the theory/observation connection lecture 1 the standard model
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The Theory/Observation connection lecture 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. - PowerPoint PPT PresentationTRANSCRIPT
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