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The Theory/Observation connection lecture 1 the standard model Will Percival The University of Portsmouth
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Lecture outline The standard model (flat Lambda CDM universe) – GR – cosmological equations – constituents of the Universe – redshifts, distances Inflation Curvature
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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)
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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
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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
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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
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General Relativity in 1-slide Metric Inverse Raise/Lower Indices with metric/inverse Christoffel Symbol Ricci (Curvature) Tensor Ricci Scalar
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Einstein’s Equations Ricci Tensor Ricci Scalar Newton’s Constant Energy Momentum Tensor Shows how matter causes changes in the metric (gravity)
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Application to Cosmology FRW metric for flat space has: So (for example) the Christoffel symbol reduces to:
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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
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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
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Decomposing the density Can write the Friedmann equation in terms of density components Measure densities relative to the critical density Where
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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
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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
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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
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Acceleration vs deceleration All matter in the Universe tends to cause deceleration BUT, we see accelerated expansion … First-Year SNLS Hubble Diagram
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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
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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
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The constituents of the Universe
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Photon energy density Cosmic Microwave Background (CMB) temperature has been extremely well measured (T = 2.35 10 -4 eV). Can turn this into a measurement of the photon density.
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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
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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)
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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
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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
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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)
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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
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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.
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Inflation: motivation Temperature of CMB is very similar in all directions. Suggests causal contact. Comoving perturbation scales fixed. Enter horizon at different times
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Inflation: motivation 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 ~10 100 larger than driving current acceleration
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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
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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)
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What if w≠-1? Constant w models
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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.htmhttp://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
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