Examining the Viability of Phantom Dark Energy · 2015. 12. 20. · Overview of Phantom Dark Energy...

Examining the Viability of Phantom Dark Energy

Kevin J. Ludwick

LaGrange College

12/20/15(11:00-11:30)

Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 12/20/15 (11:00-11:30) 1 / 38

Outline

1 Overview of Phantom Dark Energy and its Difficulty

2 Perturbation Theory

3 Constant w Models

4 Non-Constant wφ Models


Overview of Phantom Dark Energy and its Difficulty

Discovery of Dark Energy

High-z Supernova Search Team in 1998, Supernova CosmologyProject in 1999: SNIa spectra

Conclusion: dark energy, responsible for cosmic acceleration

Other evidence: galaxy surveys, late-time integrated Sachs-Wolfeeffect (evidence for the effect of dark energy on superclusters andsupervoids in the CMB)

2011 Nobel Prize: Schmidt, Riess, Perlmutter



Characteristics of Dark Energy

About 68% of our universe is dark energy

Physical intuition of the nature and dynamics of DE lacking

Strange feature: as volume increases (i.e., universe expands), DEdensity decreases at lower rate compared to that of normal matter

DE density can even stay constant or increase as volume increases



http://scienceblogs.com/startswithabang/2009/09/dark matter part i how much ma.phpKevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 12/20/15 (11:00-11:30) 5 / 38


Modeling Dark Energy

Relationship between pressure and density usually assumed to bepi = wiρi

For the cosmological constant (CC) model, wΛ = −1, and this givesconstant DE density

−1 ≤ wDE < −1/3: quintessence dark energy (density decreases orstays constant as universe expands)

wDE < −1: phantom dark energy (density increases as universeexpands; rate of acceleration increases; leads to a rip)



http://www.aip.org/png/2003/200.htmKevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 12/20/15 (11:00-11:30) 7 / 38


Observational Constraints

Planck 2015: w = −1.006± 0.045Planck 2013: w = −1.13+0.13−0.10WMAP9 (CMB+BAO+H0+SNIa): w = −1.084± 0.063Suggestive that dark energy really could be phantom



Scalar Field Dark Energy

Using the flat FLRW metric: ds2 = a2(τ)[−dτ2 + dx idxi ]

S =∫d4x√−g[

R16πG −

12g

µν∇µφ∇νφ− V (φ)]

+ Sm

homogeneous: ρφ =φ̇2

2a2+ V (φ), Pφ =

φ̇2

2a2− V (φ)

wφ ≥ −1 ⇐⇒ ρφ + Pφ ≥ 0 ⇐⇒ KEφ term = φ̇2

2a2≥ 0



Phantom DE: w < −1

w < −1 implies negative KE term:

ρφ =φ̇2

2a2+ V (φ), Pφ =

φ̇2

2a2− V (φ)

wφ < −1 ⇐⇒ ρφ + Pφ < 0 ⇐⇒ KEφ term = φ̇2

2a2< 0

NOT GOOD



As a result, what is usually done for phantom field (wφ < −1): signflip in front of the kinetic term in the action so KEφ term is positive

When this is done, the phantom field is ghostlike: phantom DE candecay to a potentially unlimited number of heavier, more energeticparticles (i.e., gravitons) along with DE particles of negative energy!

Effective field theory may be able to render this instabilityunobservable, but not without great difficulty. But perhaps there’s asimpler way....


Perturbation Theory

Alternative, Accurate Framework

φ(τ)→ φ(τ) + δφ(~x , τ)

weff ≡Pφ + δPφρφ + δρφ

=1

2a2(φ̇2 + 2φ̇ ˙δφ)− (V (φ) + V ′(φ)δφ)

12a2

(φ̇2 + 2φ̇ ˙δφ) + (V (φ) + V ′(φ)δφ)

KEeff =1

2a2(φ̇2 + 2φ̇ ˙δφ)

For wφ < −1, ρφ + Pφ < 0, but still possible for weff ≥ −1:

ρφ + δρφ + Pφ + δPφ ≥ 0 ⇐⇒ KEeff ≥ 0 ⇐⇒ weff ≥ −1


Perturbation Theory

But usually in the perturbative approach, equations hold both at 0thand 1st order. This leads to a problem if each order is represented asa scalar field:

weff ≥ −1 ⇐⇒ 12a2 (φ̇2 + 2φ̇ ˙δφ) ≥ 0 whilewφ < −1 ⇐⇒ φ̇

2

2a < 0 (which can’t be true)


Perturbation Theory

Perhaps only the full perturbed phantom fluid is the true phantom DEfield, Φ(~x , τ):

ρΦ(~x , τ) ≡ ρφ(τ) + δρφ(~x , τ) =Φ̇2

2a2− k2 Φ

2

2+ V (Φ), (1)

PΦ(~x , τ) ≡ Pφ(τ) + δPφ(~x , τ) =Φ̇2

2a2− k2 Φ

2

2− V (Φ), (2)

2KEΦ = ρΦ + PΦ =Φ̇2

a2− k2Φ2, (3)

where the term proportional to k2 is present for a field Φ(~x , τ) that is notspatially homogeneous.So for an apparent value of wDE < −1 as measured by observationalprobes, it may be the case that wΦ ≡ PΦ/ρΦ ≥ −1 and KEΦ ≥ 0,indicative of a viable scalar field theory for phantom dark energy.


Perturbation Theory

Cosmological Perturbation Theory

synchronous gauge: ds2 = a2(τ)

[−dτ2 + (δij + hij )dx idx j

]

hij (~x , τ) =

∫d3ke i

~k·~x{k̂i k̂jh(~k , τ) + (k̂i k̂j −

δij3

)6η(~k , τ)

}Perturbed stress-energy tensor:

T 00 = − (ρ+ δρ),T 0i = (ρ+ P)vi ,

T ij = (ρ+ δP)δij + Σ

ij , Σ

ii = 0.

Solve perturbed Einstein’s equation (1st-order part): δGµν = 8πGδTµνSolve conservation of energy and momentum (1st-order part): δTµν;µ = 0


Perturbation Theory

Condition for Positive KE

ρφ + δρφ + Pφ + δPφ ≥ 0 ⇐⇒ 1 + δφ + wφ + δPφ/ρφ ≥ 0

δ ≡ δρρUsing wφ ≤ −1, c2sφ = 1, Σi j = 0 (no anisotropic stress for DE), andequations from conservation of energy/momentum, this inequalitybecomes:

Hk

[2dVφda

a + Vφ

({−1− 3wφ}+

dwφda

a

1 + wφ

)]≤ −1,

where V ≡ θ/k (θ ≡ ik jvj ), and H ≡ ȧa .

Even with |Vφ| < 1, |V̇φ| = |aHdVφda | < 1, it is mathematically possible for

the inequality to be satisfied. We must solve for Vφ from perturbationequations to determine for certain.


Perturbation Theory

Relevant Length Scales

Type Ia supernovae for DE detection:z ≈ 0.3 to z ≈ 2 → 10−4 Mpc−1 . k . 8× 10−4 Mpc−1

late-time Sachs-Wolfe effect: similarly large scales

An acceptable theory of DE must be valid for at least this range oflarge scales

We use this range of k in our analysis.


Constant w Models

Radiation Era

Adiabatic initial conditions (h = A(kτ)2, k � H), H = 1τ (settingconstants of integration appearing in decaying modes to 0 to make theexpression real):

V radφ ≈ −33/2a3Ac2s k

3

211/2(πρr0)3/2(4 + 3c2s − 6w), k � H

Our k range < H during rad era

Comoving curvature perturbation : R = ±√PR = ±

√As

(k

k?

)ns−1R = −η + H

k2[ḣ + 6η̇] +HV

k

Using Planck’s constraints on As and ns , our k range, and expressions forη, h, and Vrad , we can specify A ∼ ∓10−5.


Constant w Models

Matter Era Condition

h = W (kτ)2 (so W = A), k � H, H = 2τ (setting constants ofintegration appearing in decaying modes to 0 to make the expression real):

Vmattφ ≈33/2c2s Wk

3a3/2

25/2(ρm0π)3/2 (5 + 9c2s − 15w)

Positive KE:

−3W (c2s (1 + 6w)− 5)k2a

4πρm0(5 + 9c2s − 15w)≤ −1

Only satisfied for W < 0

Best chance: max k � H, max a = aDE = ( −Ωm0ΩDE03w )−13w ,

min w ≈ −2 (with constraint on A): ∼ −3 · 10−6

NOT SATISFIED


Constant w Models

KE in Radiation and Matter Eras

NOTE: A > 0 enabled (LHS of ineq) < 0 during rad era,but A < 0 enabled (LHS of ineq) < 0 during matter era

→ Even if we could increase magnitude of A arbitrarily, still couldn’t havepositive KE in both eras.

(However, if w ≥ −1, positive KE in both eras for A > 0 or A < 0)


Constant w Models

DE Era

Exact for any k (setting some constants of integration to 0 to avoidimaginary sol for −2 . w . −1):

VDEφ = S

(2π

3ρDE0

) 13w+1

a−1

Positive KE:

( 8π3 ρDE0)1/2

ka−

1+3w2 [−S

(2π

3ρDE0

) 13w+1

(3 + 3w)a−1] ≤ −1

Only satisfied for S < 0

Best chance: min k , max a, min w ≈ −2S found from matching Vmattφ (aDE ) = V

DEφ (aDE ) for

k ∼ 10−4, − 2 . w . −1→ S ≈ ∓(10−10 − 10−8)For a = 1, LHS ∼ −10−6 (NOT SATISFIED)


Constant w Models

DE Era

Positive KE:

( 8π3 ρDE0)1/2

ka−

1+3w2 [−S

(2π

3ρDE0

) 13w+1

(3 + 3w)a−1] ≤ −1

In fact, it turns out that LHS = δDEφ . So LHS ≤ −1 would breakperturbation assumption |δDEφ | < 1.

NOT SATISFIED


Constant w Models

Isocurvature Perturbations for Constant wφ

Even when including the maximum isocurvature contribution that Planckallows (which increases magnitude of constant of integration),the LHS of the inequality for each era only changes from ∼ −10−6 to∼ −10−5, which is still not ≤ −1.

NOT SATISFIED


Constant w Models

In conclusion, for adiabatic or isocurvature initial conditions, phantom DEas a perfect fluid with constant w in 1st-order FLRW: positive KE term

not possible.

But what about non-constant w?


Non-Constant wφ Models

Little Rip and Pseudo-Rip

For constant wφ, ρφ behaves as a power law in a and lead to a big rip(ρφ →∞ in a finite time).Both of these types below: ρφ increases in a more slowly than apower law.

ρlr = ρDE0

(3α

2ρ1/2DE0

ln a + 1

)2ρpr = ρDE0

( ln[ 1f +1/a + 1b ]ln[ 1f +1 +

1b ]

)swlr and wpr are strictly less than −1 for all a, and they approach −1as a→∞.α, f , b, and s: all chosen to fit supernovae data

[parametrizations taken from P. Frampton, K.L., R. Scherrer, Phys. Rev.D 85, 083001 (2012) and P. Frampton, K.L., R. Scherrer, Phys. Rev. D84, 063003 (2011)]Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 12/20/15 (11:00-11:30) 26 / 38


Recall the condition for positive KE:

Hk

[2dVφda

a + Vφ

({−1− 3wφ}+

dwφda

a

1 + wφ

)]≤ −1

Call LHS ≡ µ.

For the following analysis, adiabatic initial conditions constrained fromPlanck’s data are used. (Isocurvature contributions were also studied in theanalysis, and the behavior in the following plots are basically unchanged.)

No longer analytic solutions: must specify extra initial conditions, beyondthe one that we had constrained with Planck data for constant w



Radiation Era

0.00002 0.00005 0.0001 0.0003a

-25

-20

-15

-10

-5

0

5

Μ

Figure : We plot µ for k = 10−4 Mpc−1 (blue lines) and k = 8× 10−4 Mpc−1 (red lines) with chosen initial conditionsV radφ (10

−5) = 10−2 and V̇ radφ (10−5) = 2× 10−4 during the radiation era for both the little rip (solid lines) and pseudo-rip

(dashed lines) parametrizations. All perturbations are sufficiently small.



Matter Era

0.001 0.005 0.010 0.050 0.100 0.500a

-30

-25

-20

-15

-10

-5

0

Μ

Figure : We plot µ for k = 10−4 Mpc−1 (blue lines) and k = 8× 10−4 Mpc−1 (red lines) with chosen initial conditionsV mattφ (aeq ) = 0.4 and V̇

mattφ (aeq ) = 2× 10

−3 during the matter era for both the little rip (solid lines) and pseudo-rip (dashed

lines) parametrizations. All perturbations are sufficiently small.



Rad and matter eras: Failed to have positive KE for the whole of the eras.

Suppose DE is a phenomenon active only during its domination era andnot before (as we have no observational evidence for DE’s existence beforeits era, strictly speaking). If DE becomes active only during DE era as theresult of, say, some spontaneous symmetry breaking, then there is nonecessary continuity with the DE perturbations we calculated for thematter era.

→ Freedom to choose initial conditions conveniently in what follows.



DE Era

0.7 0.8 0.9 1.0a

-10

-8

-6

-4

-2

Μ

Figure : We plot µ for k = 10−4 Mpc−1 (blue lines) and k = 8× 10−4 Mpc−1 (red lines). We chose V DEφ (aDE ) = 0.6,V̇ DEφ (aDE ) = −3× 10

−4, and V̈ DEφ (aDE ) = −8× 10−8 during the dark energy era for the little rip parametrization (solid

lines) and V DEφ (aDE ) = 0.6, V̇DEφ (aDE ) = −6× 10

−4, and V̈ DEφ (aDE ) = −2× 10−7 for the pseudo-rip parametrization

(dashed lines). All perturbations are sufficiently small. µ < −1 for the whole range of k and a for the psuedo-rip model. But ingeneral, for a > 1, µ becomes > −1.



More accurate: Consider an era dominated by BOTH DM and DE:H = a( 8π3 (ρDE (a) + ρc0 a

−3))1/2.

If DE considered active only from aDE onwards → Freedom to chooseinitial conditions.

But less freedom in choosing initial conditions: have to ensure thesmallness of δc along with the other perturbations



DM-DE Era

0.7 0.8 0.9 1.0a

-1.5

-1.0

-0.5

Μ

Figure : We plot µ for k = 10−4 Mpc−1 (blue lines) and k = 8× 10−4 Mpc−1 (red lines) during the DM-DE era for thelittle rip parametrization (solid lines), pseudo-rip parametrization (dashed lines), and constant wφ = −1.1 (dot-dashed lines) fora ∈ (0.61, 1), where 0.61 is close to aDE for all 3 parametrizations. We chose appropriate initial conditions forV DM−DEφ

(0.61), V̇ DM−DEφ

(0.61), V̈ DM−DEφ

(0.61), and...V

DM−DEφ

(0.61) for each line. All perturbations are sufficiently small.

In this more accurate DM-DE era, none of these (little rip, pseudo-rip, constant w) satisfies positive KE for all k and a.



Constant wφ > −1

Same equations derived for phantom case apply for quintessence case(as long as wφ 6= −1). The condition for positive KE is satisfied forconstant wφ > −1 for either sign of initial conditions forsingle-component eras (with similar magnitude of µ with similar valueof |wφ − 1|).However, for DM-DE era, situation similar to what we had withphantom case for constant wφ: same freedom of initial conditionchoices, but some ranges of k and a for which positive KE is notsatisfied



DM-DE Era (Quintessence)

0.7 0.8 0.9 1.0a

-1.0

-0.8

-0.6

-0.4

-0.2

Μ

Figure : We plot µ for k = 10−4 Mpc−1 (blue line) and k = 8× 10−4 Mpc−1 (red line) during the DM-DE era for constantwφ = −0.99 (dot-dashed lines) for a ∈ (0.61, 1) and for the same initial conditions for the constant wφ cases in previousfigure. All perturbations are sufficiently small. We see that positive kinetic energy is not satisfied for all of the DM-DE era.



Conclusion

Phantom DE: Possible to have positive KE for some relevant k and aranges in 1st-order perturbation theory, but not for all.

Quintessence DE: Possible to have negative KE for some k and all ofa for DM-DE era in 1st-order perturbation theory. We suspect thesame for non-constant parametrizations.

Side note:

Constant wφ = −1: It turns out that the relevant perturbations in theinequality are 0, and it always has positive KE in 1st-order perturbationtheory.



Conclusion

So we see that phantom and quintessence DE may not categorically havepositive and negative KE term, respectively.

If we were to consider more accurately the contributions from allcomponents in H (instead of a particular era of domination) and theirperturbative contributions in the stress-energy tensor, it would be evenmore difficult to find initial conditions giving positive KE for phantom DE(and negative KE for quintessence DE) and make all the perturbationssmall.



More to Explore

Spatial curvature

back-reaction on background

imperfect fluid (viscosity, shear)

Other space-times (Bianchi, Tolman-Bondi, etc)

Coupled DE-DM

Treat as quantum field theory in perturbed space-time (Note: Alreadyin the literature from Kahya, Onemli, Woodward: Ghost behavioravoided for phantom DE in 0th-order FLRW. What happens inperturbed FLRW?)

and more...

Based on K.L., Phys. Rev. D 92, 063019 (2015)(arXiv:1507.06492).


Overview of Phantom Dark Energy and its DifficultyPerturbation TheoryConstant w ModelsNon-Constant w Models

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