Tests of gravity with large cosmology surveys

"Tests of gravity with large cosmology surveys", talk at the University of Manchester Astrosoc on 25th November 2015

Transcript

1. Was Einstein Wrong?! Testing Gravity with combined cosmological probes using the Dark Energy Survey! Donnacha Kirk, UCL Wed 25th November 2015 UoM Astrosoc •  Why do we care about gravity? •  How we “do cosmology” – multiple probes. •  Dark Energy Survey as a multi-probe survey. •  What can multiple probes tell us about gravity?
2. COSMOLOGY
3. Einstein/Hubble Fundamental theory of gravity, dominant force on cosmic scales, provides our ‘equations of motion’ for the Universe. The first empirical observations about the scale, structure and dynamics of the Universe.
4. Einstein’s Universe •  Einstein makes initial attempts to apply his new theory of gravity to model the whole Universe. •  His General Theory of Relativity relates the curvature of space-time to the stress-energy tensor. ϵi = γi + ϵi Gµν = 8πGTµν Gµν = 8πGTµν + Λ Gµν = 8πGTµν + Λ “Mass tells spacetime how to bend, spacetime tells mass how to move.”
5. Hubble and the Big Bang •  Soon thereafter, 1931, Edwin Hubble discovers the Universe is far from static. •  Galaxies recede from each other, meaning the Universe had a beginning, the primeval atom or Big Bang.
6. Attempts to measure deceleration •  Once the Universe was known to be expanding, an obvious question was “How fast?” •  Mass attracts through gravity. •  After the big bang, this attractive force will slow the expansion. •  The future course of the Universe depends on the exact amount of mass.
7. Expansion of the Universe! •  Supernovae as Standard Candles.
8. TYPE IA SUPERNOVAE!
9. Expansion of the Universe •  Supernovae as Standard Candles. •  Measure deceleration of the Universe…
10. Expansion of the Universe •  Supernovae as Standard Candles. •  Measure deceleration of the Universe… … but find accelerating expansion.
11. Accelerated Expansion! •  What can drive the acceleration? •  Ordinary matter + general relativity? A new type of exotic matter/energy with negative pressure.
12. Accelerated Expansion! •  What can drive the acceleration? •  Ordinary matter + general relativity? No! •  Need some other source of energy to power the acceleration à Dark Energy! •  A new type of exotic matter/energy with negative pressure.
13. Accelerated Expansion! •  What can drive the acceleration? •  Ordinary matter + general relativity? No! •  Need some other source of energy to power the acceleration à Dark Energy! •  A new type of exotic matter/energy with negative pressure.
14. Problem Solved? •  QFT calculations predict space contains a vacuum energy due to billions of tiny interactions. •  This has all the properties that dark energy needs to make the Universe accelerate.
15. Cosmological Constant Problem •  Not quite- this is our own vaguely plausible physical theory and it’s a bit embarrassing. •  QFT calculations of the vacuum energy are 120 orders of magnitude out- probably the worst prediction in all the history of physics à solve this or back to the drawing board.
16. Dark Energy •  Today we have lots of energy for a “standard model” of the Universe… •  … but the resulting picture is quite strange.
17. MG for Cosmic Acceleration Observation: Accelerating Expansion Gµν = 8πGTµν Gµν +Gdark µν = 8πGTµν Gµν = 8πG Tµν +Tdark µν Gµν = 8πGTµν Gµν = 8πG Tµν +Tdark µν Gµν +Gdark µν = 8πGTµν Cannot be explained by GR + matter
18. MG for Cosmic Acceleration Observation: Accelerating Expansion Gµν = 8πGTµν Gµν +Gdark µν = 8πGTµν Gµν = 8πG Tµν +Tdark µν Gµν = 8πGTµν Gµν = 8πG Tµν +Tdark µν Gµν +Gdark µν = 8πGTµν Cannot be explained by GR + matter Add Dark Energy to the Stress-Energy tensor?
19. MG for Cosmic Acceleration Observation: Accelerating Expansion Gµν = 8πGTµν Gµν +Gdark µν = 8πGTµν Gµν = 8πG Tµν +Tdark µν Gµν = 8πGTµν Gµν = 8πG Tµν +Tdark µν Gµν +Gdark µν = 8πGTµν Cannot be explained by GR + matter Add Dark Energy to the Stress-Energy tensor? OR modify the Einstein tensor – change gravity?
20. Measurement Techniques Radio Microwave Optical X-Ray Gamma Ray Ways of slicing up Cosmology
21. Measurement Techniques Radio Microwave Optical X-Ray Gamma Ray Ways of slicing up Cosmology
22. Constituents: •  Baryons •  Radiation •  Neutrinos •  Dark Matter •  Dark Energy Ways of slicing up Cosmology
23. History Text Ways of slicing up Cosmology
24. Ways of slicing up Cosmology Observables Cosmic Microwave Background Weak Gravitational Lensing Galaxy Surveys Supernovae
25. CMB
26. Galaxy Surveys
27. Weak Gravitational Lensing
28. Isaac Newton: Gravity & Light Never calculated the effect himself. “Do not Bodies act upon Light at a distance, and by their action bend its Rays; and is this not action… strongest at the least distance”
29. Light bending round the Sun
30. Donnacha Kirk Diploma Club, 21st March 2013 Strong Lensing •  Abell plot.
31. Weak Gravitational Lensing •  We saw strong lensing due to gravity. •  Every galaxy we observe has been distorted a little.
32. Weak Gravitational Lensing •  We saw strong lensing due to gravity. •  Every galaxy we observe has been distorted a little.
35. Weak Gravitational Lensing •  We saw strong lensing due to gravity. •  Every galaxy we observe has been distorted a little. •  Like looking at the world through a frosted window.
36. Typical star used for finding telescope response Typical galaxy used for lensing
37. Formidable observational challenge
38. Breaking degeneracies •  Different cosmic probes are sensitive to fundamental parameters in different ways. •  Combining multiple probes can “break degeneracies”. •  Producing much more accurate measurements.
39. Dark Energy Survey (DES) •  First light Sept 2012 •  Optics made in UCL. •  570 megapixel camera for WGL over 5,000deg2.
41. 41 Dark Energy Survey Collaboration Fermilab, UIUC/NCSA, University of Chicago, LBNL, NOAO, University of Michigan, University of Pennsylvania, Argonne National Lab, Ohio State University, Santa-Cruz/SLAC/Stanford, Texas A&M Brazil Consor<um UK Consor<um: UCL, Cambridge, Edinburgh, NoEngham, Portsmouth, Sussex Spain Consor<um: CIEMAT, IEEC, IFAE CTIO Ludwig-Maximilians Universität LMU ETH Zurich ~300 scientists US support from DOE+NSF
42. DECam at CTIO 4
43. DES SV image of a deep SN field
44. 1st light: 12 Sept. 2012 5 Fornax cluster
45. 1st light: 12 Sept. 2012 5 Fornax cluster
46. Donnacha Kirk Diploma Club, 21st March 2013 1st light: 12 Sept. 2012 6 1 chip Fornax NGC1365 galaxy cluster, image from a single CCD chip!
48. Looking at images: invaluable!
49. DES as multi-probe survey Large-Scale Structure •  DES will measure ~300 million galaxies with photometric redshifts. Growth of structure with time and BAO for expansion rate. Supernovae •  DES will measure ~3000 type Ia supernovae standard candles. Measure of the expansion history of the Universe. Galaxy Clusters •  Identify galaxy clusters and count how the populations change with time and cluster size. Results will depend on the amount of mass in the Universe and the competition between expansion and collapse. Weak Gravitational Lensing •  Direct probe of all matter. Simultaneous probe of expansion and structure. Tomography allows us to chart structure growth with time.
50. Let’s understand this plot a bit more…
51. Sampling a Likelihood Surface •  You can calculate a likelihood. •  The likelihood depends on some parameters. •  You want to find the best-fit values of those parameters and the error.
52. Sampling a Likelihood Surface Basic Inference:
 Grids Low number of parameters
 㱺can try grid of all possible parameter combinations Number of posterior evaluations 
 = (grid points per param)(number params) Approximate PDF integrals as sums · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · x y •  You can calculate a likelihood. •  The likelihood depends on some parameters. •  You want to find the best-fit values of those parameters and the error. •  Simplest approach – the grid. •  Evaluate the likelihood
53. What is a likelihood – SNe Example s y L is given by f = L 4⇡D2 L In Big Bang cosmology it is given by DL = (1 + z)c H0 p |1 ⌦| Sk(r), z 0 dz0 p ⌦m (1 + z0)3 + ⌦v + (1 ⌦)(1 + z0)2 . on whether ⌦ ⌘ ⌦m + ⌦v is > 1, = 1, or < 1, and z is the m , ⌦v and H0 are the density parameters (today) in matter, nt. It is beyond the scope of these notes to derive this, but it ate cosmology course. mplifies to •  Take the flux of the supernova.
54. What is a likelihood – SNe Example s y L is given by f = L 4⇡D2 L In Big Bang cosmology it is given by DL = (1 + z)c H0 p |1 ⌦| Sk(r), z 0 dz0 p ⌦m (1 + z0)3 + ⌦v + (1 ⌦)(1 + z0)2 . on whether ⌦ ⌘ ⌦m + ⌦v is > 1, = 1, or < 1, and z is the m , ⌦v and H0 are the density parameters (today) in matter, nt. It is beyond the scope of these notes to derive this, but it ate cosmology course. mplifies to •  Take the flux of the supernova. •  Relate to the luminosity distance. 0 ⌦m (1 + z0)3 + ⌦v + (1 ⌦)(1 + z0)2 r, sinh r, depending on whether ⌦ ⌘ ⌦m + ⌦v is > 1, = 1, or < 1, an of the supernova. ⌦m , ⌦v and H0 are the density parameters (today) d the Hubble constant. It is beyond the scope of these notes to derive t al for an undergraduate cosmology course. erse (⌦ = 1), this simplifies to DL(z) = 3000h 1(1 + z) Z z 0 dz0 p ⌦m (1 + z0)3 + 1 ⌦m Mpc, m s 1 Mpc 1. To avoid evaluating integrals to calculate DL, we can use a lid for flat universes only), given by U.-L. Pen, ApJS, 120:4950, 1999: DL(z) = c H0 (1 + z)  ⌘(1, ⌦0 ) ⌘ ✓ 1 1 + z , ⌦0 ◆ = 2 p s3 + 1  1 a4 0.1540 s a3 + 0.4304 s2 a2 + 0.19097 s3 a + 0.066941s4 1/8 /⌦0 . This is claimed to be accurate to better than 0.4% for 0.2  ⌦0  1 •  This depends on cosmology.
55. What is a likelihood – SNe Example s y L is given by f = L 4⇡D2 L In Big Bang cosmology it is given by DL = (1 + z)c H0 p |1 ⌦| Sk(r), z 0 dz0 p ⌦m (1 + z0)3 + ⌦v + (1 ⌦)(1 + z0)2 . on whether ⌦ ⌘ ⌦m + ⌦v is > 1, = 1, or < 1, and z is the m , ⌦v and H0 are the density parameters (today) in matter, nt. It is beyond the scope of these notes to derive this, but it ate cosmology course. mplifies to U.-L. Pen, ApJS, 120:4950, 1999: ⌘ ✓ 1 1 + z , ⌦0 ◆ 4 s2 a2 + 0.19097 s3 a + 0.066941s4 1/8 o better than 0.4% for 0.2  ⌦0  1. m = 2.5 log10 F+constant. The distance itude, which is the value of m if the source is og10 ✓ D⇤ L Mpc ◆ know what the absolute magnitude (or luminosity) of th known distances, we don’t. In fact M and h are assume Hubble’s law to give us the distance. For the •  Take the flux of the supernova. •  Relate to the luminosity distance. 0 ⌦m (1 + z0)3 + ⌦v + (1 ⌦)(1 + z0)2 r, sinh r, depending on whether ⌦ ⌘ ⌦m + ⌦v is > 1, = 1, or < 1, an of the supernova. ⌦m , ⌦v and H0 are the density parameters (today) d the Hubble constant. It is beyond the scope of these notes to derive t al for an undergraduate cosmology course. erse (⌦ = 1), this simplifies to DL(z) = 3000h 1(1 + z) Z z 0 dz0 p ⌦m (1 + z0)3 + 1 ⌦m Mpc, m s 1 Mpc 1. To avoid evaluating integrals to calculate DL, we can use a lid for flat universes only), given by U.-L. Pen, ApJS, 120:4950, 1999: DL(z) = c H0 (1 + z)  ⌘(1, ⌦0 ) ⌘ ✓ 1 1 + z , ⌦0 ◆ = 2 p s3 + 1  1 a4 0.1540 s a3 + 0.4304 s2 a2 + 0.19097 s3 a + 0.066941s4 1/8 /⌦0 . This is claimed to be accurate to better than 0.4% for 0.2  ⌦0  1 •  This depends on cosmology. •  Convert flux to magnitudes.
56. What is a likelihood – SNe Example s y L is given by f = L 4⇡D2 L In Big Bang cosmology it is given by DL = (1 + z)c H0 p |1 ⌦| Sk(r), z 0 dz0 p ⌦m (1 + z0)3 + ⌦v + (1 ⌦)(1 + z0)2 . on whether ⌦ ⌘ ⌦m + ⌦v is > 1, = 1, or < 1, and z is the m , ⌦v and H0 are the density parameters (today) in matter, nt. It is beyond the scope of these notes to derive this, but it ate cosmology course. mplifies to z) = c H0 (1 + z)  ⌘(1, ⌦0 ) ⌘ ✓ 1 1 + z , ⌦0 ◆  1 a4 0.1540 s a3 + 0.4304 s2 a2 + 0.19097 s3 a + 0.066941s4 1/8 claimed to be accurate to better than 0.4% for 0.2  ⌦0  1. ed in magnitudes, where m = 2.5 log10 F+constant. The d M is the absolute magnitude, which is the value of m if the so n Mpc1, this is µ = 25 5 log10 h + 5 log10 ✓ D⇤ L Mpc ◆ exercise here; we assume we know what the absolute magnitude (or lumin nless we have supernovae with known distances, we don’t. In fact M a redshift supernovae where we assume Hubble’s law to give us the distance. t. U.-L. Pen, ApJS, 120:4950, 1999: ⌘ ✓ 1 1 + z , ⌦0 ◆ 4 s2 a2 + 0.19097 s3 a + 0.066941s4 1/8 o better than 0.4% for 0.2  ⌦0  1. m = 2.5 log10 F+constant. The distance itude, which is the value of m if the source is og10 ✓ D⇤ L Mpc ◆ know what the absolute magnitude (or luminosity) of th known distances, we don’t. In fact M and h are assume Hubble’s law to give us the distance. For the mula (valid for flat universes only), given by U.-L. Pen, ApJ DL(z) = c H0 (1 + z)  ⌘(1, ⌦0 ) ⌘ ✓ 1 1 + z , ⌦ ⌘(a, ⌦0 ) = 2 p s3 + 1  1 a4 0.1540 s a3 + 0.4304 s2 a2 + 0.19097 1 ⌦0 )/⌦0 . This is claimed to be accurate to better than are usually expressed in magnitudes, where m = 2.5 lo s µ = m M, where M is the absolute magnitude, which i nce 10pc. With DL in Mpc1, this is µ = 25 5 log10 h + 5 log10 ✓ D⇤ L Mpc ◆ a simplification in the exercise here; we assume we know what the a rnovae are, but in fact unless we have supernovae with known distan •  Take the flux of the supernova. •  Relate to the luminosity distance. 0 ⌦m (1 + z0)3 + ⌦v + (1 ⌦)(1 + z0)2 r, sinh r, depending on whether ⌦ ⌘ ⌦m + ⌦v is > 1, = 1, or < 1, an of the supernova. ⌦m , ⌦v and H0 are the density parameters (today) d the Hubble constant. It is beyond the scope of these notes to derive t al for an undergraduate cosmology course. erse (⌦ = 1), this simplifies to DL(z) = 3000h 1(1 + z) Z z 0 dz0 p ⌦m (1 + z0)3 + 1 ⌦m Mpc, m s 1 Mpc 1. To avoid evaluating integrals to calculate DL, we can use a lid for flat universes only), given by U.-L. Pen, ApJS, 120:4950, 1999: DL(z) = c H0 (1 + z)  ⌘(1, ⌦0 ) ⌘ ✓ 1 1 + z , ⌦0 ◆ = 2 p s3 + 1  1 a4 0.1540 s a3 + 0.4304 s2 a2 + 0.19097 s3 a + 0.066941s4 1/8 /⌦0 . This is claimed to be accurate to better than 0.4% for 0.2  ⌦0  1 •  This depends on cosmology. •  Convert flux to magnitudes. •  Observers talk in terms of the distance modulus.
57. What is a likelihood – SNe Example sely accurate), and a distance modulus µi, with associated measu SN.dat) contains one header line and there are n = 291 supernova SN z mu sigma quality SN90O 0.030 35.90 0.21 Gold SN90T 0.040 36.38 0.20 Gold SN90af 0.050 36.84 0.22 Gold . . these data on a graph. Exercise an MCMC code to estimate h and ⌦m from the supernova dat nd the errors are gaussian, i.e. assume that the likelihood is L / exp " 1 2 n X i=1 (µi µth (zi))2 2 i # s y L is given by f = L 4⇡D2 L In Big Bang cosmology it is given by DL = (1 + z)c H0 p |1 ⌦| Sk(r), z 0 dz0 p ⌦m (1 + z0)3 + ⌦v + (1 ⌦)(1 + z0)2 . on whether ⌦ ⌘ ⌦m + ⌦v is > 1, = 1, or < 1, and z is the m , ⌦v and H0 are the density parameters (today) in matter, nt. It is beyond the scope of these notes to derive this, but it ate cosmology course. mplifies to z) = c H0 (1 + z)  ⌘(1, ⌦0 ) ⌘ ✓ 1 1 + z , ⌦0 ◆  1 a4 0.1540 s a3 + 0.4304 s2 a2 + 0.19097 s3 a + 0.066941s4 1/8 claimed to be accurate to better than 0.4% for 0.2  ⌦0  1. ed in magnitudes, where m = 2.5 log10 F+constant. The d M is the absolute magnitude, which is the value of m if the so n Mpc1, this is µ = 25 5 log10 h + 5 log10 ✓ D⇤ L Mpc ◆ exercise here; we assume we know what the absolute magnitude (or lumin nless we have supernovae with known distances, we don’t. In fact M a redshift supernovae where we assume Hubble’s law to give us the distance. t. U.-L. Pen, ApJS, 120:4950, 1999: ⌘ ✓ 1 1 + z , ⌦0 ◆ 4 s2 a2 + 0.19097 s3 a + 0.066941s4 1/8 o better than 0.4% for 0.2  ⌦0  1. m = 2.5 log10 F+constant. The distance itude, which is the value of m if the source is og10 ✓ D⇤ L Mpc ◆ know what the absolute magnitude (or luminosity) of th known distances, we don’t. In fact M and h are assume Hubble’s law to give us the distance. For the mula (valid for flat universes only), given by U.-L. Pen, ApJ DL(z) = c H0 (1 + z)  ⌘(1, ⌦0 ) ⌘ ✓ 1 1 + z , ⌦ ⌘(a, ⌦0 ) = 2 p s3 + 1  1 a4 0.1540 s a3 + 0.4304 s2 a2 + 0.19097 1 ⌦0 )/⌦0 . This is claimed to be accurate to better than are usually expressed in magnitudes, where m = 2.5 lo s µ = m M, where M is the absolute magnitude, which i nce 10pc. With DL in Mpc1, this is µ = 25 5 log10 h + 5 log10 ✓ D⇤ L Mpc ◆ a simplification in the exercise here; we assume we know what the a rnovae are, but in fact unless we have supernovae with known distan •  Take the flux of the supernova. •  Relate to the luminosity distance. 0 ⌦m (1 + z0)3 + ⌦v + (1 ⌦)(1 + z0)2 r, sinh r, depending on whether ⌦ ⌘ ⌦m + ⌦v is > 1, = 1, or < 1, an of the supernova. ⌦m , ⌦v and H0 are the density parameters (today) d the Hubble constant. It is beyond the scope of these notes to derive t al for an undergraduate cosmology course. erse (⌦ = 1), this simplifies to DL(z) = 3000h 1(1 + z) Z z 0 dz0 p ⌦m (1 + z0)3 + 1 ⌦m Mpc, m s 1 Mpc 1. To avoid evaluating integrals to calculate DL, we can use a lid for flat universes only), given by U.-L. Pen, ApJS, 120:4950, 1999: DL(z) = c H0 (1 + z)  ⌘(1, ⌦0 ) ⌘ ✓ 1 1 + z , ⌦0 ◆ = 2 p s3 + 1  1 a4 0.1540 s a3 + 0.4304 s2 a2 + 0.19097 s3 a + 0.066941s4 1/8 /⌦0 . This is claimed to be accurate to better than 0.4% for 0.2  ⌦0  1 •  This depends on cosmology. •  Convert flux to magnitudes. •  Observers talk in terms of the distance modulus. •  Likelihood can then be constructed just like our toy example.
58. Marginalisation
59. Let’s understand this plot a bit more…
60. Monte Carlo Markov Chains (MCMC)
61. Monte Carlo Markov Chains (MCMC) •  Chain: A series of samples in our parameter space. Not regular like a grid.
62. Monte Carlo Markov Chains (MCMC) •  Chain: A series of samples in our parameter space. Not regular like a grid. •  Markov: Any point in the chain depends only on the immediately preceding point, not any of the other previous points. Produces a “memory-less sequence”.
63. Monte Carlo Markov Chains (MCMC) •  Chain: A series of samples in our parameter space. Not regular like a grid. •  Markov: Any point in the chain depends only on the immediately preceding point, not any of the other previous points. Produces a “memory-less sequence”. •  Monte Carlo: There is a random element in the sequence of “jumps” which produces the chain.
64. How#do#I#get#error dimensi • Read#Numerical#Recipes Monte Carlo Markov Chains (MCMC) •  Chain: A series of samples in our parameter space. Not regular like a grid. •  Markov: Any point in the chain depends only on the immediately preceding point, not any of the other previous points. Produces a “memory-less sequence”. •  Monte Carlo: There is a random element in the sequence of “jumps” which produces the chain. à Goal: Produce a series of points whose density is proportional to the local likelihood.
65. Monte Carlo Markov Chains (MCMC) Metropolis-Hastings xn the Metropolis–Hastings algorithm •  Take one point, x_n.
66. Monte Carlo Markov Chains (MCMC) Metropolis-Hastings xn Metropolis-Hastings Q ( pn | xn) the Metropolis–Hastings algorithm •  Take one point. •  Use a proposal distribution to select another point, p_n.
67. Metropolis-Hastings P ( pn) > P ( xn) Definite accept Monte Carlo Markov Chains (MCMC) the Metropolis–Hastings algorithm •  Take one point. •  Use a proposal distribution to select another point, p_n. •  If P(p_n)/P(x_n)>1 then accept the new point.
68. Metropolis-Hastings P ( pn) > P ( xn) Definite accept Metropolis-Hastings P ( pn) < P ( xn) Let ↵ = P ( pn) /P ( xn) Generate u ⇠ U (0 , 1) Accept if ↵ > u Monte Carlo Markov Chains (MCMC) the Metropolis–Hastings algorithm •  Take one point. •  Use a proposal distribution to select another point, p_n. •  If P(p_n)/P(x_n)>1 then accept the new point. •  If P(p_n)/P(x_n)<1 then sometimes accept the new point. This is where the Monte Carlo element appears.
69. dimensio • Read#Numerical#Recipes, Monte Carlo Markov Chains (MCMC) the Metropolis–Hastings algorithm
70. Back to tests of gravity… Simpson et al. 2013
71. Expansion vs. Growth of Structures Dark Energy & Modified Gravity can predict the same accelerated expansion… … but they predict different growth rates as matter collapses due to gravity.
72. Universe was 0.2 Gyr old
73. Universe was 1 Gyr old
74. Universe was 4.7 Gyrs old
75. Today (13.6 Gyr)
77. There are very many ways to modify the Einstein equations. Two simple examples: MG Theories ds = (1 + 2ψ)dt + (1 − 2φ)a (t)dx k2(φ + ψ) == 4πGeff (k, z)¯ ρMG a2δMG φ = ψη(k, z) ( d4x √ −g 1 16πGN R → d4x √ −g 1 16πGN f(R) ( 1 f(R) Simplest generalisation of the Einstein-Hilbert action Can produce late-time acceleration. Sub-class of chameleon/scalar-tensor
78. There are very many ways to modify the Einstein equations. Two simple examples: MG Theories ds = (1 + 2ψ)dt + (1 − 2φ)a (t)dx k2(φ + ψ) == 4πGeff (k, z)¯ ρMG a2δMG φ = ψη(k, z) ( d4x √ −g 1 16πGN R → d4x √ −g 1 16πGN f(R) ( 1 DGP two functions, F and U. This additional freedom allows for greater flexibil servational and theoretical constraints. However, the price we pay is additio arbitrariness. The f(R) theories have one arbitrary function, and here the U(ψ). There is no preferred choice of these functions from fundamental the Modifications of the Einstein-Hilbert action, which lead to fourth-order struggle to meet the minimum requirements in the simplest cases, or contain arbitrary choices than quintessence models in general relativity. Therefore, appears to be a serious competitor to quintessence in general relativity. B. BRANE-WORLD MODELS Modifications to general relativity within the framework of quantum grav violet corrections that must arise at high energies in the very early universe black hole. The leading candidate for a quantum gravity theory, string theory infinities of quantum field theory and unify the fundamental interactions, i there is a price – the theory is only consistent in 9 space dimensions. Brane of higher dimension than strings, and play a fundamental role in the theory on which open strings can end. Roughly speaking, the endpoints of open s the standard model particles like fermions and gauge bosons, are attached closed strings of the gravitational sector can move freely in the higher-dime time. Classically, this is realised via the localization of matter and radiatio with gravity propagating in the bulk (see Fig. 4). e− e+ γ G FIG. 4: The confinement of matter to the brane, while gravity propagates in t f(R) Simplest generalisation of the Einstein-Hilbert action Embed 3+1 spacetime in 5D bulk. Can produce late-time acceleration. Sub-class of chameleon/scalar-tensor z Crossover scale 4 4 (5) (5) 1 16 m R d x gR d x gL G S     ³ ³ c r (Dvali, Gabadadze,Porrati) braneworld model Gravity ‘leaks’ off brane, weakened at large scales z Crossover scale 4D Newtonian gravity c r r  c r 5 4 4 (5) (5) 1 1 32 16 m c S d x g R d x gR d x gL Gr G S S      ³ ³ ³ c r (Dvali, Gabadadze,Porrati) Example 2. braneworld model
79. There are very many ways to modify the Einstein equations. Two simple examples: MG Theories ds = (1 + 2ψ)dt + (1 − 2φ)a (t)dx k2(φ + ψ) == 4πGeff (k, z)¯ ρMG a2δMG φ = ψη(k, z) ( d4x √ −g 1 16πGN R → d4x √ −g 1 16πGN f(R) ( 1 DGP two functions, F and U. This additional freedom allows for greater flexibil servational and theoretical constraints. However, the price we pay is additio arbitrariness. The f(R) theories have one arbitrary function, and here the U(ψ). There is no preferred choice of these functions from fundamental the Modifications of the Einstein-Hilbert action, which lead to fourth-order struggle to meet the minimum requirements in the simplest cases, or contain arbitrary choices than quintessence models in general relativity. Therefore, appears to be a serious competitor to quintessence in general relativity. B. BRANE-WORLD MODELS Modifications to general relativity within the framework of quantum grav violet corrections that must arise at high energies in the very early universe black hole. The leading candidate for a quantum gravity theory, string theory infinities of quantum field theory and unify the fundamental interactions, i there is a price – the theory is only consistent in 9 space dimensions. Brane of higher dimension than strings, and play a fundamental role in the theory on which open strings can end. Roughly speaking, the endpoints of open s the standard model particles like fermions and gauge bosons, are attached closed strings of the gravitational sector can move freely in the higher-dime time. Classically, this is realised via the localization of matter and radiatio with gravity propagating in the bulk (see Fig. 4). e− e+ γ G FIG. 4: The confinement of matter to the brane, while gravity propagates in t f(R) Simplest generalisation of the Einstein-Hilbert action Embed 3+1 spacetime in 5D bulk. Can produce late-time acceleration. Sub-class of chameleon/scalar-tensor z Crossover scale 4 4 (5) (5) 1 16 m R d x gR d x gL G S     ³ ³ c r (Dvali, Gabadadze,Porrati) braneworld model Gravity ‘leaks’ off brane, weakened at large scales z Crossover scale 4D Newtonian gravity c r r  c r 5 4 4 (5) (5) 1 1 32 16 m c S d x g R d x gR d x gL Gr G S S      ³ ³ ³ c r (Dvali, Gabadadze,Porrati) Example 2. braneworld model Brans-Dicke TeVeS Degravita<on EBI gravity K-infla<on f(G) Horava- Lifschitz Kine<c Gravity Braiding Einsten- Aether Fab Four Covariant Galileons Massive Gravity Horndeski General Scalar-Tensor Randall- Sundrum Cascading gravity MOND Bimetric theories and more…
80. •  Pick a constant in GR à Free parameter à measure any deviation. •  Labour-saving for both theoreticians & observers. •  Can’t use expansion alone, need growth of structure. e.g. Gravity-testing Programme: Parameterisation 2 parameter formalism- weak-field, large scale bert = 1 16πG d4x √ −gR (7) f(R) = R − µ4 R (8) δ, θv , Ψ, Φ (9) + 2Ψ)dt2 + (1 − 2Φ)a2(t)dx2 (10) − 1 2 gµν R = − 8πG c4 Tµν (11) − 1 gµν R = −8πGTµν (12) 4 variables, 2 eqns from conservation of energy- momentum, 2 eqns from a theory of gravity. Einstein-Hilbert = 1 16πG d4x −gR (7) f(R) = R − µ4 R (8) δ, θv , Ψ, Φ (9) = −(1 + 2Ψ)dt2 + (1 − 2Φ)a2(t)dx2 (10) k2Ψ = −4πGa2ρ∆ (11) Φ = Ψ (12) f(R) = R − µ4 R (8) δ, θv , Ψ, Φ (9) −(1 + 2Ψ)dt2 + (1 − 2Φ)a2(t)dx2 (10) k2Ψ = −4πGa2ρ∆ (11) Φ = Ψ (12) k2Ψ = −4πµ(k, z)Ga2ρ∆ (13) eterise deviations from GR in a phenomenological manner. ct, we simply wish to ask: is the strength of gravity the same mological scales as it is here on Earth? If not, this may mod- motions of both relativistic and non-relativistic particles, but essarily in the same manner. Figure 1 illustrates why we are g a modified gravity signal on cosmological scales; it is the in which the classical Newtonian attraction is overwhelmed repulsive force associated with dark energy. he perturbed Friedmann-Robertson-Walker metric may be sed in terms of the scale factor a(t), the Newtonian poten- and the curvature potential ds2 = (1 + 2 ) dt2 a2(t) (1 2 ) d x2 . (1) each of the two gravitational potentials and , which mplicit spatial and temporal dependencies, may be deter- from the distribution of matter in the Universe. In Fourier his is given by k2 GR = 4⇡Ga2 ¯ ⇢ , (2) we have introduced the wavenumber k, the mean cosmic The metric:
81. SEinstein-Hilbert = 16πG d4x −gR (7 f(R) = R − µ4 R (8 δ, θv , Ψ, Φ (9 s2 = −(1 + 2Ψ)dt2 + (1 − 2Φ)a2(t)dx2 (10 k2Ψ = −4πµ(k, z)Ga2ρ∆ (11 Φ = γ(k, z)Ψ (12 f(R) = R − µ4 R (8) δ, θv , Ψ, Φ (9) = −(1 + 2Ψ)dt2 + (1 − 2Φ)a2(t)dx2 (10) k2Ψ = −4πµ(k, z)Ga2ρ∆ (11) Φ = γ(k, z)Ψ (12) Rµν − 1 gµν R = − 8πG 4 Tµν (13) •  Pick a constant in GR à Free parameter à measure any deviation. •  Labour-saving for both theoreticians & observers. •  Can’t use expansion alone, need growth of structure. e.g. Gravity-testing Programme: Parameterisation 2 parameter formalism- weak-field, large scale bert = 1 16πG d4x √ −gR (7) f(R) = R − µ4 R (8) δ, θv , Ψ, Φ (9) + 2Ψ)dt2 + (1 − 2Φ)a2(t)dx2 (10) − 1 2 gµν R = − 8πG c4 Tµν (11) − 1 gµν R = −8πGTµν (12) 4 variables, 2 eqns from conservation of energy- momentum, 2 eqns from a theory of gravity. In GR: δ, θv , Ψ, Φ ds2 = −(1 + 2Ψ)dt2 + (1 − 2 k2Ψ = −4πµ(k, z)Ga Φ = γ(k, z)Ψ µ(k, z) = 1 γ(k, z) = 1 1 δ, θv , Ψ, Φ ds2 = −(1 + 2Ψ)dt2 + (1 − 2Φ k2Ψ = −4πµ(k, z)Ga2ρ Φ = γ(k, z)Ψ µ(k, z) = 1 γ(k, z) = 1 1 No agreement on form of these parameters, let alone notation. eterise deviations from GR in a phenomenological manner. ct, we simply wish to ask: is the strength of gravity the same mological scales as it is here on Earth? If not, this may mod- motions of both relativistic and non-relativistic particles, but essarily in the same manner. Figure 1 illustrates why we are g a modified gravity signal on cosmological scales; it is the in which the classical Newtonian attraction is overwhelmed repulsive force associated with dark energy. he perturbed Friedmann-Robertson-Walker metric may be sed in terms of the scale factor a(t), the Newtonian poten- and the curvature potential ds2 = (1 + 2 ) dt2 a2(t) (1 2 ) d x2 . (1) each of the two gravitational potentials and , which mplicit spatial and temporal dependencies, may be deter- from the distribution of matter in the Universe. In Fourier his is given by k2 GR = 4⇡Ga2 ¯ ⇢ , (2) we have introduced the wavenumber k, the mean cosmic The metric:
82. Why is one sensitive to WL, one to RSD? tial experienced by non-relativistic particles and the lensing po- tential ( + ) experienced by relativistic particles, are now mod- ulated by the parameters ⌃ and µ (k, a) = [1 + µ(k, a)] GR (k, a) , (4) [ (k, a) + (k, a)] = [1 + ⌃(k, a)] [ GR (k, a) + GR (k, a)] , (5) where we have adopted notation similar to that of Amendola et al. (2008), as used more recently by Zhao et al. (2010) and Song et al. (2011). This parameterisation has the advantage of separat- ing the modified behaviour of non-relativistic particles, as dictated by µ(k, a), from modifications to the deflection of light, as given by ⌃(k, a). By being able to reproduce a wide range of observa- tional outcomes, in terms of the two-point weak lensing correla- tion functions and the growth of large scale structure, we ensure a (k, a) = [1 + µ(k, a)] GR (k, a) , (4) [ (k, a) + (k, a)] = [1 + ⌃(k, a)] [ GR (k, a) + GR (k, a)] , (5) where we have adopted notation similar to that of Amendola et al. (2008), as used more recently by Zhao et al. (2010) and Song et al. (2011). This parameterisation has the advantage of separat- ing the modified behaviour of non-relativistic particles, as dictated by µ(k, a), from modifications to the deflection of light, as given by ⌃(k, a). By being able to reproduce a wide range of observa- tional outcomes, in terms of the two-point weak lensing correla- tion functions and the growth of large scale structure, we ensure a broad sensitivity to different types of deviations from GR. There is some flexibility in how we choose to parameterise the scale and time dependence of these two parameters, much like the dark en- ergy equation of state w(z). Previous works have often chosen a scale independent model with a parameterised time variation of ulated by the parameters ⌃ and µ (k, a) = [1 + µ(k, a)] GR (k, a) , (4) [ (k, a) + (k, a)] = [1 + ⌃(k, a)] [ GR (k, a) + GR (k, a)] , (5) where we have adopted notation similar to that of Amendola et al. (2008), as used more recently by Zhao et al. (2010) and Song et al. (2011). This parameterisation has the advantage of separat- ing the modified behaviour of non-relativistic particles, as dictated by µ(k, a), from modifications to the deflection of light, as given by ⌃(k, a). By being able to reproduce a wide range of observa- tional outcomes, in terms of the two-point weak lensing correla- tion functions and the growth of large scale structure, we ensure a [ (k, a) + (k, a)] = [1 + ⌃(k, a)] [ GR (k, a) + GR (k, a)] , (5) where we have adopted notation similar to that of Amendola et al. 2008), as used more recently by Zhao et al. (2010) and Song et al. (2011). This parameterisation has the advantage of separat- ng the modified behaviour of non-relativistic particles, as dictated by µ(k, a), from modifications to the deflection of light, as given by ⌃(k, a). By being able to reproduce a wide range of observa- ional outcomes, in terms of the two-point weak lensing correla- ion functions and the growth of large scale structure, we ensure a broad sensitivity to different types of deviations from GR. There s some flexibility in how we choose to parameterise the scale and ime dependence of these two parameters, much like the dark en- ergy equation of state w(z). Previous works have often chosen a scale independent model with a parameterised time variation of s s Sensitive to deflection of non- relativistic particles. •  Simpson et al. parameterisation. Sensitive to deflection of light. Rela<vis<c par<cles collect equal contribu<ons from the two poten<als, since they traverse equal quan<<es of space and <me.
83. 1 Parameterizations 1, 2 Parameters 2 3 6 8 1 1 1 2 14 15 16 17 20 21 24 25 30 33 34 35 2-Parameters 1-Parameter γ, Growth Parameter 4 5 9 13 18 26 28 31 36 37 38 40 41 42 43 22 27 Other Owen et al. 2013 (in prep.)
84. Results… a work in progress CFHTLenS: Testing the Laws of Gravity 9 •  No detected deviation from GR but still early days… •  WGL tells us about expansion and growth. •  This allows us to tell the difference between dark energy and modified gravity. Simpson et al. 2013
85. Results… a work in progress CFHTLenS: Testing the Laws of Gravity 9 •  No detected deviation from GR but still early days… x15 more accurate… •  WGL tells us about expansion and growth. •  This allows us to tell the difference between dark energy and modified gravity. •  DES will give us the first percent-level measurements of gravity on cosmic scales. Simpson et al. 2013
86. SUMMARY •  We know that we don’t understand most of the stuff that the Universe is made from. •  To remedy this we need to simultaneously deploy many different cosmic probes. •  This requires a sophisticated statistical framework… the devil’s in the details. •  Was Einstein wrong? We don’t know but we should have an answer by 2020…