The Theory/Observation connection lecture 1 the standard model

1. The Theory/Observation connectionlecture 1the standard model Will Percival The University of Portsmouth

2. Lecture outline • The standard model (flat Lambda CDM universe) • GR • cosmological equations • constituents of the Universe • redshifts, distances • Inflation • Curvature

3. 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)

4. 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

5. 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

6. 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 Q Example: infinitesimal vector PQ xa+dxa P A covariant tensor of rank (order) 1 transforms under a change of coordinates according to xa Higher rank = more derivatives in transform e.g. contravariant tensor of rank 2 transforms as Can form mixed tensors

7. General Relativity in 1-slide Metric Inverse Raise/Lower Indices with metric/inverse Christoffel Symbol Ricci (Curvature) Tensor Ricci Scalar

8. Einstein’s Equations Ricci Tensor Newton’s Constant Ricci Scalar Energy Momentum Tensor Shows how matter causes changes in the metric (gravity)

9. Application to Cosmology FRW metric for flat space has: So (for example) the Christoffel symbol reduces to:

10. 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

11. 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

12. Decomposing the density Measure densities relative to the critical density Where Can write the Friedmann equation in terms of density components

13. Evolution of energy densities To see how a material behaves, we need to assume conservation of energy (conservation of the energy-momentum tensor) Fundamental property of a material: its Equation of state Density at present day

14. 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

15. Relativistic particles Bosons such as photons have Bose-Einstein distributions. For photons, E=p Pressure and density equations then lead to Conservation of energy gives Evolution is consistent with dilution with expanding Universe and energy loss due to frequency shift

16. Acceleration vs deceleration First-Year SNLS Hubble Diagram All matter in the Universe tends to cause deceleration BUT, we see accelerated expansion …

17. Dark Energy In standard model, dark energy is caused by a cosmological constant with w=-1 Need component with w < - 1/3 for acceleration Conservation of energy gives Empty space contains energy

18. 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

19. The constituents of the Universe

20. Photon energy density 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.

21. Photon energy density Energy density of gas of bosons in equilibrium Spin states Bose-Einstein condensation Sum over phase space For relativistic material, E=p

22. redshift Define stretching factor of light due to cosmological expansion as redshift For low redshifts, z ≈ v/c, so redshift directly measures recession velocity Animation from Wayne Hu Original Hubble diagram (Hubble 1929)

23. 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

24. 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

25. Distances: angular diameter distance Given apparent size of object, can we measure its distance? l Comoving size of object is l/a, so comoving angle of distant object (on Euclidean grid) is  dA If no Euclidean picture (not flat)

26. 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

27. 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.

28. Inflation: motivation Temperature of CMB is very similar in all directions. Suggests causal contact. Comoving perturbation scales fixed. Enter horizon at different times

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

30. 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

31. Beyond the “standard model”: curvature Friedmann equation can be written in the form Remove flatness constraint in FRW metric, then get extra term in Friedmann equation gives evolution of densities relative to critical density (evolution of critical density gives E2 terms)

32. 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)

33. What if w≠-1? Constant w models

34. 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