II Russian-Spanish Congress Particle and Nuclear Physics at allscales, Astroparticle Physics and Cosmology, S. Petersburg

Soft singularity crossing and

transformation of matter properties

A.Yu. Kamenshchik

University of Bologna and INFN, BolognaL.D. Landau Institute for Theoretical Physics, Moscow

October 1-4, 2013

Based on

Z. Keresztes, L.A. Gergely, A.Y. Kamenshchik, V. Gorini andD. Polarski,Soft singularity crossing and transformation of matterproperties,Phys. Rev. D 88 (2013) 023535

Previous papers:

V. Gorini, A.Y. Kamenshchik, U. Moschella and V. Pasquier,Tachyons, scalar fields and cosmology,Phys. Rev. D 69 (2004) 123512

Z. Keresztes, L.A. Gergely, V. Gorini, U. Moschella andA.Y. Kamenshchik, Tachyon cosmology, supernovae data andthe Big Brake singularity,Phys. Rev. D 79 (2009) 083504

Z. Keresztes, L.A. Gergely, A.Y. Kamenshchik, V. Gorini andD. Polarski,Will the tachyonic Universe survive the Big Brake?,Phys. Rev. D 82 (2010) 123534

Z. Keresztes, L.A. Gergely and A.Y. Kamenshchik,The paradox of soft singularity crossing and its resolution bydistributional cosmological quantitities,Phys. Rev. D 86 (2012) 063522

Review:A.Yu. Kamenshchik,Quantum cosmology and late-time singularities,Class. Quantum Grav. 30 (2013) 173001

Content

1. Introduction

2. Description of the tachyon cosmological model

3. The Big Brake cosmological singularity and more generalsoft singularities

4. Crossing of the soft singularity in the model with theanti-Chaplygin gas and dust

5. Crossing the Big Brake singularity and the future of theuniverse in the tachyon model in the presence of dust

6. Numerical simulations for the tachyon model

7. Conclusions

8. Postscript

Introduction

I The general relativity connects the geometrical propertiesof the spacetime to its matter content.The matter tells to the spacetime how to curve itself, thespacetime geometry tells to the matter how to move.

I The cosmological singularities constitute one of the mainproblems of modern cosmology.

I The discovery of the cosmic acceleration stimulated thedevelopment of “exotic” cosmological models of darkenergy; some of these models possess the so called soft orsudden singularities characterized by the finite value ofthe radius of the universe and its Hubble parameter.

I “Traditional” or “hard” singularities are associated withthe zero volume of the universe (or of its scale factor),and with infinite values of the Hubble parameter, of theenergy density and of the pressure –Big Bang and BigCrunch

I In some models interplay between the geometry and thematter forces the matter to change some of its basicproperties, such as equation of state for fluids and eventhe form of the Lagrangian.

I Tachyons (Born-Infeld fields) is a natural candidate for adark energy

I The toy tachyon model, proposed in 2004 has twoparticular features:Tachyon field transforms itself into a pseudo-tachyon field,The evolution of the universe can encounter a new typeof singularity - the Big Brake singularity.

I The Big Brake singularity is a particular type of the socalled “soft” cosmological singularities - the radius of theuniverse is finite, the velocity of expansion is equal tozero, the deceleration is infinite.

I The predictions of the model do not contradictobservational data on supenovae of the type Ia(2009,2010)

I The Big Brake singularity is a particular one - it ispossible to cross it (2010)

I Open questions: other soft singularities - is itpossible to cross them ?

I What is more important: matter or geometry ?

Description of the tachyon model

The flat Friedmann universe

ds2 = dt2 − a2(t)dl2

The tachyon Lagrange density

L = −V (T )√

1− T 2

The energy density

ρ =V (T )√1− T 2

The pressure

p = −V (T )√

1− T 2

The Friedmann equation

H2 ≡ a2

a2= ρ

The equation of motion for the tachyon field

T

1− T 2+ 3HT +

V,T

V= 0

In our model

V (T ) =Λ

sin2[

32

√Λ (1 + k) T

]

×√

1− (1 + k) cos2[3

2

√Λ (1 + k) T

],

where k and Λ > 0 are the parameters of the model. Thecase k > 0 is more interesting.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

T

sP

P′

Q

Q′

I II III

IV

V III

IV

IV

I

II

IV

σ

ψ χ

τ ξ

IV

III V

I II III

I

II

IV

Phase portrait of the model for a positive k.

Some trajectories (cosmological evolutions) finish in theinfinite de Sitter expansion. In other trajectories the tachyonfield transforms into the pseudotachyon field with theLagrange density, energy density and positive pressure.

L = W (T )√

T 2 − 1,

ρ =W (T )√T 2 − 1

,

p = W (T )√

T 2 − 1,

W (T ) =Λ

sin2[

32

√Λ (1 + k) T

]

×√

(1 + k) cos2

[3

2

√Λ (1 + k) T − 1

]

What happens with the Universe after the

transformation of the tachyon into the

pseudotachyon ?

It encounters the Big Brake

cosmological singularity.

The Big Brake cosmological singularity

and other soft singularities

t → tBB < ∞a(t → tBB) → aBB < ∞

a(t → tBB) → 0

a(t → tBB) → −∞R(t → tBB) → +∞

T (t → tBB) → TBB , |TBB | < ∞|T (t → tBB)| → ∞

ρ(t → tBB) → 0

p(t → tBB) → +∞If a(tBB) 6= 0 it is more general soft singularity.

Crossing the Big Brake singularity and the future of

the universe

At the Big Brake singularity the equations forgeodesics are regular, because the Christoffelsymbols are regular (moreover, they are equal tozero).

Is it possible to cross the Big Brake ?

Let us study the regime of approaching the BigBrake.

Analyzing the equations of motion we find that approachingthe Big Brake singularity the tachyon field behaves as

T = TBB +

(4

3W (TBB)

)1/3

(tBB − t)1/3.

Its time derivative s ≡ T behaves as

s = −(

4

81W (TBB)

)1/3

(tBB − t)−2/3,

the cosmological radius is

a = aBB − 3

4aBB

(9W 2(TBB)

2

)1/3

(tBB − t)4/3,

its time derivative is

a = aBB

(9W 2(TBB)

2

)1/3

(tBB − t)1/3

and the Hubble variable is

H =

(9W 2(TBB)

2

)1/3

(tBB − t)1/3.

All these expressions can be continued in the regionwhere t > tBB ,which amounts to crossing the BigBrake singularity. Only the expression for s issingular at t = tBB but this singularity is integrableand not dangerous.

Once reaching the Big Brake, it is impossible for

the system to stay there because of the infinite

deceleration, which eventually leads to the decreaseof the scale factor. This is because after the Big

Brake crossing the time derivative of the

cosmological radius and Hubble variable change

their signs. The expansion is then followed by a

contraction, culminating in the Big Crunchsingularity.

Crossing of the soft singularity in the model with the

anti-Chaplygin gas and dustOne of the simplest cosmological models revealing the BigBrake singularity is the model based on the anti-Chaplygin gaswith an equation of state

p =A

ρ, A > 0

Such an equation of state arises in the theory of wiggy strings( B. Carter, 1989, A. Vilenkin, 1990).

ρ(a) =

√B

a6− A

At a = a∗ =(

BA

)1/6the universe encounters the Big Brake

singularity.

The anti-Chaplygin gas plus dust

The energy density and the pressure are

ρ(a) =

√B

a6− A +

M

a3, p(a) =

A√Ba6 − A

.

Due to the dust component, the Hubble parameter has anon-zero value at the encounter with the singularity, thereforethe dust implies further expansion. With continued expansionhowever, the energy density and the pressure of theanti-Chaplygin gas would become ill-defined.

In principle one can solve the paradox by redefining theanti-Chaplygin gas in a distributional sense (Keresztes, Gergely,Kamenshchik, 2012) . Then a contraction could follow theexpansion phase at the singularity at the price of a jump in theHubble parameter. Although such an abrupt change is notcommon in any cosmological evolution, we explicitly show thatthe set of Friedmann, Raychaudhuri and continuity equationsare all obeyed both at the singularity and in its vicinity.The jump in the Hubble parameter

H → −H

leaves intact the first Friedmann equation H2 = ρ, thecontinuity equations and the equations of state, however, itbreaks the validity of the second Friedmann (Raychaudhuri)equation H = −3

2(ρ + p).

H(t) = HSsgn(tS − t)

+

√3A

2HSa4S

sgn(tS − t)√|tS − t| ,

H = −2HSδ(tS − t)−√

3A

8HSa4S

sgn(tS − t)√|tS − t| .

To restore the validity of the Raychaudhuri equation we add asingular δ -term to the pressure of the anti-Chaplygin gas

p =

√A

6HS |tS − t| +4

3HSδ(tS − t).

To preserve the equation of state we also modify theexpression for its energy density:

ρ =A√

A6HS |tS−t| +

43HSδ(tS − t)

.

Change of the equation of state at soft singularity

crossingsThe abrupt transition from the expansion to the contraction ofthe universe does not look natural. There is analternative/complementary way of resolving the paradox.

One can tray to change the equation of state of theanti-Chaplygin gas at passing the soft singularity.

There is some analogy between the transition from anexpansion to a contraction of a universe and an absolutelyelastic bounce of a ball from a wall in classical mechanics.There is also an abrupt change of the direction of the velocity(momentum).

However, we know that really the velocity is changedcontinuously due to the deformation of the ball and of the wall.

The pressure of the anti-Chaplygin gas

p =A√

Ba6 − A

tends to +∞ when the universe approaches the softsingularity.Requiring the expansion to continue into the region a > aS ,while changing minimally the equation of state, we assume

p =A√

| Ba6 − A|

,

p =A√

A− Ba6

, for a > aS .

It implies the energy density

ρ = −√

A− B

a6.

The anti-Chaplygin gas transforms itself into Chaplygin gaswith negative energy density.The pressure remains positive, expansion continues. Thespacetime geometry remains continuous.The expansion stops at a = a0, where

M

a30

−√

A− B

a60

= 0.

Then the contraction of the universe begins.

At the moment when the energy density of theChaplygin gas becomes equal to zero

(again a soft singularity), the Chaplygin gastransforms itself into the anti-Chaplygin gas

and the contraction continues to culminate in theencounter with the Big Crunch singularitya = 0.

Crossing the Big Brake singularity and the future of

the universe in the tachyon model in the presence of

dust.

What happens with the Born-Infeld typepseudo-tachyon field in the presence of a dustcomponent? Does the universe still run into a softsingularity?Yes !

T = TS ±√

2

3HS

√tS − t, HS =

√ρm,0

a3S

.

How can the universe cross this singularity ?

A pseudo-tachyon field with a constant potential

is equivalent to the anti-Chaplygin gas.To the change of the equation of state of the anti-Chaplygingas corresponds the following transformation of theLagrangian of the pseudo-tachyon field:

L = W0

√g ttT 2 + 1,

p = W0

√T 2 + 1

ρ = − W0√T 2 + 1

.

It is a new type of Born-Infeld field, which we may call“quasi-tachyon”.

For an arbitrary potential the Lagrangian

reads

L = W (T )

√g ttT 2 + 1, a > aS

T

T 2 + 1+ 3HT − W,T

W= 0,

ρ = − W (T )√T 2 + 1

,

p = W (T )√

T 2 + 1.

In the vicinity of the soft singularity thefriction term 3HT in the equation of motiondominates over the potential term W,T/W .Hence, the dependence of W (T ) on itsargument is not essential and apseudo-tachyon field approaching thissingularity behaves like one with a constantpotential. Thus, it is reasonable to assumethat upon crossing the soft singularity thepseudo-tachyon transforms itself into aquasi-tachyon for any potential W (T ).

The dynamics of the model with trigonometric

potential in the presence of dust.After the soft singularity crossing the absolute value of thenegative contribution to the energy density of the universeinduced by the quasi-tachyon grows while the energy density ofthe dust decreases due to the expansion of the universe. Thus,at some moment the total energy density vanishes and theuniverse reaches the point of maximal expansion, after whichthe expansion is replaced by a contraction and the Hubblevariable changes sign.At some finite moment of time the universe hits again the softsingularity. Upon crossing this singularity the quasi-tachyontransforms back to pseudo-tachyon.After this the universe continues its contraction until it hitsthe Big Crunch singularity.

Numerical simulations for the tachyon model.

Comparing the prediction of our model with the Supernovae IaUnion2 Dataset, we have found the subset of accessible initialconditions (T , T , Ωm).

Starting from this initial conditions we have simulated futureevolutions of the universe.

Some of the trajectories go towards de Sitter attractive node.

Other trajectories go towards the transformationtachyon-pseudo-tachyon, the first crossing the soft singularity,the turning point, the second soft singularity crossing, andfinally, the encounter with the Big Crunch.

yx

-1

-0.8

-0.6

-0.4

-0.2

0

z

Ωm=0.03

(1,0,-1)

-1-0.5

0 0.5

1 0

0.2 0.4

0.6 0.8

1

z

yx

-1

-0.8

-0.6

-0.4

-0.2

0

z

Ωm=0.09

(1,0,-1)

-1-0.5

0 0.5

1 0

0.2 0.4

0.6 0.8

1

z

yx

-1

-0.8

-0.6

-0.4

-0.2

0

z

Ωm=0.15

(1,0,-1)

-1-0.5

0 0.5

1 0

0.2 0.4

0.6 0.8

1

z

yx

-1

-0.8

-0.6

-0.4

-0.2

0

z

Ωm=0.21

(1,0,-1)

-1-0.5

0 0.5

1 0

0.2 0.4

0.6 0.8

1

z

yx

-1

-0.8

-0.6

-0.4

-0.2

0

z

Ωm=0.27

(1,0,-1)

-1-0.5

0 0.5

1 0

0.2 0.4

0.6 0.8

1

z

yx

-1

-0.8

-0.6

-0.4

-0.2

0

z

Ωm=0.33

(1,0,-1)

-1-0.5

0 0.5

1 0

0.2 0.4

0.6 0.8

1

z

The future evolution of those universes, which are in a 68.3%

confidence level fit with the supernova data.

Conclusions.

I The general relativity contains a lot of surprisesconcerning relations between the matter andgeometry. It is enough to take it seriously.

I The next direction of investigations - to includea spatial curvature.

I More difficult problem - to consider spatiallyinhomogeneous case.

P.S. Spanish Influence

A. Feinstein, Power law inflation from the rolling tachyon,Phys. Rev. D 66 (2002) 063511.

It was shown that the tachyon potential 1T 2 can provide a

power law expansion of a flat Friedmann universe.

L. Fernandez-Jambrina and R. Lazkoz, Geodesic behaviour ofsudden future singularities, Phys. Rev. D 70 (2004) 121503;Classification of cosmological milestones, Phys. Rev. D 74(2006) 064030.

It was shown that soft cosmological singularities can becrossed.