Has elasticity anything to do with cosmology?

Angelo TartagliaRELGRAV

February 17 2011

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“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

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The strain is described by the differential change of u

N

nNa

,....,1

,....,1

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Metricity

dddd

XXdXdXdl

ba

abba

ab

2

dxdxdxdxg

dxdxxx

XXdXdXdl

ba

abba

ab

2

'''''2

What about space-time?

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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?

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A four-dimensional manifold

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Minkowski (flat) space-time

dxdxdzdydxdtcds 222222

General (curved) space-time

dxdx2dxdxgds2

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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)

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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

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13

Isotropic medium

C

Lamé coefficients

2

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Elastic energy

2

21

21

V

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The Lagrangian density

xdg221

RS 4matter

2 L

“Kinetic” term Potential term

Geometry

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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?

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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

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Fitting the data (307 SnIa)

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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

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Bayesian posterior probability

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Optimal value of the parameters

2521

a

3270m

252

m100600120B

mkg10150452

m10080282B

0

..

/..

..

N. Radicella, M. Sereno, A. Tartaglia

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Schwarzschild symmetry

22222222 dsinrdrhdrfdds

2222222

22 dwdwdrdrdw

dds

sin

Natural frame

Reference frame (Minkowski)

Gauge function

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The strain tensor

222

22

2

00

2

2

2

21

sinrw

rw

h'w

f

rr

drdw

w'

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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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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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Approximate solutions

2

200

21

1

21

r

rm

hg

rrm

fg

rr

, = functions of , ~ ,

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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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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