular the Robertson-Walker metric that we will need to describe the universe.

Read and follow these sections, but we don’t have the lecture time to go into much detail with the perspective of observational astronomers in a mixed galaxies/cosmology course.

Theoretical Framework

Section 5.5 covers observables. We’re going to jump to the chase momentarily and walk through Hogg (2000), which integrates this material with world models (Chapter 7, and a Sandage review article I will probably assign soon.)

Theoretical Framework

Theoretical Framework

Chapter 6 introduces General Relativity, which I won’t go over in class. Again, read through it for your own benefit.

Friedman World Models

Einstein’s Field Equations

Under ideas discussed previously (cosmological principle, Weyl’s postulate, isotropy, homogeneity) the field equations reduce to the simple pair of independent equations:

R is the scale factor, ρ is total inertial mass density of matter & radiation, p the associated pressure. Script R is the radius of curvature, and there’s lambda.

The Standard Dust Models (Λ=0)

“Dust” means pressureless fluid, p=0

Field equations then reduce to:

Have a Newtonian Analog:

The Standard Dust Models (Λ=0)

Have a Newtonian Analog:

Replace x by comoving value r using the scale factor R, x = Rr, and express density in terms of its value at the present epoch ρ = ρ₀R^-3, then:

Which matches eq. 7.1 for dust and lambda=0. Multiplying by the derivative of R and integrating gives us essentially eq. 7.2

The Standard Dust Models (Λ=0)

Critical Density and the Density Parameter:

The ratio of the current density to the critical density is “omega-naught” Ω₀:

Often use different subscripts on omega to denote density contributions from baryons, dark matter, etc.

The Standard Dust Models (Λ=0)

Inserting Ω₀ into eq. 7.1 and 7.2:

Setting eq. 7-18 to present epoch, t =t₀, R=0, and derivative of R is H₀, then:

And we see that curvature and density are intimately related.

The Standard Dust Models (Λ=0)

Dynamics: using the previous equations, we can rewrite eq. 7-18:

And in the limit of large values of R, we get:

This is easy to interpret.

When Ω₀ < 1, universe is open, hyperbolic, and expands to infinity with finite velocity.

When Ω₀ > 1, universe is closed, spherical, and eventually collapse after reaching a maximum size after a time:

The Standard Dust Models (Λ=0)

This is easy to interpret.

When Ω₀ < 1, universe is open, hyperbolic, and expands to infinity with finite velocity.

When Ω₀ > 1, universe is closed, spherical, and eventually collapse after reaching a maximum size after a time:

Recollapses after a time t = 2 t_max.

When Ω₀ = 1, universe is critical, flat, and expands to infinity with velocity approaching zero. Einstein-de Sitter model:

The Standard Dust Models (Λ=0)

The Standard Dust Models (Λ=0)

Deceleration Parameter, q₀:

Substituting into the first of the dynamics equation (7-20) we immediately can write:

Keep in mind this (and all these results so far) are for universes with zero cosmological constant.

The Standard Dust Models (Λ=0)

Cosmic Time-Redshift Relation:

Because R = (1 + z)^-1, eq. 7-20 gives us

Which can be integrated to give cosmic time since the big bang. For different types of universes need different forms of the equations:

The Standard Dust Models (Λ=0)

The Flatness Problem

Can determine how Hubble’s “constant” changes with time, from eq. 7-20 and writing R = (1+z)^-1:

Similarly for Omega, using the general definition Ω=8πGρ/3H², and expressing the density ρ = ρ₀(1+z)³, then

And rewriting...

Notice the behavior at high z. This is the origin of the problem.

The Standard Dust Models (Λ=0)

Distance Measures as a function of redshift

Radial comoving distance coordinate r incremental is

Integrate from redshift 0 to z:

The Standard Dust Models (Λ=0)

Distance Measures as a function of redshift

Then to find the proper distance D, recall from chapter 5 that D = R sin(r/R) where script R is given by eq. 7-19. For an exercise you could derive the general expression:

Notice what happens for an empty universe. Deriving this was a closed book exam question I had in grad school (prof assumed we had the curiosity to check this for ourselves).

The Standard Dust Models (Λ=0)

Observed Properties of Standard Objects in the Friedman World Models with zero cosmological constant (cf. Hogg 2000, chapter 5, Ned Wright’s calculator).

Angular Diameters (need Angular Distance f(z))

Flux Densities (need luminosity distance f(z))

Comoving volume within redshift z

In particular covered in more detail in section 7.2.8

The Standard Dust Models (Λ=0)

Models for which Λ is not 0

Einstein originally used lambda to create a static (non-expanding, non-contracting) universe according to his preconceptions.

Such models also popular in 1930s when the Hubble constant was thought to be 500 km/s/Mpc, creating problems with the age of the universe (less than age of Earth).

Supernova results, WMAP results, both favor non-zero cosmological constant.

Models for which Λ is not 0

Einstein field equations become

Eq. 7.56 indicates even in an empty universe there is a net force on a test particle (+ or -).

Models for which Λ is not 0

For those interested, there is an interpretation of scalar Higgs fields under quantum field theory (see Zeldovich 1986).

Zero point vacuum fluctations associated with zero point energies of quantum fields results in a negative energy equation of state (having “tension” rather than “pressure”). Quantum field theory can then make predictions about the value of a cosmological constant – and is off by some 120 orders of magnitude! Works for inflationary period, but not now.

Models for which Λ is not 0

Can rewrite field equations in terms of mass-energy densities:

Models for which Λ is not 0

Can then identify lambda with vacuum mass density:

So now can interpret lambda in terms of “omega – lambda” which is often used in discussions. What of q, the deceleration parameter, in these models?

Models for which Λ is not 0

Equations 7.60 and 7.62 now give us:

And can rewrite the dynamical equations (again!)

Models for which Λ is not 0

Substituting the values of R, dR/dt, and R = 1 at the present epoch, we can solve for curvature of space given the contributions to Omega:

Dynamics of Models with Λ not 0

If Lambda < 0, Omega_Lambda is less than zero, and the term will enhance gravity. In all cases expansion is eventually reversed.

Models with Lambda > 0, we essentially incorporate a repulsive force that opposes gravity.

Some of the mathematical details in the text.

Dynamics of Models with Λ not 0

Inhomogeneous World Models

Because the real world is not perfectly homogeneous, is it? These perturbations cause deviations of the paths of light rays and must be taken into account for some applications.

Inhomogeneous World Models

How one observable changes with homogeneity.

Lensing effects are also a result of inhomogeneities.

Sandage Review vs. Chapter 8

Geometry (curvature) tests

Number counts (z,m,q,Evolution!)

Hubble Diagrams

Galaxy evolution plus K-corrections

Theta-z tests

Timescale tests

Other items (e.g. Malmquist bias)

Tests of World Models


	Lecture slides based on Longair, Ch. 5-8
	- More detail than Combes et al.
	- May explore after Thanksgiving



	Unless noted, all figs and eqs from Longair.


	Section 5.1, the “Cosmological Principle”
		Isotropic, homogenous, uniform expansion
		Can write relativistic equations in different forms (famous names cited here…)
		Weyl’s postulate: “The particles of the substratum (representing the nebulae) lie in space-time on a bundle of geodesics diverging from a porint in the (finite or infinite) past.”


	Section 5.1, the “Cosmological Principle”
		Geodesics are “world-lines” of galaxies and do not intersect except at a singular point in the past. Weyl’s idea predates Hubble’s law.
		Fundamental observers on each world line, each with standard clock measuring cosmic time from that singular point.

	“We are not located at any special location in the universe.”


	Sections 5.1-5.4 cover underpinnings of GR (curved spaces, space-time metrics) and in particular the Robertson-Walker metric that we will need to describe the universe.
	Read and follow these sections, but we don’t have the lecture time to go into much detail with the perspective of observational astronomers in a mixed galaxies/cosmology course.


	Section 5.5 covers observables. We’re going to jump to the chase momentarily and walk through Hogg (2000), which integrates this material with world models (Chapter 7, and a Sandage review article I will probably assign soon.)


	Chapter 6 introduces General Relativity, which I won’t go over in class. Again, read through it for your own benefit.


	Einstein’s Field Equations
		Under ideas discussed previously (cosmological principle, Weyl’s postulate, isotropy, homogeneity) the field equations reduce to the simple pair of independent equations:





		R is the scale factor, ρ is total inertial mass density of matter & radiation, p the associated pressure. Script R is the radius of curvature, and there’s lambda.


	“Dust” means pressureless fluid, p=0
		Field equations then reduce to:



		Have a Newtonian Analog:


	Have a Newtonian Analog:


		Replace x by comoving value r using the scale factor R, x = Rr, and express density in terms of its value at the present epoch ρ = ρ₀R^-3, then:


		Which matches eq. 7.1 for dust and lambda=0. Multiplying by the derivative of R and integrating gives us essentially eq. 7.2


	Critical Density and the Density Parameter:


	The ratio of the current density to the critical density is “omega-naught” Ω₀:


	Often use different subscripts on omega to denote density contributions from baryons, dark matter, etc.


	Inserting Ω₀ into eq. 7.1 and 7.2:




	Setting eq. 7-18 to present epoch, t =t₀, R=0, and derivative of R is H₀, then:


	And we see that curvature and density are intimately related.


	Dynamics: using the previous equations, we can rewrite eq. 7-18:


	And in the limit of large values of R, we get:


	This is easy to interpret.
		When Ω₀ < 1, universe is open, hyperbolic, and expands to infinity with finite velocity.
		When Ω₀ > 1, universe is closed, spherical, and eventually collapse after reaching a maximum size after a time:


	Deceleration Parameter, q₀:



	Substituting into the first of the dynamics equation (7-20) we immediately can write:



	Keep in mind this (and all these results so far) are for universes with zero cosmological constant.


	Cosmic Time-Redshift Relation:
		Because R = (1 + z)^-1, eq. 7-20 gives us



		Which can be integrated to give cosmic time since the big bang. For different types of universes need different forms of the equations:


	The Flatness Problem
		Can determine how Hubble’s “constant” changes with time, from eq. 7-20 and writing R = (1+z)^-1:


		Similarly for Omega, using the general definition Ω=8πGρ/3H², and expressing the density ρ = ρ₀(1+z)³, then


		And rewriting...



		Notice the behavior at high z. This is the origin of the problem.


	Distance Measures as a function of redshift
		Radial comoving distance coordinate r incremental is






		Integrate from redshift 0 to z:


	Distance Measures as a function of redshift
		Then to find the proper distance D, recall from chapter 5 that D = R sin(r/R) where script R is given by eq. 7-19. For an exercise you could derive the general expression:



		Notice what happens for an empty universe. Deriving this was a closed book exam question I had in grad school (prof assumed we had the curiosity to check this for ourselves).


Observed Properties of Standard Objects in the Friedman World Models with zero cosmological constant (cf. Hogg 2000, chapter 5, Ned Wright’s calculator).

	Angular Diameters (need Angular Distance f(z))

	Flux Densities (need luminosity distance f(z))

	Comoving volume within redshift z
		In particular covered in more detail in section 7.2.8


	Einstein originally used lambda to create a static (non-expanding, non-contracting) universe according to his preconceptions.
	Such models also popular in 1930s when the Hubble constant was thought to be 500 km/s/Mpc, creating problems with the age of the universe (less than age of Earth).
	Supernova results, WMAP results, both favor non-zero cosmological constant.


	Einstein field equations become







	Eq. 7.56 indicates even in an empty universe there is a net force on a test particle (+ or -).


	For those interested, there is an interpretation of scalar Higgs fields under quantum field theory (see Zeldovich 1986).
	Zero point vacuum fluctations associated with zero point energies of quantum fields results in a negative energy equation of state (having “tension” rather than “pressure”). Quantum field theory can then make predictions about the value of a cosmological constant – and is off by some 120 orders of magnitude! Works for inflationary period, but not now.


	Can rewrite field equations in terms of mass-energy densities:


	Can then identify lambda with vacuum mass density:







	So now can interpret lambda in terms of “omega – lambda” which is often used in discussions. What of q, the deceleration parameter, in these models?


	Equations 7.60 and 7.62 now give us:



	And can rewrite the dynamical equations (again!)


	Substituting the values of R, dR/dt, and R = 1 at the present epoch, we can solve for curvature of space given the contributions to Omega:


	If Lambda < 0, Omega_Lambda is less than zero, and the term will enhance gravity. In all cases expansion is eventually reversed.

	Models with Lambda > 0, we essentially incorporate a repulsive force that opposes gravity.

	Some of the mathematical details in the text.


	Because the real world is not perfectly homogeneous, is it? These perturbations cause deviations of the paths of light rays and must be taken into account for some applications.


	How one observable changes with homogeneity.
	Lensing effects are also a result of inhomogeneities.


	Geometry (curvature) tests
	Number counts (z,m,q,Evolution!)
	Hubble Diagrams
	Galaxy evolution plus K-corrections
	Theta-z tests
	Timescale tests
	Other items (e.g. Malmquist bias)