Post on 03-Feb-2016
description
Has elasticity anything to do with cosmology?
Angelo TartagliaRELGRAV
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
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
February 17 2011
RELGRAV 4
The strain is described by the differential change of u
N
nNa
,....,1
,....,1
February 17 2011
RELGRAV 5
Metricity
dddd
XXdXdXdl
ba
abba
ab
2
dxdxdxdxg
dxdxxx
XXdXdXdl
ba
abba
ab
2
'''''2
What about space-time?
February 17 2011
6RELGRAV
Space-time/Matter-energy
TG What is this?
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
A four-dimensional manifold
February 17 2011
RELGRAV 8
Minkowski (flat) space-time
dxdxdzdydxdtcds 222222
General (curved) space-time
dxdx2dxdxgds2
February 17 2011
RELGRAV 9
Defects in continua
Flat reference frame
Curved natural frame
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
February 17 2011
RELGRAV 11
Strained space-time
2
g
dxdxgdsnat
2
dr0
dr
unstrained
strained
g
Strain tensor
dxdxdsref
2
12
The “elastic” approach
C
Elastic modulus tensor
Stress tensor
Hooke’s law
February 17 2011
RELGRAV
13
Isotropic medium
C
Lamé coefficients
2
February 17 2011
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Elastic energy
2
21
21
V
February 17 2011
RELGRAV 14
The Lagrangian density
xdg221
RS 4matter
2 L
“Kinetic” term Potential term
Geometry
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
Has the universe a R-W symmetry?
February 17 2011
RELGRAV 16
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)
18
The distance modulus of bright objects
z
0 'zH'dz
z1log525Mm
Observed magnitude
Absolute magnitudeHubble parameter
Distances in Mpc
February 17 2011
RELGRAV
February 17 2011
RELGRAV 191919
Fitting the data (307 SnIa)
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.
Nucleosynthesis
40
22
0
16
zBB
cH ra
In the early stages the universe is radiation-dominated
4009
80
aB ra XBoost
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)
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
February 17 2011
RELGRAV 24
Bayesian posterior probability
February 17 2011
RELGRAV 25
Optimal value of the parameters
2521
a
3270m
252
m100600120B
mkg10150452
m10080282B
0
..
/..
..
N. Radicella, M. Sereno, A. Tartaglia
February 17 2011
RELGRAV 26
Schwarzschild symmetry
22222222 dsinrdrhdrfdds
2222222
22 dwdwdrdrdw
dds
sin
Natural frame
Reference frame (Minkowski)
Gauge function
February 17 2011
RELGRAV 27
The strain tensor
222
22
2
00
2
2
2
21
sinrw
rw
h'w
f
rr
drdw
w'
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
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
February 17 2011
RELGRAV 30
Approximate solutions
2
200
21
1
21
r
rm
hg
rrm
fg
rr
, = functions of , ~ ,
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
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