Examining the Viability of Phantom Dark Energy

Transcription

1 Examining the Viability of Phantom Dark Energy Kevin J. Ludwick LaGrange College 11/12/16 Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 1 / 28

2 Outline 1 Overview of Phantom Dark Energy and its Difficulty 2 Perturbation Theory 3 Non-Constant w φ Models Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 2 / 28

3 Overview of Phantom Dark Energy and its Difficulty Discovery of Dark Energy High-z Supernova Search Team in 1998, Supernova Cosmology Project in 1999: SNIa spectra Conclusion: dark energy, responsible for cosmic acceleration Other evidence: galaxy surveys, late-time integrated Sachs-Wolfe effect (evidence for the effect of dark energy on superclusters and supervoids in the CMB) 2011 Nobel Prize: Schmidt, Riess, Perlmutter Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 3 / 28

4 Overview of Phantom Dark Energy and its Difficulty 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), DE density decreases at lower rate compared to that of normal matter DE density can even stay constant or increase as volume increases Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 4 / 28

5 Overview of Phantom Dark Energy and its Difficulty Modeling Dark Energy Relationship between pressure and density usually assumed to be p i = w i ρ i For the cosmological constant (CC) model, w Λ = 1, and this gives constant DE density 1 w DE < 1/3: quintessence dark energy (density decreases or stays constant as universe expands) w DE < 1: phantom dark energy (density increases as universe expands; rate of acceleration increases; leads to a rip) Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 5 / 28

6 Overview of Phantom Dark Energy and its Difficulty Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 6 / 28

7 Overview of Phantom Dark Energy and its Difficulty Observational Constraints Planck 2015: w = ± Planck 2013: w = WMAP9 (CMB+BAO+H 0 +SNIa): w = ± Suggestive that dark energy really could be phantom Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 7 / 28

8 Overview of Phantom Dark Energy and its Difficulty Scalar Field Dark Energy Using the flat FLRW metric: ds 2 = a 2 (τ)[ dτ 2 + dx i dx i ] S = d 4 x g [ R 16πG 1 2 g µν µ φ ν φ V (φ) ] + S m φ homogeneous: ρ φ = 2 φ + V (φ), P 2a 2 φ = 2 V (φ) 2a 2 w φ 1 ρ φ + P φ 0 KE φ term = φ 2 2a 2 0 Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 8 / 28

9 Overview of Phantom Dark Energy and its Difficulty Phantom DE: w < 1 Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 9 / 28

10 Overview of Phantom Dark Energy and its Difficulty Phantom DE: w < 1 w < 1 implies negative KE term: φ ρ φ = 2 φ + V (φ), P 2a 2 φ = 2 V (φ) 2a 2 w φ < 1 ρ φ + P φ < 0 KE φ term = φ 2 2a 2 < 0 Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 9 / 28

11 Overview of Phantom Dark Energy and its Difficulty Phantom DE: w < 1 w < 1 implies negative KE term: φ ρ φ = 2 φ + V (φ), P 2a 2 φ = 2 V (φ) 2a 2 w φ < 1 ρ φ + P φ < 0 KE φ term = NOT GOOD φ 2 2a 2 < 0 Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 9 / 28

12 Overview of Phantom Dark Energy and its Difficulty As a result, what is usually done for phantom field (w φ < 1): sign flip in front of the kinetic term in the action so KE φ term is positive When this is done, the phantom field is ghostlike: phantom DE can decay to a potentially unlimited number of heavier, more energetic particles (i.e., gravitons) along with DE particles of negative energy! Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 10 / 28

13 Overview of Phantom Dark Energy and its Difficulty As a result, what is usually done for phantom field (w φ < 1): sign flip in front of the kinetic term in the action so KE φ term is positive When this is done, the phantom field is ghostlike: phantom DE can decay to a potentially unlimited number of heavier, more energetic particles (i.e., gravitons) along with DE particles of negative energy! Effective field theory may be able to render this instability unobservable, but not without great difficulty. But perhaps there s a simpler way... Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 10 / 28

14 Perturbation Theory Alternative, Accurate Framework w eff P φ + δp φ ρ φ + δρ φ = φ(τ) φ(τ) + δφ( x, τ) 1 ( 2a 2 1 ( 2a 2 φ φ δφ) (V (φ) + V (φ)δφ) φ φ δφ) + (V (φ) + V (φ)δφ) KE eff = 1 2a 2 ( φ φ δφ) For w φ < 1, ρ φ + P φ < 0, but still possible for w eff 1: ρ φ + δρ φ + P φ + δp φ 0 KE eff 0 w eff 1 Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 11 / 28

15 Perturbation Theory Perhaps only the full perturbed phantom fluid is the true phantom DE field, Φ( x, τ): ρ Φ ( x, τ) ρ φ (τ) + δρ φ ( x, τ) = Φ 2 P Φ ( x, τ) P φ (τ) + δp φ ( x, τ) = Φ 2 Φ2 k2 2a2 2 Φ2 k2 2a2 2 2KE Φ = ρ Φ + P Φ = Φ 2 + V (Φ), (1) V (Φ), (2) a 2 k2 Φ 2, (3) where the term proportional to k 2 is present for a field Φ( x, τ) that is not spatially homogeneous. So for an apparent value of w DE < 1 as measured by observational probes, it may be the case that w Φ P Φ /ρ Φ 1 and KE Φ 0, indicative of a viable scalar field theory for phantom dark energy. Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 12 / 28

16 Perturbation Theory Cosmological Perturbation Theory ] synchronous gauge: ds 2 = a 2 (τ) [ dτ 2 + (δ ij + h ij )dx i dx j h ij ( x, τ) = { d 3 ke i k x ˆk i ˆk j h( k, τ) + ( ˆk i ˆk j δ } ij 3 )6η( k, τ) Perturbed stress-energy tensor: T 0 0 = (ρ + δρ), T 0 i = (ρ + P)v i, T i j = (P + δp)δ i j + Σ i j, Σ i i = 0. Solve perturbed Einstein s equation (1st-order part): δg µν = 8πGδT µν Solve conservation of energy and momentum (1st-order part): δt µν ;µ = 0 Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 13 / 28

17 Perturbation Theory Relevant Length Scales Type Ia supernovae for DE detection: z 0.3 to z Mpc 1 k Mpc 1 late-time Sachs-Wolfe effect: similarly large scales An acceptable theory of phantom DE must be valid for at least this range of large scales We use this range of k in our analysis. Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 14 / 28

18 Perturbation Theory Long story short: for adiabatic or isocurvature initial conditions, phantom DE as a perfect fluid with constant w in 1st-order FLRW: positive KE term not possible when constrained by data from Planck. Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 15 / 28

19 Perturbation Theory Long story short: for adiabatic or isocurvature initial conditions, phantom DE as a perfect fluid with constant w in 1st-order FLRW: positive KE term not possible when constrained by data from Planck. But what about non-constant w? Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 15 / 28

20 Non-Constant w φ Models Little Rip and Pseudo-Rip For constant w φ, ρ φ behaves as a power law in a and lead to a big rip (ρ φ in a finite time). Both of these types below: ρ φ increases in a more slowly than a power law. ρ lr = ρ DE0 ( 3α 2ρ 1/2 DE0 ) 2 ln a + 1 ( ln[ 1 f +1/a ρ pr = ρ + 1 b ] DE0 1 ln[ f b ] w lr and w pr are strictly less than 1 for all a, and they approach 1 as 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, (2012) and P. Frampton, K.L., R. Scherrer, Phys. Rev. D 84, (2011)] Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 16 / 28 ) s

21 Non-Constant w φ Models In the following plots, the condition for positive KE: µ 1 No longer analytic solutions like we had for constant w φ. Must specify extra initial conditions, beyond the one that we had constrained with Planck data for constant w φ. Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 17 / 28

22 Radiation Era Non-Constant w φ Models a Figure : We plot µ for k = 10 4 Mpc 1 (blue lines) and k = Mpc 1 (red lines) with chosen initial conditions Vφ rad (10 5 ) = 10 2 rad and V φ (10 5 ) = during the radiation era for both the little rip (solid lines) and pseudo-rip (dashed lines) parametrizations. All perturbations are sufficiently small. Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 18 / 28

23 Matter Era Non-Constant w φ Models a Figure : We plot µ for k = 10 4 Mpc 1 (blue lines) and k = Mpc 1 (red lines) with chosen initial conditions Vφ matt matt (aeq) = 0.4 and V φ (aeq) = during the matter era for both the little rip (solid lines) and pseudo-rip (dashed lines) parametrizations. All perturbations are sufficiently small. Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 19 / 28

24 Non-Constant w φ Models 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 and not before (as we have no observational evidence for DE s existence before its era, strictly speaking). If DE becomes active only during DE era as the result of, say, some spontaneous symmetry breaking, then there is no necessary continuity with the DE perturbations we calculated for the matter era. Freedom to choose initial conditions conveniently in what follows. Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 20 / 28

25 DE Era Non-Constant w φ Models a Figure : We plot µ for k = 10 4 Mpc 1 (blue lines) and k = Mpc 1 (red lines). We chose Vφ DE (a DE ) = 0.6, V φ DE (a DE ) = , and V φ DE (a DE ) = during the dark energy era for the little rip parametrization (solid lines) and Vφ DE (a DE DE ) = 0.6, V φ (a DE ) = DE, and V φ (a DE ) = 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 in general, for a > 1, µ becomes > 1. Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 21 / 28

26 Non-Constant w φ Models More accurate: Consider an era dominated by BOTH DM and DE: H = a( 8π 3 (ρ DE (a) + ρ c0 a 3 )) 1/2. If DE considered active only from a DE onwards Freedom to choose initial conditions. But less freedom in choosing initial conditions: have to ensure the smallness of δ c along with the other perturbations Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 22 / 28

27 Non-Constant w φ Models DM-DE Era (Non-Constant and Constant w φ ) a Figure : We plot µ for k = 10 4 Mpc 1 (blue lines) and k = Mpc 1 (red lines) during the DM-DE era for the little rip parametrization (solid lines), pseudo-rip parametrization (dashed lines), and constant w φ = 1.1 (dot-dashed lines) for a (0.61, 1), where 0.61 is close to a DE for all 3 parametrizations. We chose appropriate initial conditions for V DM DE DM DE DM DE (0.61), V (0.61), V (0.61), and... φ φ φ V DM DE (0.61) for each line. All perturbations are sufficiently small. φ In this more accurate DM-DE era, none of these (little rip, pseudo-rip, constant w) satisfies positive KE for all k and a. Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 23 / 28

28 Non-Constant w φ Models Constant w φ > 1: w φ > 1 satisfies positive KE for radiation, matter, and DE eras. However, for DM-DE era, situation similar to what we had with phantom case for constant w φ : same freedom of initial condition choices, but there are some ranges of k and a for which positive KE is not satisfied Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 24 / 28

29 Non-Constant w φ Models DM-DE Era (Quintessence) a Figure : We plot µ for k = 10 4 Mpc 1 (blue line) and k = Mpc 1 (red line) during the DM-DE era for constant w φ = 0.99 (dot-dashed lines) for a (0.61, 1) and for the same initial conditions for the constant w φ cases in previous figure. All perturbations are sufficiently small. Negative KE for quintessence: µ > 1. We see that positive kinetic energy is not satisfied for all of the DM-DE era. Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 25 / 28

30 Conclusion Non-Constant w φ Models Phantom DE: Possible to have positive KE for some relevant k and a ranges in 1st-order perturbation theory, but not for all. Quintessence DE: Possible to have negative KE for some k and all of a for DM-DE era in 1st-order perturbation theory. We suspect the same for non-constant parametrizations. Side note: Constant w φ = 1: It turns out that the relevant perturbations in the inequality µ 1 are 0, and it always has positive KE in 1st-order perturbation theory. Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 26 / 28

31 Conclusion Non-Constant w φ Models So we see that phantom and quintessence DE may not categorically have positive and negative KE term, respectively. If we were to consider more accurately the contributions from all components in H (instead of a particular era of domination) and their perturbative contributions in the stress-energy tensor, it would be even more difficult to find initial conditions giving positive KE for phantom DE (and negative KE for quintessence DE) and make all the perturbations small. Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 27 / 28

32 Non-Constant w φ Models 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 FLRW space-time (in progress) and more... Based on K.L., Phys. Rev. D 92, (2015) (arxiv: ). Kevin J. Ludwick (LaGrange College) Examining the Viability of Phantom Dark Energy 11/12/16 28 / 28