The Theory/Observation connectionlecture 1

the standard model

Will Percival

The University of Portsmouth

Lecture outline

The standard model (flat Lambda CDM universe)– GR

– cosmological equations

– constituents of the Universe

– redshifts, distances

Inflation Curvature

The Universe is expanding

Scale factor a quantifies expansion

Figure from Dodelson “modern cosmology” (as are a number of the explanatory diagrams in this talk)

Metrics

Coordinate differences on expanding grid are comoving distances.

To get a physical distance dl, from a Set of coordinate differences, use the metric.

The metric for distances on the surface of a sphere is well known

The FRW metric

The scale factor a(t) is the key function in the Friedmann-Robsertson-Walker metric

In a flat Universe, k=0, and the metric reduces to

Note: summation convention

Assume c=1

Tensors in 1-slide

A contravariant tensor of rank (order) 1 is a set of quantities, written Xa in the xa coordinate system, associated with a point P, which transform under a change of coordinates according to

Example: infinitesimal vector PQ Q

PA covariant tensor of rank (order) 1 transforms under a change of coordinates according to

Higher rank = more derivatives in transform e.g. contravariant tensor of rank 2 transforms as

xa

xa+dxa

Can form mixed tensors

General Relativity in 1-slide

Metric Inverse

Raise/Lower Indices with metric/inverse

Christoffel Symbol

Ricci (Curvature) Tensor

Ricci Scalar

Einstein’s Equations

Ricci Tensor

Ricci Scalar

Newton’s Constant

Energy Momentum Tensor

Shows how matter causes changes in the metric (gravity)

Application to Cosmology

FRW metric for flat space has:

So (for example) the Christoffel symbol reduces to:

Time-time component of Einstein’s equations

Similar simplifications give

So time-time component of Einstein’s equations reduces to

Giving Friedmann equation for cosmological evolution

Space-space component of Einstein’s equations

Similar analysis to that for the time-time component leads to

Where P is the diagonal space-space component of the energy-momentum tensor

Combine with the Friedmann equation to give

Deceleration, unless +3P<0

Decomposing the density

Can write the Friedmann equation in terms of density components

Measure densities relative to the critical density

Where

Evolution of energy densities

Fundamental property of a material: its Equation of state

To see how a material behaves, we need to assume conservation of energy (conservation of the energy-momentum tensor)

Density at present day

Non-relativistic matter (dust)

Pressure of material is very small compared with energy density, so effective w=0

Evolution is consistent with simple dilution with expanding Universe

Relativistic particles

Bosons such as photons have Bose-Einstein distributions. For photons, E=p

Evolution is consistent with dilution with expanding Universe and energy loss due to frequency shift

Pressure and density equations then lead to

Conservation of energy gives

Acceleration vs deceleration

All matter in the Universe tends to cause deceleration

BUT, we see accelerated expansion …

First-Year SNLS Hubble Diagram

Dark Energy

In standard model, dark energy is caused by a cosmological constant with w=-1

Conservation of energy gives Empty space contains energy

Need component with w < - 1/3 for acceleration

Decomposing the density

Can write the Friedmann equation in terms of density components

Evolution of Universe depends on contents and will go through phases as each becomes dominant

The constituents of the Universe

Photon energy density

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Cosmic Microwave Background (CMB) temperature has been extremely well measured (T = 2.35 10-4eV). Can turn this into a measurement of the photon density.

Photon energy density

Energy density of gas of bosons in equilibrium

Spin statesSum over phase space

Bose-Einstein condensation

For relativistic material, E=p

redshift

Animation from Wayne Hu

Define stretching factor of light due to cosmological expansion as redshift

For low redshifts, z ≈ v/c, so redshift directly measures recession velocity

Original Hubble diagram (Hubble 1929)

Distances: comoving distance

In a time dt, light travels a distance dx = cdt/a on a comoving grid

Define comoving distance from us to a distant object as

For flat cosmologies, with matter domination,

Can use this distance measure to place galaxies on a comoving grid. BEWARE: this only works for flat cosmologies SDSS

Conformal time

Comoving distance a light particle could have travelled since the big bang

In expanding Universe, this is a monotonically increasing function of time, so we can consider it a time variable

Called conformal time

Comoving size of object is l/a, so comoving angle of distant object (on Euclidean grid) is

Distances: angular diameter distance

dA

l

Given apparent size of object, can we measure its distance?

If no Euclidean picture (not flat)

Distances: luminosity distance

Given apparent flux from an object (actual luminosity L), can we measure its distance?

On a comoving grid,

But, expansion means that the number of photons crossing (in a fixed time interval) the shell is lower by a factor a. Also get a factor of a from energy change (redshift).

Again, we need to adjust this for non-flat cosmologies, where we can not use an Euclidean grid

Inflation: motivation

Comoving Horizon

Comoving distance particles can travel up to time t: defines distances over which causal contact is possible

Can rewrite as function of Hubble radius (aH)-1

Hubble radius gives (roughly) the comoving distance travelled as universe expands by factor ~2. The comoving horizon is logarithmic integral of this.

Inflation: motivation

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Temperature of CMB is very similar in all directions. Suggests causal contact.

Comoving perturbation scales fixed. Enter horizon at different times

Inflation: motivation

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Inflation in early Universe allows causal contact at early times: requires Hubble radius to decrease with time

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Inflation = early dark energy

Decreasing Hubble radius means that we need acceleration

Dark Energy dominated the expansion of the Universe. Magnitude needs to be ~10100 larger than driving current acceleration

Beyond the “standard model”: curvature

Friedmann equation can be written in the form

gives evolution of densities relative to critical density(evolution of critical density gives E2 terms)

Remove flatness constraint in FRW metric, then get extra term in Friedmann equation

Beyond the “standard model”: curvature

Critical densities are parameteric equations for evolution of universe as a function of the scale factor a

All cosmological models will evolve along one of the lines on this plot (away from the EdS solution)

What if w≠-1?

Constant w models

Further reading

Dodelson, SLAC lecture notes (formed basis for the first part of this lecture, and a number of the explanatory diagrams). Available online at

– http://www-conf.slac.stanford.edu/ssi/2007/lateReg/program.htm

Dodelson, “Modern Cosmology”, Academic Press Peacock, “Cosmological Physics”, Cambridge University Press For a review of the effect of dark energy see

– Percival et al (2005), astro-ph/0508156