has elasticity anything to do with cosmology?
DESCRIPTION
Has elasticity anything to do with cosmology?. Angelo Tartaglia RELGRAV. r. r’. X a. “Elastic” continua. N+n. N. ξ a. x μ. N. Geometry and elasticity. In a strained medium each point is in one to one correspondence with points in the unstrained state. - PowerPoint PPT PresentationTRANSCRIPT
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Has elasticity anything to do with cosmology?
Angelo TartagliaRELGRAV
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February 17 2011
RELGRAV 2
“Elastic” continua
u
xμ
Xa
ξa
r
r’
'u
021 nNX,...,X,Xf
0X,...,X,Xh nN21
N+nN
N
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3
Geometry and elasticity
In a strained medium each point is in one to one correspondence with points in the unstrained
state
XuX'rXr
u, r and r’ are (N+n)-vectors in the flat embedding space
February 17 2011
RELGRAV
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February 17 2011
RELGRAV 4
The strain is described by the differential change of u
N
nNa
,....,1
,....,1
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February 17 2011
RELGRAV 5
Metricity
dddd
XXdXdXdl
ba
abba
ab
2
dxdxdxdxg
dxdxxx
XXdXdXdl
ba
abba
ab
2
'''''2
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What about space-time?
February 17 2011
6RELGRAV
Space-time/Matter-energy
TG What is this?
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7
• Is it a mathematical artifact to describe the gravitational field and the global properties of the universe?
• Is it something real endowed with physical properties?
What is space-time?
February 17 2011
RELGRAV
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A four-dimensional manifold
February 17 2011
RELGRAV 8
Minkowski (flat) space-time
dxdxdzdydxdtcds 222222
General (curved) space-time
dxdx2dxdxgds2
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February 17 2011
RELGRAV 9
Defects in continua
Flat reference frame
Curved natural frame
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10
What consequences from a defect?
• The defect fixes the global symmetry
• A spontaneous strain tensor εμν (or displacement vector field ua) appears
• All this must show up in the Lagrangian of the strained manifold (space-time)
February 17 2011
RELGRAV
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February 17 2011
RELGRAV 11
Strained space-time
2
g
dxdxgdsnat
2
dr0
dr
unstrained
strained
g
Strain tensor
dxdxdsref
2
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12
The “elastic” approach
C
Elastic modulus tensor
Stress tensor
Hooke’s law
February 17 2011
RELGRAV
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13
Isotropic medium
C
Lamé coefficients
2
February 17 2011
RELGRAV
Elastic energy
2
21
21
V
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February 17 2011
RELGRAV 14
The Lagrangian density
xdg221
RS 4matter
2 L
“Kinetic” term Potential term
Geometry
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February 17 2011
RELGRAV 15
Robertson-Walker symmetry
222222 dldbdldrdsref
2222 dladdsnat
g21
z
r
r l
l
=f(r)
21 2
00
b
21 2a
ii
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Has the universe a R-W symmetry?
February 17 2011
RELGRAV 16
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February 17 2011
RELGRAV 17
The Hubble parameter from the Einstein equations
21
0r0m3
2
20
2
z1z1B3
8
a
z113
16B
caa
H
/
2
24
162
BGc
A. Tartaglia and N. Radicella, CQG, 27, 035001 (2010)
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18
The distance modulus of bright objects
z
0 'zH'dz
z1log525Mm
Observed magnitude
Absolute magnitudeHubble parameter
Distances in Mpc
February 17 2011
RELGRAV
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February 17 2011
RELGRAV 191919
Fitting the data (307 SnIa)
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February 17 2011
RELGRAV 20
Other cosmological tests
• Primordial nucleosynthesis (correct proportion between He, D and hydrogen)
• CMB acoustic horizon• Structure formation after the
recombination era.
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Nucleosynthesis
40
22
0
16
zBB
cH ra
In the early stages the universe is radiation-dominated
4009
80
aB ra XBoost
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Large Scale Stractures
Particle horizon at the equality epoch (z=3150)
Constraint from LSS:
Boost
truemapparm
X
hh 0
0
Ωm0 : matter density in units of GH
cr
83 2
0
h: Hubble constant in units of km/(sMpc)
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Acoustic scale of the CMB
The power spectrum of the CMB depends on the expansion rate of the universe
lss
lsAlsA zr
zDzl 1
zls 1090 last scattering
DA:angular diameter distance
rs:sound horizon
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February 17 2011
RELGRAV 24
Bayesian posterior probability
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February 17 2011
RELGRAV 25
Optimal value of the parameters
2521
a
3270m
252
m100600120B
mkg10150452
m10080282B
0
..
/..
..
N. Radicella, M. Sereno, A. Tartaglia
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February 17 2011
RELGRAV 26
Schwarzschild symmetry
22222222 dsinrdrhdrfdds
2222222
22 dwdwdrdrdw
dds
sin
Natural frame
Reference frame (Minkowski)
Gauge function
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February 17 2011
RELGRAV 27
The strain tensor
222
22
2
00
2
2
2
21
sinrw
rw
h'w
f
rr
drdw
w'
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February 17 2011
RELGRAV 28
Three field equations
02432428
2434216
1
222
222422
4
422222
2
2
22
22
2
2
2
2
'hwfrw
hf'wfhrw
hfhffhhf
r
'fwrw
fhfhh'fwhfhrw
fhhf
rh'h
rh
02432248
2432416
222
222422
4
422222
2
2
22
22
2
2
2
22
'hwfrw
hf'wffhrw
hfhfhfr
'fwrw
fhfhh'fwrw
fhfhhfr
rf'f
hhh
0122222
323
1244
4242
143
4243
2
2
23
22
22
3
2
2
2222
22
2
32
2222222
wrw
'wf'f
h'hr
hr
'wf'fr
h'h
rrh
''wh'wfhr
wfr
w'wr
fr
h'h
frw
rf'f
fr
rw
h
'wrh'h
fr'f
hr
'wwh
''wfhwfhr'wfrhrfh
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February 17 2011
RELGRAV 29
Weak strain
rm
hf
www
hhh
fff
211
00
10
10
10
1r,rr
w,h,f;1
rm 221
11
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February 17 2011
RELGRAV 30
Approximate solutions
2
200
21
1
21
r
rm
hg
rrm
fg
rr
, = functions of , ~ ,
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February 17 2011
RELGRAV 31
Post-Keplerian circular orbits
Looks like the effect of dark matter
Light rays
23223
23
23
2
22
1637232c
RM
GcRM
G
2
622
2
22
br
rM
21brrM
21br
ddr
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February 17 2011
RELGRAV 32
Conclusion
• The strained space-time theory introduces a strain energy of vacuum depending on curvature
• The idea of a cosmic defect explains why the symmetry of the universe should be R-W (or anything else)
• The theory accounts for the accelerated expansion of the universe and is consistent with BBN, structure formation, acoustic scale of the CMB and SnIa’s luminosity