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8/7/2019 Geometry & Cosmology--Doran & Lasenby
http://slidepdf.com/reader/full/geometry-cosmology-doran-lasenby 1/17
Geometry and CosmologyChris Doran
Anthony Lasenby
www.mrao.cam.ac.uk/~clifford
8/7/2019 Geometry & Cosmology--Doran & Lasenby
http://slidepdf.com/reader/full/geometry-cosmology-doran-lasenby 2/17
Geometry and Cosmology
Non-Euclidean Geometry
• Axioms of geometry date back to
Euclid’s Elements• Among these is the parallel postulate:
– Given a line l and a point P not on l , thereexists a unique line m in the same plane as
P and l which passes through P and does
not meet l
• Non-Euclidean geometry arises by
removing the uniqueness requirement
8/7/2019 Geometry & Cosmology--Doran & Lasenby
http://slidepdf.com/reader/full/geometry-cosmology-doran-lasenby 3/17
Geometry and Cosmology
Non-Euclidean Geometry
• Developed by Lobachevskii (1792-
1856) and Bolyai (1802-1860)• In modern terminology this defines
hyperbolic geometry• A homogeneous, isotropic, unbounded
space of constant negative curvature
• An elegant view of this geometry was
constructed by Poincaré (1854-1912)
8/7/2019 Geometry & Cosmology--Doran & Lasenby
http://slidepdf.com/reader/full/geometry-cosmology-doran-lasenby 4/17
Geometry and Cosmology
Poincaré Disc
• Points contained in a disc of unit radius
• Boundary of the disk represents set of points at infinity
• Lines (geodesics) are represented bycircles which intersect the unit circle at
right angles
• All geodesics through the origin are
straight lines (in Euclidean sense)
8/7/2019 Geometry & Cosmology--Doran & Lasenby
http://slidepdf.com/reader/full/geometry-cosmology-doran-lasenby 5/17
Geometry and Cosmology
Poincaré Disc
Disc
Right-angle
intersection
Set of lines
through A
which miss l
l
Plot constructed
in Cinderella
8/7/2019 Geometry & Cosmology--Doran & Lasenby
http://slidepdf.com/reader/full/geometry-cosmology-doran-lasenby 6/17
Geometry and Cosmology
Distance
• The metric in the disc representation is
• This is a conformal representation –only differs from flat by a single factor
• Distortions get greater as you moveaway from the centre
• Can define tesselations
8/7/2019 Geometry & Cosmology--Doran & Lasenby
http://slidepdf.com/reader/full/geometry-cosmology-doran-lasenby 7/17
Geometry and Cosmology
Circle Limit 3M. Escher
8/7/2019 Geometry & Cosmology--Doran & Lasenby
http://slidepdf.com/reader/full/geometry-cosmology-doran-lasenby 8/17
Geometry and Cosmology
de Sitter Space
• Now suppose that the underlying
signature is Lorentzian• Construct a homogeneous, isotropic
space of constant negative curvature• This is de Sitter space
• 2D version from embedding picture
8/7/2019 Geometry & Cosmology--Doran & Lasenby
http://slidepdf.com/reader/full/geometry-cosmology-doran-lasenby 9/17
Geometry and Cosmology
Embedding View
time
space
null geodesic –
straight line in
embedding space
8/7/2019 Geometry & Cosmology--Doran & Lasenby
http://slidepdf.com/reader/full/geometry-cosmology-doran-lasenby 10/17
Geometry and Cosmology
Lorentzian View
Circle mapped onto a line via a
stereographic projection.
Extend out assuming null
trajectories are at 45o
8/7/2019 Geometry & Cosmology--Doran & Lasenby
http://slidepdf.com/reader/full/geometry-cosmology-doran-lasenby 11/17
Geometry and Cosmology
Lorentzian View
Boundary
Timelike geodesic
Spacelike
geodesic Null cone
Always at 45o
Hyperbolae
‘Perpendicular’
intersection
8/7/2019 Geometry & Cosmology--Doran & Lasenby
http://slidepdf.com/reader/full/geometry-cosmology-doran-lasenby 12/17
Geometry and Cosmology
The Cosmological Constant
• Start with FRW equations
• Introduce the dimensionless ratios
8/7/2019 Geometry & Cosmology--Doran & Lasenby
http://slidepdf.com/reader/full/geometry-cosmology-doran-lasenby 13/17
Geometry and Cosmology
The Cosmological Constant
• Write
• Evolution equations now
• Define trajectories via
8/7/2019 Geometry & Cosmology--Doran & Lasenby
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Geometry and Cosmology
Cosmic Trajectories
Dust Radiation
Big Bang
de Sitter phase
8/7/2019 Geometry & Cosmology--Doran & Lasenby
http://slidepdf.com/reader/full/geometry-cosmology-doran-lasenby 15/17
Geometry and Cosmology
The de Sitter Phase
• End of the universe enters a de Sitter
phase• Should really be closed for pure de Sitter
• Only get a natural symmetric embeddingonto entire de Sitter topology if
• Says that a photon gets ¼ of the way
across the universe
8/7/2019 Geometry & Cosmology--Doran & Lasenby
http://slidepdf.com/reader/full/geometry-cosmology-doran-lasenby 16/17
Geometry and Cosmology
A Preferred Model
Critical
Dust
Current Observations
Arrive at a model
quite close toobservation
For dust (η =0) predicta universe closed at
about the 10% level
8/7/2019 Geometry & Cosmology--Doran & Lasenby
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Geometry and Cosmology
Summary
• de Sitter geometry is a natural
extension of non-Euclidean geometry• Has a straightforward construction in a
Lorentzian space
• Can form a background space for agauge theory of gravity
• Appears to pick out a preferredcosmological model
• But is this causal?