GUSTAVO NIZ University of Nottingham - C
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e r i a l
Imp ge
C o l l e
. ' 0 9
Oct Can M-theory
resolve the Big
Crunch/Big Bang
singularity?
GUSTAVO NIZ
University of Nottingham
In collaboration with N. Turok and E. Copeland
© www.phidelity.com
Plan Why a bang? A simple M-theory model of the BC/BB String description Classical propagation Quantum analysis Conclusions
Observational
evidence
Hubble's law
Figure:
Freedman et al
(2000)
Galaxies moving away from us, with
Velocity=H*distance
The universe is expanding! Hot and dense past!!!
Observational
evidence
As a consequence: there should be “glow” from this hotter
epoch, with a black body radiation profile of a few Kelvin in
temperature... (Gamow; Alpher and Herman)
Discovered by
Penzias and Wilson
in 1969
COBE team
Observational
evidence
Highly homogeneous map WMAP
fluctuations of ~ 10-5 around T = 2.7 K
Many other observations support the idea of a Hot Big Bang
(abundances of light elements, age of stars, etc.)
Theoretical
motivation
A simple model (FRWL)
works well...
But what happens when
One hits the “initial” singularity!
Is this generic?
Or is it a measure zero in the phase space of solutions?
(cf Oppenheimer-Snyder solution)Theoretical
motivation
Singularity theorems
Initial data (assuming some energy conditions) can lead,
unavoidable, to geodesically incomplete space-times.
Penrose, Hawking (60-70's)
Global statement.
What about the analytical structure of fields near the
singularity?Theoretical
motivation
Belinskii, Khalatnikov and Lifshitz (BKL), 1969
Assumed ultralocality :
spatial gradients are not as important as time derivatives!
System reduces to 1d, but may have strong
dependence on the initial conditions!
Chaos
Big Bang (e.g. Mixmaster) Misner '69Theoretical
motivation
All depends on matter content:
*scalar fields tend to remove chaos Belinskii and
Khalatnikov '73
*gauge fields (p-forms) may restore it
Cosmological billiards
Hamiltonian: Damour et al '03
Near t=0,Theoretical
motivation
Away from
walls:
Kasner
metricTheoretical
motivation
Away from
walls:
Milne
Universe
Kasner p1=1
metric Pi=0 (i≠1)The big bang
singularity
What is the nature of the big bang singularity?
Is this singularity the beginning of space and time?
Or was there a pre-big bang phase in our Universe?
Cyclic / Ekpyrotic
e.g. model
Khoury, Ovrut, Steinhardt and Turok, 2001.
Steinhardt and Turok, 2002.
In these models one needs to explain how
information maps through the bounce/singularityM-theory
model
(In Heterotic M-theory) A big crunch/big bang transition
can be modeled using an orbifold collision
Horava-Witten
G
N
Lukas et al
A
B
Hull & Townsend
Khoury et al.
Near the singularity, an effective field theory (in d≥4) should
break down, because massive modes will get excited!M-theory
model
Consider M2
excitations
Perry, Steinhardt and Turok, 2004
Berman and Perry, 2006
Near t=0 (i.e. small orbifold separation),
there are two decoupled modes:
10d PICTURE
(orbifold)
● Winding membranes Strings Perturbative
(IIA, Het.)
Light Gravity
● KK (bulk) modes “Black Holes”
(D0's in IIA)
HeavyM-theory
model
Near the singularity the effective metric is:
Compactified 2d “Milne” Universe
X
9d flat spacetime.
Orbifold Rapidity
in case of [0, π]M-theory
model
Near the singularity the effective metric is:
Maybe,
Compactified 2d “Milne” the
Universe
X simplest
singularity
9d flat spacetime.
Orbifold Rapidity
in case of [0, π]M-theory
model
Winding membranes (strings): Nambu-Goto action
11d Milne
Winding membranes field independent of:
Efective TensionDyNamics
Two equivalent descriptions
1 2
● String living on flat ● String living on FRWL
spacetime
● Tension: ● Fixed tension:
● Tensionless at t=0 ● Speed of light at t=0
● Like harmonic oscillator ● Better to study classical
with a time-dependent behaviour
frequency (quantum)DyNamics
Bare in mind...
Quantum corrections:
● String interactions are suppressed ( )
● a'-corrections are under control GN & Turok.
Small perturbations (ripples on the orbifolds) lead to 11d
Kasner backgrounds:Classical
evolution
Solutions are regular across the singularity if
Tolley 2006
and can be described by different series expansions:
GN & Turok 2006
M2
-ts -tx t=0 tx ts
11d 11dClassical
evolution
GN & String breaks into bits!!
Turok t 0
2006
Like ultra-locality
in BKL analysis
Consider Hamiltonian for winding membrane
Arbitrary Interaction term,
function coupling ~
|t|>>1 expansion
|t|String
Spectrum
IN state OUT state
t=-ts t=0 t=+ts
Classical evolution evolution of Heisenberg's
operator to leading order in
IN state: Eqns. of motion are asymptotically like
strings in Minkowski space:String
Spectrum
Left/right-mover decomposition:
d-1
Constraint curves in S
Kibble & Turok, 1982
String's massless bosonic sector:
spin: 0 2 1String
Spectrum
Left/right-mover decomposition:
Will quantise
the dilaton
Constraint mode curves
only! in S d-1
Kibble & Turok, 1982
String's massless bosonic sector:
spin: 0 2 1String
Spectrum
GN & Turok 2006
Ang.
0 maximum in between Mom.Rotor (classically)
Rotor (classically)
Rotor (classically)
Rotor (classically)
loop
quantization
Copeland, GN & Turok.
In the case of circular loops (with no CM momentum),
the Hamiltonian constraint is:
R(t )
String wave function
c.f. Harmonic oscillatorString wave function
Particle
production
Hermite polynomials
for large |t|:
So can send IN vacuum (positive frequency mode) and read
the OUT state in terms harmonic oscillator states.
Particle production:
Bogoliubov coeffs.Particle
production
R
R
R
R
R
R
Finite Particle Production ( decays exp. with n )
Independent of orbifold rapidity if small CM momentumParticle
production
R
R
R
Exponential
decay
Powerlaw
growth
R
R
R
This is for
(~10% speed of light)Particle
production
R
R
R
Exponential
decay
Powerlaw
growth
R
R
R
For a gas of cosmic strings
roughly Albrech & Turok,
Scherrer & Pressremarks
R
R
R
R
R
Unitarity is preserved if:conclusions
● Classical string (winding membranes) are regular across a
Kasner 11d singularity, provided
– String travels at the speed of light at t=0
– Higher oscillation modes “eat” the divergences
– Ultralocal behaviour (string “breaks” into bits)
● Circular loops can be quantised and there is finite particle
production
● Unitarity is preserved
● What about quantising non-circular loops?
● Fermions? Non-winding modes?
● Backreaction and chaos?You can also read
