Modern Cosmology (II)fma.if.usp.br/~abramo/semidiv/Mini-curso-cosmologia-2.pdf · 2016-02-04 ·...

30
Modern Cosmology (II) Raul Abramo Physics Institute, Univ. of São Paulo [email protected] Outline II: Review of General Relativity Einstein’s equations Kinematics & dynamics of FLRW spacetimes III: Cosmological distances Cosmography IV: Observational Cosmology Dark matter, dark energy and the Big Questions

Transcript of Modern Cosmology (II)fma.if.usp.br/~abramo/semidiv/Mini-curso-cosmologia-2.pdf · 2016-02-04 ·...

Modern Cosmology (II)Raul Abramo

Physics Institute, Univ. of São [email protected]

Outline II:

• Review of General Relativity• Einstein’s equations• Kinematics & dynamics of FLRW spacetimes III:

• Cosmological distances• Cosmography IV:

• Observational Cosmology• Dark matter, dark energy and the Big Questions

Relativity

Special Relativity: the fundamental object is the light cone

Bibliography:H. Lorentz, A. Einstein, H. Minkowski & H. Weyl, “The Principle of Relativity”S. Weinberg, “Gravitation and Cosmology”

space

space

time

past light cone

future light cone

t=const hypersurface

(“present”)

Light cone:

ds2 = �c2 dt2 + d�x2 ) 0

ct

0 = �(ct� v

c

x)

x

0 = �(x� v

c

ct)

Lorentz transformation:

) ds02 = ds2 !!!

4-D distance invariant!

� =1p

1� v2/c2

Einstein’s equivalence principle rehabilitates accelerated observers➠ covariance under general coordinate transformations

time

space

Stationary observer in gravitational field

Acceleratedobserver (free fall)

=inertial observer

ds2 = �c2 dt2 + d�x2

) ds

2 = gµ� dxµdx

metric: geometry of spacetime

General Relativity is based on the equivalence principle!

In these theories, the metric of spacetime (i.e., its geometry) has a dual role: it both causes the motions of bodies... and it is affected by them.

• What is the geometry of spacetime?• What causes it?• How can we make measurements to test our theories?

Cosmology: the simplest possible geometry!

We believe that the Universe is, to a very good approximation, homogeneous and isotropic.

Homogeneity: space has the same properties at all points

Homogeneity without isotropy

Isotropy: space looks the same in all directions

Isotropy without homogeneity

Cosmological principle: • the universe is about the same (density, temperature, etc.) everywhere• it looks the same in all directions

Now, with Mathematics!

Curvature!

Georg Bernard Riemann, in 1854, proposed in his Ph.D. thesis a global view of Geometry as a study of manifolds of any number of dimensions, in any kind of space.

These geometries are essentially non-Euclidean: the distance between two points is given in terms of a metric, which can itself be an arbitrary, differentiable function.

ds

2 = gµ�dxµdx

The metric has a dual role:

i) it can be used to measure the invariant distances between any two points; and

ii) it determines (through the affine connections) how

to transport geometrical data along any smooth path on the manifold - e.g.:

d Vµ

d�= ��

µ⇥d x⇥

d�V� �Vµ =

1

2R

�⇥µ⇤V�

Idx

µx

Curved

Flat!

Curved

Some examples:

,

c=1

However, if space is curved, the derivatives of the connections cannot be made to vanish...

Hence, curvature cannot be “gauged away”

The freedom to choose coordinates means that, on any given point, we can always use the “Einstein elevator” and go to a system where the metric is locally Minkowski, and the connections vanish:

gµ� ! �µ� ��µ⇥ ! 0

@⇥ ��µ⇤ 6= 0 !

R�⇥µ⇤ =

���⇥µ

�x⇤�

���⇥⇤

�xµ+ ��

⌅⇤�⌅⇥µ � ��

⌅µ�⌅⇥⇤

��µ⇤ =

1

2g�⇥

✓� g⇥µ�x⇤

+� g⇥⇤�xµ

� � gµ⇤�x⇥

◆DV �

Dx⇥= V �

;⇥ =� V �

� x⇥+ ��

µ⇥Vµ

DV�

Dx⇥= V� ; ⇥ =

� V�

� x⇥� �µ

�⇥Vµ

Newtonian limitSmall velocities...

Geodesic equation

Suppose we are given a space with some metric. What defines a freely falling (“inertial”) observer at any point in that space?

➟ Acceleration over paths that go through that point should vanish:

D2X↵

D�2= 0 ) d2 X�

d�2+ ��

µ⇥dXµ

d�

dX⇥

d�= 0 X↵(�)

The geodesic equation determines both the spatial coordinates and the time coordinate of the inertial observer.➟ “Proper time” is the X0 = τ along a geodesic!

d2x�

d�2+ ��

µ⇥dxµ

d�

dx⇥

d�⇡ d2x�

d�2+ ��

00dt

d�

dt

d�= 0

Einstein’s Equations

Matter and gravity must be locked into a self-consistent dynamics

➟ Fundamental symmetries imply conservation laws (Noether’s theorem)

Matter curves space, determines metric...

... metric determines the kinematics of

matter...

Matter and metric jointly determine the dynamics

Symmetries and conservation laws

Invariance under time translations/reparametrizations ➟ Energy conservation

Invariance under spatial translations/reparametrizations ➟ Momentum conservation

Invariance under spatial rotations ➟ Angular momentum conservation

But what about boosts (t-x rotations)? They are a symmetry as well...Moreover, they mix energy and momentum! Pµ = mUµ , P 0µ =

�x0µ

�x�P�

Energy conservation for classical point particle (non-relativ.):

S =

ZdtL(q, q) , L(q, q) =

1

2q2 � V (q)

t ! t+ �t

q ! q + q �t

q ! q + q�t+ q�t

�S =

Zdt

⇥L

⇥q�q +

⇥L

⇥q�q

�=

Zdt

q(q �t+ q �t)� dV

dqq �t

Energy = conserved “charge”

t → t + δt : “global” symmetry�Sfi =

⇥q2�t

⇤fi�

Z f

idt

d

dt

1

2q2 + V (q)

��t = 0

But since:

The dynamics of matter should be independent of the coordinate system, and therefore the matter action should remain invariant under a coordinate transformation:

V µ;µ =

1⇤�g

�µ�⇤

�g V µ�

⇥Z

Vd4x

⇤�g V µ

;µ =�⇤

�g V µNµ

�Nµ:S(V)

Sm =

Zd

4x

⇥�gLm

�⇤Sm =

Zd4x

⇥⇥�gLm

⇥gµ⇥�gµ⇥ +

⇥⇥�gLm

⇥(⇥�gµ⇥)⇥��g

µ⇥

�⇥Sm =

Zd4x

p�g

2Tµ�(⇥

µ;� + ⇥�;µ) =

Zd4x

p�g Tµ�⇥µ;�

z }| { z }| {

Tµ��µ;� = (Tµ��µ);� � Tµ�;��µ

Why is it safe to assume that this vanishes??...

Therefore, we get the conservation law: Tµ�;� = 0

Tμν :• Energy• Momentum• Stresses/energy flows

=1p�g

⇥��p

�g Tµ��µ�� Tµ�

;��µ

�⇥Sm =

Zd4x

⇥⇤�

�p�g Tµ�⇥µ

�� Tµ�

;�⇥µ⇤! 0

The energy-momentum tensor (or stress-energy tensor)

In general, it turns out to be more instructive to construct the EMT from first principles.

For a continuous media, the relevant quantities are: the 4-velocity, the energy density, the isotropic pressure, and the shear stress.Consider a fluid element:

Tii - pressure p:forces normal to surface

T0i - energy flow:flux of energy/momentum

across surface i

T00:energy density

Tij - shear stress σij:forces parallel to surface

⇢• Symm., traceless part: shear

• Anti-symm. part: anisotropic stress

Uμ - 4-velocity:displacement of

the fluid element

In Minkowsky spacetime a fluid at rest, without any stresses, is given completely in terms of its energy density and pressure:

T 00 = � T 0i = 0

Uµ = (1, 0, 0, 0)T ij = p �ij

Or, in terms of the 4-velocity: Tµ� = (⇥+ p)UµU� + p �µ�

A fluid in motion is still given by the same expression, if we replace the 4-velocity by:

Uµ ! �(v)(1,⇥v)

We then get that:

T 00 = �2(⇥+ p)� p =⇥+ p v2

1� v2

T 0i =�+ p

1� v2vi

T ij =⇥+ p

1� v2vi vj + p�ij

) ⇥µTµ0 =

⇥t

�+ p v2

1� v2+

⇥xi

(�+ p) vi

1� v2

The non-relativistic limit [i.e., neglect O(v2) terms], the conservation of the stress-energy tensor is the so-called continuity equation:

⇥µTµ0 =

⇥t

�+ p v2

1� v2+

⇥xi

(�+ p) vi

1� v2

' �+ ⇥r[(�+ p)⇥v] ' �+ (�+ p) ⇥r⇥v

Why is it OK to neglect ?⇥�(�+ p) · ⇥v

But this is simply the well-known thermodynamic equation for energy conservation:

dE + p dV = 0

) 1

V

d(�V )

dt+ p

1

V

dV

dt=

d�

dt+ (�+ p)

1

V

dV

dt= 0

where the volume changes according to the divergence of the velocity, 1

V

dV

dt= �� · �v

Conservation of the stress-energy tensor:• Energy conservation• Euler equation{

The metric counterpart of the stress-energy tensor must be conserved as well. In fact, the Einstein-Hilbert action satisfies this constraint, and we have that:

Einstein’s field equations, finally!

S =

Zd4x

p�g

R

16�G+ Lm

��! Gµ� = Rµ� � 1

2gµ�R = �8�GTµ�

Matter curves space, determines metric...

... metric determines the kinematics of

matter...

Einstein’s field equations and Cosmology

Because of the Cosmological Principle (homogeneity and isotropy), the left- and the right-hand sides of this equation must be (approximately) functions of time only.

The only free parameters are spacetime constants:

Metric:

Spatial curvatureCosmological constant

Matter:

MassesCoupling constants

(and, e.g., ratios of densities)

S =

Zd4x

p�g

R

16�G+ Lm

��! Gµ� = Rµ� � 1

2gµ�R = �8�GTµ�

A. Friedmann (1922-24), G. Lemaitre (1927), H. P. Robertson (1935-36), A. G. Walker(1937)

Kinematics and dynamics of FLRW spacetimes

Homogeneous & isotropic, spatial sections of constant curvature:

ds2 = �dt2 + a2(t) d�2

k=0

k=-

k=1

FLRW spacetimes

Def.: a(t0)=1

d�2 =dr2

1� kr2+ r2 d⇥2

k = ± (R0)-2

Some other popular coordinates used to express FLRW spatial sections:

Polar coordinates: d�2 =dr2

1� kr2+ r2 d⇥2

k = ± (R0)-2

Hyperspherical coordinates:

r =1pk

sin⇣p

k �⌘

) d�2 = d�2 +1

ksin2

⇣pk �

⌘d⇥2

Conformal-Cartesian coordinates:

r =R

1 + k4R

2) d�2 =

dR2 +R2d⇥2

(1 + k4R

2)2

The most common choice is the second one, since in hyperspherical coordinates the radial geodesics are trivial.

Spatial sections are homogeneous,

isotropic

Geodesics in FLRW spacetime

Let’s take the FLRW metric in conformal-Cartesian coordinates:

ds2 = �dt2 + a2(t) d�2

d�2 = �ij dxi dxj , �ij = ⇥ij

✓1 +

1

4k ⇤x 2

◆�2

The connections are: �000 = �i

00 = �0i0 = 0

�0ij = a a �ij , �i

0j =a

a⇥ij

�kij =

k

2

1

1 + k4⇥x

2

!⇥��ijx

k � �ikxj � �jkx

i�

The solution to this set of equations is: Uµ = Uµ0 = (1, 0, 0, 0)

A particle initally at rest in these coordinates has the 4-velocity:

Uµ0 =

dxµ

d�

����0

= (1, 0, 0, 0)

The geodesic equation is: dUµ

d�+ �µ

�⇥U�U⇥ = 0

which means the following set of equations:

µ = k ! dUk

d�+ 2�k

0i U0U i + �k

ij UiU j = 0

µ = 0 ! dU0

d�+ �0

ij UiU j = 0

Hence, a particle at rest in any point in this spacetime will remain in the same position! This is the practical meaning of “homogeneity and isotropy”!

ds2 = �dt2 + a2(t) d�2 d�2 = �ij dxi dxj , �ij = ⇥ij

✓1 +

1

4k ⇤x 2

◆�2

Grid coordinates: comoving coordinates

Physical coordinates: dp = a(t) dc

Comoving distances v. physical distances

The speed with which these particles (“at rest”!) are separating is given by:

v =d

dt�l =

a

a�l

FLRW metric and expansion

Consider two particles at rest on different spatial locations.

The physical distance between them is given, at time t, by:

ds20 = �dt2 + a2(t) d�2 = 0

Consider now a light ray propagating in the radial direction - and let’s use the hyperspherical coordinates for this. We have:

) � =

Z t0+�t

t0

dt

a(t)

�s2 = �l2(t) = a2(t)(�x0 � �x1)2 = a2(t)��x 2

x

i0 x

i1

a(t)constant!

t0

t0+Δt

a

a⌘ H

Hubble parameter

Cosmological redshift

Suppose we have a light source emitting radiation with a frequency ν0 at the radial position r0, and at the instant t0. A time T0 later, t0+T0 , the light source will be emitting radiation at the same phase (+2π) as in t0 .

The light ray which was emitted at t0 is then observed at a position r1 .

t0

t0+Δt

r0 r1

t0+T0

t0+Δt+T1

� =

Z t1

t0

dt

a(t)=

Z t1+T1

t0+T0

dt

a(t)

)Z t0+T0

t0

dt

a(t)=

Z t1+T1

t1

dt

a(t)) T0

a(t0)' T1

a(t1)) �0

�1=

a(t1)

a(t0)

emmitted here

observed here

In terms of the wavelength of the light, we have: �obs

�emm

=a(t

obs

)

a(temm

)

The “redshift” (or “blueshift”) is defined as: 1 + z =�obs

�emm

=⇥emm

⇥obs

) z =�obs

� �emm

�emm

=⇥emm

� ⇥obs

⇥obs

Absorption lines at the Sun

Same lines at a distant galaxy

Any emmission or absorption line can be used to compute the redshift!

Typically, we observe here on Earth (r=χ=0, t=0) the light emitted by distant galaxies at time t.

Since by convention the scale factor today is a0 =a(t0)=1, we have that the redshift of those distant galaxies is given by:

Example: SDSS galaxy at z=0.1003

At rest, some of these lines are:

OII : 3727 A

z =�obs

� �

�=

a0

a(t)� 1 =

1

a(t)� 1

H↵ : 6563 A

H� : 4861 A

Example: quasar

(“quasi-stellar object”)

Model (average quasar, restframe) Actual quasar (2dF survey)

Cosmological redshift and Doppler effect are manifested in the same way!

�obs

= (1 + z)�rest

�obs

= (1 + z)�rest

a(t)

a0= a(t) =

1

1 + z