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

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 X a in the x a coordinate system, associated with a point P, which transform under a change of coordinates according to Example: infinitesimal vector PQ Q P A 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 xaxa x a +dx a 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 givesEmpty 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 Cosmic Microwave Background (CMB) temperature has been extremely well measured (T = 2.35  eV). Can turn this into a measurement of the photon density.

Photon energy density Energy density of gas of bosons in equilibrium Spin states Sum 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  dAdA 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 Temperature of CMB is very similar in all directions. Suggests causal contact. Comoving perturbation scales fixed. Enter horizon at different times

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

Inflation = early dark energy Decreasing Hubble radius means that we need acceleration Dark Energy dominated the expansion of the Universe. Magnitude needs to be ~ 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 E 2 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 –  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/