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Closed Universes, de Sitter Space and Inflation
Chris Doran Cavendish Laboratory
Based on astro-ph/0307311 by Lasenby and Doran


The Cosmological Constant
· Dark energy responsible for around 70% of the total energy density · But scale of far too small to be predicted by qft effects · Suggests that the origin may be geometric after all · Leads us to consider spaces of constant curvature · These have many attractive features ...


Evolution Equations with
· Start with FRW equations

· Introduce the dimensionless ratios


Evolution Equations with
· Now write · Evolution equations now

· Define trajectories via


Cosmic Trajectories
Dust =0 de Sitter phase Radiation =1/3

Initial singularity


Cosmic Trajectories
· For a range of initial conditions, models re-focus around the flat case · For decoupled matter and radiation can solve the equations exactly · Solutions governed by two constants:


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, there exists a unique line m in the same plane as P and l which passes through P and does not meet l

· For many years mathematicians attempted to prove this from the remaining postulates · Non-Euclidean geometry arises by removing the uniqueness requirement


Non-Euclidean Geometry
· Replace a unique line by an infinite number of lines · Developed by Lobachevskii (17921856) 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 PoincarÈ (1854-1912)


PoincarÈ Disc
· Points contained in a disc of unit radius · Boundary of the disk represents set of points at infinity · Lines (geodesics) are represented by circles that intersect the unit circle at right angles · All geodesics through the origin are straight lines (in Euclidean sense) · Angles are faithfully represented


PoincarÈ Disc
Set of lines through A which miss l Disc

l

Right-angle intersection

Plot constructed in Cinderella


Distance
· The metric in the disc representation is

· This is a conformal representation ­ only differs from flat by a single factor · Distortions from flat get greater as you move away from the centre · A feature of the representation · Can define tesselations


Circle Limit 3 by M. Escher

d lines


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 · Spatially closed, but exists for infinite time


Embedding View

time

null geodesic ­ straight line in embedding space

space


Lorentzian View
Seek an equivalent construction to the PoincarÈ disk

Circle mapped onto a line via a stereographic projection. Extend out assuming null trajectories are at 45o


Lorentzian View
t `Perpendicular' intersection Timelike geodesic Boundary x Hyperbolae Spacelike geodesic Null cone Always at 45

o


Lorentzian View
· All null vectors are straight lines at 45o · Geodesics are hyperbolae perpendicular to the boundary · Can be represented as 3-vectors in a 4dimensional algebra · A similar view exists for adS space · Metric has the conformal structure


The de Sitter Phase
· Universe tends towards a final de Sitter phase · Universe should really be closed for pure de Sitter · Only get a natural symmetric embedding onto entire de Sitter topology if the cosmic singularity lies at t=0 · Says that a photon gets ¼ of the way across the universe


A Cosmic Boundary Condition
· Derive a boundary condition by insisting that

· A total conformal time of /2 · Can express this in terms of a differential equation for the conformal line element · Equivalent to demanding a given fall-off as


A Preferred Model
Current Observations Dust Arrive at a model quite close to observation For dust (=0) predict a universe closed at about the 10% level

Critical


Inflation
· de Sitter model with boundary condition gets close to observed values · But all the standard problems remain · Still require an inflationary phase · But we need to be able to locate the initial singularity to apply the boundary condition · Can we do this for an inflationary model? · Start with the simplest case


Inflation
· Assume a quadratic potential

· Equations of motion are

· Dimensionless, with


Scale Invariance
· Solutions can be scaled to define new solutions via (constant ) · Many quantities are invariant under this, including conformal time · Take advantage of this in numerical computation · Scaling property does not survive quantisation ­ significant for vacuum fluctuations


Series Expansion
· Want to expand around initial singularity · Find that we need a power series in u and ln(u)

· By specifying H0, 0 and 1, all remaining terms determined recursively · Need a series in u1/3 to generate curvature


Model Parameters
· Expansion around singularity governed by two constants:

· Under scaling these go as


Curvature
· The constant b4 controls the curvature

· Total volume of the Universe goes as

· Total energy contained in the (closed) Universe is


Entry Into inflationary Region

H emerges from singularity as 1/3t

Inflationary Region


Hubble Function Exit
Effective big bang

Inflationary regime

Dust cosmology


Evolution of Conformal Time
Saturation as R increases rapidly Horizon problem removed

Any future increase in takes a very long time


Slow Roll Estimates
· Can make reasonable estimates for start and end of inflation:

· Find total number of e-foldings: · Total elapsed conformal time:


The Cosmic See-saw
· Increased initial curvature increased b4 greater elapsed universe today closer to flat · Can also estimate by matching onto matter

· Predicts a suitably small !


Evolution of
Growth from big bang Inflationary region

Universe driven close to flat


The Hubble Horizon
· Closed universes have a natural distance scale ­ useful in calculations! · Suppose that at current epoch a scale occupies x of the Hubble horizon · Equal to Hubble horizon at R when

· So can write


The Curvature Spectrum
· Follow usual slow roll approximation
Evaluated at x


The Curvature Spectrum
· Combine the two preceding graphs:
Numerical prediction (solid)

Power law with exponential cut-off


Power Law
· During slow roll, find that
Best fit power law

· Different from the standard power law relation · Gives a running spectral index

WMAP best fit


CMB Prediction I

=1.02 using WMAP best fit parameters


CMB Prediction II

Better fit with =1.04 and lower H0


Discussion I
· Based on astro-ph/0307311 · de Sitter geometry is a natural extension of non-Euclidean geometry · Has a straightforward construction in a Lorentzian space · Appears to pick out a preferred class of cosmological models · But the boundary condition appears to be acausal (but fits data) · No problem constructing closed inflationary models


Discussion II
· Get 50-60 e-foldings, consistent with other estimates · No problem with trans-Planckian effects · Find total energy · Universe behaves classically ­ predictions robust to quantum gravity effects · Not said anything about reheating · Need improved numerics for inflationary era and parameter estimation