The Universe Before and After The Big Bangvvanchur/2013PHYS1021/vanchurin.pdf · The Universe...

MOTIVATIONS THEORETICAL PHYSICS THEORETICAL COSMOLOGY COSMIC INFLATION ETERNAL INFLATION CONCLUSIONS

The Universe Before and AfterThe Big Bang

VITALY VANCHURIN

University of Minnesota, DuluthSeptember 5, 2013


OUTLINE

MOTIVATIONS

THEORETICAL PHYSICS

Language of PhysicsGenres of Physics

THEORETICAL COSMOLOGY

General RelativityBig Bang Theory

COSMIC INFLATION

Scalar inflationVector inflation

ETERNAL INFLATION

Quantum FluctuationsMultiverse

CONCLUSIONS


PROBING COSMOLOGY AFTER THE BIG BANGI (1929) Cosmological redshift→ cosmological expansion

I (1962) Ultra high energy cosmic rays→ remains a puzzleI (1975) Galactic rotation curves→ dark matterI (1998) Supernova data→ dark energy→ 2011 Nobel prizeI (201?) CMB discontinuities? → cosmic strings?


PROBING COSMOLOGY BEFORE THE BIG BANGI CMB Experiments

I (1964) Thermal radiation→ 1978 Nobel prizeI (1992) Anisotropies→ 2006 Nobel prize

I (2001) Acoustic peaks→ primordial spectrumI (2006) Spectral tilt→ inflationI (201?) Tensor modes? → chaotic inflation?I (201?) Non-gaussianities? → eternal inflation?

I 21 cm Experiments?


LANGUAGE OF PHYSICSI WRITE DOWN AN ACTION, e.g.

S [x(t)] =

∫dt

(12

(∂x∂t

)2

− 12ω2x2

)

I 1) CLASSICAL ANALYSIS:I Derive equations from δS[x]

δx = 0, e.g. ∂2x∂t2 + ω2x = 0

I Solve classical equation, e.g. x(t) = A sin(ωt + φ)I Calculate predictions for given initial conditions x(0), ∂x(0)

∂tI 2) QUANTUM ANALYSIS:

I Promote x and ∂x∂t to operators, e.g. x and −i~ ∂

∂x

I Solve quantum equation, e.g.(ω2

2 x2 − ~2

2∂2

∂x2

)|ψ〉 = i~ ∂∂t |ψ〉

I Calculate predictions for a given initial state |ψ(0)〉I 3) STATISTICAL ANALYSIS:

I Study evolution of a large number of particles (or strings)I Make assumptions, e.g. equilibrium, molecular chaos, etc.I Calculate macroscopic predictions (e.g. thermodynamics)



S [x(t)] =

∫dt

(12

(∂x∂t

)2

− 12ω2x2

)


δx = 0, e.g. ∂2x∂t2 + ω2x = 0


∂t

I 2) QUANTUM ANALYSIS:I Promote x and ∂x

∂t to operators, e.g. x and −i~ ∂∂x


2 x2 − ~2

2∂2

∂x2

)|ψ〉 = i~ ∂∂t |ψ〉





S [x(t)] =

∫dt

(12

(∂x∂t

)2

− 12ω2x2

)


δx = 0, e.g. ∂2x∂t2 + ω2x = 0




∂x


2 x2 − ~2

2∂2

∂x2

)|ψ〉 = i~ ∂∂t |ψ〉

I Calculate predictions for a given initial state |ψ(0)〉

I 3) STATISTICAL ANALYSIS:I Study evolution of a large number of particles (or strings)I Make assumptions, e.g. equilibrium, molecular chaos, etc.I Calculate macroscopic predictions (e.g. thermodynamics)



S [x(t)] =

∫dt

(12

(∂x∂t

)2

− 12ω2x2

)


δx = 0, e.g. ∂2x∂t2 + ω2x = 0




∂x


2 x2 − ~2

2∂2

∂x2

)|ψ〉 = i~ ∂∂t |ψ〉




GENRES OF PHYSICSI FIELD THEORY: CLASSICAL, QUANTUM, STATISTICAL

I

S =

∫dx4

12

∑µ=0,1,2,3

∂ϕ

∂xµ∂ϕ

∂xµ− 1

2m2ϕ2

I Add scalar, spinor, vector fields? Add potentials, interactions?

I GENERAL RELATIVITY: CLASSICAL, SOME QUANTUM/STATISTICAL

I

S =

∫dx4√−det(g)

(−R(g)

2

)I Add a cosmological constant Λ? Add higher order corrections R2?

I STRING THEORY: CLASSICAL, SOME QUANTUM/STATISTICAL

I

S =1

2πα′

∫dζ0dζ1

√−det(γ)

(−1

2∂xµ

∂ζ0

∂ xµ∂ζ0− 1

2∂xµ

∂ζ1

∂ xµ∂ζ1

)I Add fields, interactions? Add extra dimensions?


GENRES OF PHYSICSI FIELD THEORY: CLASSICAL, QUANTUM, STATISTICAL

I

S =

∫dx4(

12∂ϕ

∂xµ∂ϕ

∂xµ− 1

2m2ϕ2

)I Add scalar, spinor, vector fields? Add potentials, interactions?

I GENERAL RELATIVITY: CLASSICAL, SOME QUANTUM/STATISTICAL

I

S =

∫dx4√−det(g)

(−R(g)

2

)I Add a cosmological constant Λ? Add higher order corrections R2?

I STRING THEORY: CLASSICAL, SOME QUANTUM/STATISTICAL

I

S =1

2πα′

∫dζ0dζ1

√−det(γ)

(−1

2∂xµ

∂ζ0

∂ xµ∂ζ0− 1

2∂xµ

∂ζ1

∂ xµ∂ζ1

)I Add fields, interactions? Add extra dimensions?


DIFFERENTIAL GEOMETRY IN A NUTSHELLBy measuring distances one can calculate the line element:

(ds)2 = gxx(dx)2 + gxy(dx)(dy) + gyx(dy)(dx) + gyy(dy)2

or metric:g =

(gxx gxy

gyx gyy

)

I Pos. curved: (ds)2 = (dx)2 + sin(x)2(dy)2

g =

(1 00 sin(x)2

)I Neg. curved: (ds)2 = (dx)2 + sinh(x)2(dy)2

g =

(1 00 sinh(x)2

)I Flat space: (ds)2 = (dx)2 + (dy)2

g =

(1 00 1

)


GENERAL RELATIVITY IN A NUTSHELLIn GR metric is caluclated by solving Einstein equations:

Gµν(gµν) = 8πTµν(gµν)

Minkowski solution (i.e. our space-time locally):

ds2 = dt2 − (dx2 + dy2 + dz2).

Friedmann solution (i.e. our space-time globally):

ds2 = dt2 − a(t)2(dx2 + dy2 + dz2).

Then physical distance between points (t, x, y,A) and (t, x, y,B)∫ B

Ads =

∫ B

Aa(t)dz = a(t)(B− A)

increases as the scale factor a(t) grows with time t.


THE BIG BANG THEORY IN A NUTSHELL

S =

∫dx4√−g

(−R

2− 1

4FµνFµν +

12∂µϕ∂

µϕ− 12

m2ϕ2 − Λ

)

Let’s fill universe with energy:ρr of radiation (e.g. light)ρm of matter (e.g. stars or dark matter)ρΛ of dark energy (e.g. cosmological constant)

Let’s solve e.o.m. for metric: ds2 = dt2 − a(t)2(dx2 + dy2 + dz2)We find that during:

radiation domination: ρr ∝ a(t)−4 and a(t) ∝ t12

matter domination: ρm ∝ a(t)−3 and a(t) ∝ t23

dark energy domination: ρΛ ∝ const and a(t) ∝ e√

Λ3 t

Conclusions:At time t = 0 the metric becomes singular→ BANG!After time t ∼ Λ−1/2 the vacuum must dominate. NOW!?


THE BIG BANG THEORY IN A NUTSHELL

S =

∫dx4√−g

(−R

2− 1

4FµνFµν +

12∂µϕ∂

µϕ− 12

m2ϕ2 − Λ

)Let’s fill universe with energy:

ρr of radiation (e.g. light)ρm of matter (e.g. stars or dark matter)ρΛ of dark energy (e.g. cosmological constant)

Let’s solve e.o.m. for metric: ds2 = dt2 − a(t)2(dx2 + dy2 + dz2)We find that during:

radiation domination: ρr ∝ a(t)−4 and a(t) ∝ t12

matter domination: ρm ∝ a(t)−3 and a(t) ∝ t23

dark energy domination: ρΛ ∝ const and a(t) ∝ e√

Λ3 t

Conclusions:At time t = 0 the metric becomes singular→ BANG!After time t ∼ Λ−1/2 the vacuum must dominate. NOW!?


SCALAR INFLATION [STAROBINSKY, GUTH, LINDE]

Consider a massive scalar minimally coupled to gravity:

S =

∫dx4√−g

(−R

2+

12∂µϕ∂

µϕ− 12

m2ϕ2)

Background equations of motion are given by:

ϕ+ 3Hϕ+ m2ϕ = 0

H2 ≡(

aa

)2

=13

(12ϕ2 +

12

m2ϕ2)

On the background level inflation solves horizon, flatness,homogeneity and isotropy problems of the Big Bang theory.Can we have a similar scenario with high spin fields? Vectors?




S =

∫dx4√−g

(−R

2+

12∂µϕ∂

µϕ− 12

m2ϕ2)


ϕ+ 3Hϕ+ m2ϕ = 0

H2 ≡(

aa

)2

=13

(12ϕ2 +

12

m2ϕ2)

On the background level inflation solves horizon, flatness,homogeneity and isotropy problems of the Big Bang theory.

Can we have a similar scenario with high spin fields? Vectors?




S =

∫dx4√−g

(−R

2+

12∂µϕ∂

µϕ− 12

m2ϕ2)


ϕ+ 3Hϕ+ m2ϕ = 0

H2 ≡(

aa

)2

=13

(12ϕ2 +

12

m2ϕ2)

On the background level inflation solves horizon, flatness,homogeneity and isotropy problems of the Big Bang theory.Can we have a similar scenario with high spin fields? Vectors?


VECTOR INFLATION [GOLOVNEV, MUKHANOV, V.V. (2008)]Consider a massive vector non-minimally coupled to gravity:

S =

∫dx4√−g

(−R

2− 1

4FµνFµν +

12

(m2 +

R6

)AµAµ

)where Fµν = ∂µAν − ∂νAµ.

Define:

Bi =Ai

a= aAi.


A0 = 0 and Bi + 3aa

Bi + m2Bi = 0.

For a triad of vector fields

H2 ≡(

aa

)2

=

(12

Bi2

+12

m2B2i

)


VECTOR INFLATION [GOLOVNEV, MUKHANOV, V.V. (2008)]Consider a massive vector non-minimally coupled to gravity:

S =

∫dx4√−g

(−R

2− 1

4FµνFµν +

12

(m2 +

R6

)AµAµ

)where Fµν = ∂µAν − ∂νAµ. Define:

Bi =Ai

a= aAi.


A0 = 0 and Bi + 3aa

Bi + m2Bi = 0.

For a triad of vector fields

H2 ≡(

aa

)2

=

(12

Bi2

+12

m2B2i

)


VECTOR INFLATION [GOLOVNEV, MUKHANOV, V.V. (2008)]

Consider a consider N randomly oriented vector fields.

A completely diagonal energy-momentum tensor

T00 ≈ T11 ≈ T22 ≈ T33 ∼ H2

becomes slightly non-diagonal Tij ∼ H2√

NB2 for i 6= j.Hence, solutions are self-consistent only when H2

√NB2 � H2

orB < N−1/4

At the same time the field rolls slowly when H > m or

B > N−1/2.

Thus, the maximal number of e-folds of inflation is 2π√

N.(Note, in one e-fold the volume grows by a factor of e3 ≈ 20).Observations suggest that there were & 60 e-folds or N & 100.


VECTOR INFLATION [GOLOVNEV, MUKHANOV, V.V. (2008)]

Consider a consider N randomly oriented vector fields.A completely diagonal energy-momentum tensor

T00 ≈ T11 ≈ T22 ≈ T33 ∼ H2

becomes slightly non-diagonal Tij ∼ H2√

NB2 for i 6= j.Hence, solutions are self-consistent only when H2

√NB2 � H2

orB < N−1/4

At the same time the field rolls slowly when H > m or

B > N−1/2.

Thus, the maximal number of e-folds of inflation is 2π√

N.(Note, in one e-fold the volume grows by a factor of e3 ≈ 20).Observations suggest that there were & 60 e-folds or N & 100.


QUANTUM FLUCTUATIONS

I Heisenberg taught us that every field must fluctuate.I The inflaton field is not an exception.I Fluctuations of ϕ or Aµ source fluctuations of gµν .

I These fluctuations are responsible for:- structure formation(Harrison, Peebles, Yu, Zel’dovich)- fluctuation in CMB(Mukhanov, Chibisov, Starobinsky)

I In addition, vector inflation predicts(Golovnev, Mukhanov, V.V.)- anisotropy of order ∼ 1/

√N

- blue spectrum of tensor modes- corr. of scalar and tensor modes

I Tensor modes might be detected very soon ...


MULTIVERSEI Quantum fluctuations ”freeze-out” when modes with

wavelength λ exist apparent horizon λ > H−1.

I The amplitude of fluctuation on time scales δt = H−1:

δϕquantum ≈H2π∼ mϕ

I Classical drift during the same time interval δt = H−1:

δϕclassical ≈ ϕH−1 = −m2ϕ

3H2 ∼1ϕ

I Inflation is eternal when δϕquantum & δϕclassical or ϕ & 1√m .

I How many universes are in the multiverse? [Linde, V.V. (2009)]I Inflationary theory allows ∼ 101010000000

universes.I String theory allows ∼ 1010375

universes.I But humans can distinguish only ∼ 101016

universes.I How to assign probabilities to these universes is a subject of a

very active research in cosmology as well as in string theory.



wavelength λ exist apparent horizon λ > H−1.I The amplitude of fluctuation on time scales δt = H−1:




3H2 ∼1ϕ













3H2 ∼1ϕ





universes.

I How to assign probabilities to these universes is a subject of avery active research in cosmology as well as in string theory.







3H2 ∼1ϕ








LANDSCAPE OF STRING THEORYThe potential energy has 10120, 10500 or even 101000 minima:

There are many minima with potential energy Λ ∼ 10−120.There are many minima where inflaton mass is m ∼ 10−5.Can the Landscape unify string theory and inflation?


SUMMARY

I General Relativity is the most elegant theory ever!

I Big Bang is one of its predictions but there many others

I Inflation solves many problems of the Big Bang theory

I Vector and scalar inflations are the two simplest models

I Eternal inflation might unify string theory and inflation

UPCOMING EXPERIMENTS WILL TEST ALL OF THESE IDEAS


OUTLOOK:I OBSERVATIONAL TOPICS:

I Design and built a 21 cm array

I PHENOMENOLOGICAL TOPICS:I Signatures from cosmic strings and inflation

I NUMERICAL TOPICS:I Kinetic theory simulations (ask Jacob Balma)I String fluid simulations (ask Hitao Shang)

I THEORETICAL TOPICS:I Fluid mechanics of strings (ask Daniel Schubring)I Non-canonical topological defects (ask Mudit Jain)I Hagedorn phase transitionI Effective field theory of strings

I MATHEMATICAL TOPICS:I Non-commutative probabilitiesI Hyperbolic dynamical systemsI Markov random fields

I PHILOSOPHICAL TOPICS:I Paradoxes in cosmology and quantum gravity



I Design and built a 21 cm arrayI PHENOMENOLOGICAL TOPICS:

I Signatures from cosmic strings and inflation

I NUMERICAL TOPICS:I Kinetic theory simulations (ask Jacob Balma)I String fluid simulations (ask Hitao Shang)






I Design and built a 21 cm arrayI PHENOMENOLOGICAL TOPICS:

I Signatures from cosmic strings and inflationI NUMERICAL TOPICS:

I Kinetic theory simulations (ask Jacob Balma)I String fluid simulations (ask Hitao Shang)





TO LEARN MORE ABOUT

COSMOLOGY GROUP AT UMDGO TO:

HTTP://D.UMN.EDU/COSMOLOGY

The Universe Before and After The Big Bangvvanchur/2013PHYS1021/vanchurin.pdf · The Universe...

Documents

Transcript of The Universe Before and After The Big Bangvvanchur/2013PHYS1021/vanchurin.pdf · The Universe...