Le-Huy Nguyen and Rajesh R. Parwani- Nonlinear Quantum Cosmology

8/3/2019 Le-Huy Nguyen and Rajesh R. Parwani- Nonlinear Quantum Cosmology

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Nonlinear Quantum Cosmology

Le-Huy Nguyen and Rajesh R. Parwani

[email protected]

Department of Physics,

National University of Singapore,

Kent Ridge,

Singapore.

Nonlinear uantum Cosmolo . 1/


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Motivation

Classical cosmologies typically have singularities, leadingto a loss of predictability.E.g. the Friedmann equation for an isotropic and

homogeneous universe with the cosmological constant = 3/a20 modelling inflationionary sources in the earlyuniverse is,

a2

+ (k a2

/a2

0) = 0 .(1)For k = 0 it has the expanding solution,

a = exp(t/a0

)(2)

which shows the a 0 singularity at early time t .

(a(t) is the scale factor).



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Standard Quantum Cosmology

Of course when the universe is small there is no reason toexpect classical physics to be accurate.Where is the quantum ?

In the minisuperspace scheme the quantisation is applied

to restricted situations such as the FRW model.From the action one gets the canonical momentum

p = La = aa p = i a .Quantising the Friedmann equation leads one to the WDW

equation in minisuperspace,

2

a2

+ V(a)(a) = 0 ,(3)Nonlinear uantum Cosmolo . 4/


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Tunneling for k=1

That is just a one-dimesional, 0 a < , time-independentSchrodinger equation for a particle of mass m = 1/2 movingin a potential

V(a) = a2(k a2/a20)(4)

For spherical geometry, k = 1, the potential barrier betweena = 0 and a = a0 means that there is quantum tunneling

[Vilenkin,Atkatz].

A classical universe emerges and exists at finite size(a

a0).

The original singularity at a = 0 is screened by the barrier.



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k=0

For k = 0 there is no barrier.

Standard quantisation of the k = 0 geometry does not help:

When one chooses the outgoing wavefunction as a solutionto the WDW, representing an expanding universe, one stillencounters the singularity at a = 0 in the sense that thewavefunction allows a universe of arbitrarily small size, and

thus arbitrarily large energy densities, to exist.



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Modifying the WDW Equation

It is generally believed that the functional WDW equation isonly an approximate, perhaps semi-classical, description ofquantum gravitational effects.

Various modified WDW equations have been investigated:

For example, through a postulated non-commutativity [H.Garcia-Compean et.al]or by using ideas inspired by loop quantum gravity[Bojowald; Ashtekar et.al].

In the latter, a discretised WDW equation emerges which is

found to avoid the classical singularity through a bounce.



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New Phyics?

It is as yet unknown which, if any, of the suggestedmodifications to classical general relativity is an accuratedescription of likely new physics at the Planck scale.

We therefore adopt the maximum uncertainty (entropy)principle" [Jaynes], or information-theoretic perspective, to

modify the WDW equation and study the consequences.

The philosophy of the information-theoretic approach is that

one should minimise any bias when choosing probabilitydistributions, while still satisfying relevant constraints

Used for example for deriving the form of probabilitydistributions in statistical mechanics.



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Maximum Uncertainty Method Example

The Gibbs-Shannon entropy (or information)

IGS =p(x) lnp(x) dx(5)

is maximised under given constraints to determine the formfor the probability distribution p(x).

For example, if the mean energy of the system

E =

(x) p(x) dx is specified, then introducing theLagrange multiplier and maximising IGS E with respectto variations in p(x) gives the well known canonicalprobability distribution p(x)

exp(

(x)).



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Nonlinear Schrodinger Equation

The maximum uncertainty approach may also be used tomotivate the usual Schrodinger equation[Friden,Reginatto,Parwani] and its nonlinear deformations

[Parwani].

We postulate that the unknown new physics as shortdistances may be modelled by a nonlinear correction as inprevious works so that the modified equation becomes

2a2

+ V(a) + F(p)

(a) = 0(6)



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Definitions

where

F(p) Q1NL Q ,(7)with

Q1NL =1

2L2

4 ln p

(1 )p + p++ 1

(1 )p

(1 )p + p+

(1 )(8)

Q = 1

p2

p

a2 .(9)

Here p(a) = (a)(a) and p(a) p(a L). L > 0 is thenonlinearity scale and 0 < < 1 is a parameter that labels afamily of nonlinearisations



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At the level of the action, F(p) is obtained by varying theKullback-Liebler information measure IKL which in the limitL

0 reduces to the Fisher information measure

responsible for the usual linear Schrodinger equation.

IKL(p,r) =

p(a) ln

p(a)

r(a)

da(10)

r(a) = p(a + L)

Note that the nonlinearity F preserves the invariance of theSchrodinger equation to .



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Tunneling for k=0

Setting a = lb, with l = a1/30

,

2

b2 b4

+ l2

F(p(lb))

(b) = 0 ,(11)

We assume that the nonlinearity is small. Then perturbing

and iterating to leading nontrivial order in L, one has alinear Schrodinger equation with an effective potential

Veff =

b4 + (3

4)f0(b) .(12)

For small b, f0 0.1b and so for < 3/4 there is an effectivepotential barrier that screens the original classicalsingularity, a finite size universe coming into being throughquantum tunneling.



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Tunneling Probability

For the k = 0 case the WKB formula applied to the effectiveSchrodinger equation gives

Pk=0 exp2

b1

0

db

Veff(b)

(13)

with b1 the point where Veff(b1) = 0. We may estimate

Pk=0 exp[0.1(3 4))] ,(14)

so that for fixed < 3/4 one may say that small values of are preferred", a conclusion which is self-consistent withour approximation 1



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Free Massless Scalar Field

As an internal clock

As the other extreme to the cosmological constant:Now we have matter with KE but no PE.



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Define = ln a. The k = 0 classical FRW action is

S = dtL = 12dt N e3

2

N2+

2

N2(15)

Classical solution:

= 13

log t + C ,(16)

=1

3

log t(17)

C = constant. As t 0, a 0, the classical universearises from an initial singularity.


Q ti ti


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Quantisation

The classical constraint becomes the Hamiltonainconstraint H = 0 giving us the WDW equation

22

+ 2

2

(, ) = 0 .(18)

Use as internal clock.Wavepacket:

= a1b2

exp

( )

2

4

exp(id( )) ,(19)

and d being constants. State is clearly localised near for large .But we also see that the universe would have been

arbitrarily small, , in the distant past .Nonlinear uantum Cosmolo . 17/

N li WDW


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Nonlinear WDW

We now proceed to nonlinearise the Klien-Gordon equationfollowing the information-theoretic approach

22

+ 2

2 F(p) + F(p)

(, ) = 0 .(20)

where F and F have the same form as the F in but with

generally distinct nonlinear parameters L > 0 and L > 0corresponding to the gravitational and matter degrees offreedom.


Eff ti P t ti l


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Effective Potential

As in the previous section, we proceed by perturbation anditeration to lowest nontrivial order to get an effective linearequation

22

+ 2

2+ Veff

(, ) = 0(21)

with

Veff = u

2(2 2) 3 ( ) ,(22)u

(3 4)(L + L)4

12.(23)


Eff ti Cl i l D i


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Effective Classical Dynamics

This effective classical action gives the following modifiedevolution equations (in the N = 1 gauge)

+ 3 + 12

e6Veff

= 0 ,(24)

2 + 3 2 + 32 + e6 3VeffVeff

= 0 ,(25)

2 + 2 + e6Veff = 0 .(26)

Note that

Veff

= Veff

= 3u

2(2 2)2 1

.(27)


Anal tical Sol tions


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Analytical Solutions

A subset of solutions to the new coupled equations can bestudied analytically by assuming the correlation = at alltimes. The constraint equation is then automatically

satisfied and the other equations reduce to

3 2 + 3u2

e6 = 0(28)

which is easily analysed. Taking < 3/4 as before (that is,u > 0) gives

a amin = exp(C

3u )(29)

Such universes have a minimum nonzero size, whichdepends on the initial conditions that fix C, and hence the

Big Bang singularity is avoided, being replaced by abounce.


Numerical Study


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Numerical Study

We studied more general initial conditions for which = .We keep t0 = 27 and u = 0.01.

The Figures show some examples, in all cases a bounceoccurs at some nonzero size that depends on the initialconditions.



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Figure 1: Results for 0 = log(2.5): The dashed line repre-sents the trajectory of the classical while the solid line corre-


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Figure 2: Results for 0 = log(1.5): The dashed line repre-sents the trajectory of the classical while the solid line corre-

Summary


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Summary

We have used an information-theoretic perspective tomodel the unknown new physics that is generally believedto exist near the Plack scale.

For simplicity we assumed the nonlinearity to be weak.Results are self-consistent to leading order in perturbationtheory.

We found that the nonlinearity could help in avoiding the BigBang singularity in k = 0 FRW models with a cosmological

constant or a free massless scalar field.In the modified dynamics, a classically contracting universeundergoes a bounce.


Outlook


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Outlook

State-dependenceBeyond perturbation theoryInflation

Phenomenology parameter Beyond FRW metricBlack/worm holesCurrent acceleration

Thank you for your attention.


References


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References

A. Vilenkin, Phys. Lett. B117 (1982) 25; D. Atkatz, Am.J. Phys. 62 (7) 1994.

M. Bojowald, Living Rev. Relativity 8, 11 (2005); A.

Ashtekar, arXiv:gr-qc/0702030v2.

Jaynes E.T. Jaynes, Probability Theory, The Logic ofScience (Cambridge University Press, 2004).

M. Reginatto, Phys. Rev. A58, 1775 (1998); B.R.Frieden, Am. J. Phys. 57 (1989) 1004; R. Parwani, J.Phys. A:Math. Gen. 38, 6231 (2005);

R. Parwani, Ann. Phys. 315, 419 (2005). R. Parwaniand O. K. Pashaev, J. Phys. A: Math. Theor. 41 (2008)235207.

L.H. Nguyen, H.S. Tan and R. Parwani,arXiv:0801.0183;


Le-Huy Nguyen and Rajesh R. Parwani- Nonlinear Quantum Cosmology

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