H. K. Jassal IISER Mohali Ftag 2013, IIT Gandhinagar
Outline 1 Overview Present day Observations Constraints on cosmological parameters 2 Theoretical Issues Clustering dark energy Integrated Sachs Wolfe effect 3 Summary
Present day Observations Introduction Cosmological equations: = 8πG 3 ρ ȧ 2 a 2 ä a = 4πG 3 (ρ + 3P) ρ = ρ DM + ρ B + ρ RAD + ρ DE Acceleration for P < 1 3ρ dark energy Figure courtesy WMAP team.
Constraints on cosmological parameters Various Models of Dark Energy Cosmological constant - Simplest explanation for dark energy Fluid Models: Barotropic Fluid; Chaplygin gas Scalar Fields Quintessence, phantom fields, K-essence, tachyon Various other models proposed DE Models
Constraints on cosmological parameters Parameterisation of DE Equation of state A nearly model independent analysis (for background cosmology) - writing the evolution of dark energy equation of state in a parameterised form Two crucial parameters Present day value of w Its derivative with respect to redshift
Constraints on cosmological parameters Effects of Dark Energy Dark energy affects rate of expansion of the universe. Affects luminosity distance and angular diameter distance The rate of expansion influences growth of density perturbations in the universe. Abundance of rich clusters of galaxies, their evolution, Integrated Sachs Wolfe effect
Constraints on cosmological parameters UNION, WMAP and BAO data constraints
Constraints on cosmological parameters SNLS, WMAP5 and BAO data
Given a scalar field as a function it is always possible to construct scalar field potentials of different classes Distance measurements are insufficient to distinguish models with the same background evolution To break degeneracy of cosmological models, one needs to look for other methods How structures evolve in the presence of dark energy especially if dark energy contributes to structure formation
Clustering dark energy Gravitational instability The smallness of CMB anisotropies suggests that the density inhomogeities at that redshift must have had very small amplitudes The amplitude of density perturbations today are much larger Universe became more inhomogeneous in the course of its evolution
Clustering dark energy Gravitational instability Density constrast δ(r, t) ρ(r, t) ρ(t) ρ(t) δ > 0 - Overdense region Density constrast in this region will increase This effect of gravitational instability leads to increase in density fluctuations evolution of structure in the universe
Clustering dark energy Perturbations in Dark Energy If Dark energy is not a cosmological constant has to cluster. Since dark energy alters the way structures grow, it can provide an observational signature apart from geometric effects Both fluid and scalar field model background evolution equivalently, this is not the case with perturbations
Clustering dark energy Synchronous gauge ds 2 = dt 2 a 2 (t) [(δ αβ + h αβ dx α )dx β] Newtonian gauge ds 2 = (1 + 2Φ)dt 2 a 2 (t) [(1 2Φ)δ αβ dx α dx β] The scalar Φ in Newtonian limit is the gravitational potential Gauge invariant quantity
Clustering dark energy Perturbation Equations ( 3ȧ Φ + ȧ ) a a Φ + k 2 a 2 Φ = 4πG [ρ NRδ NR + ρ DE δ DE ] Φ + ȧ a Φ = 4πG ρ A v A Φ + 4ȧ a Φ + 2ä a Φ + ȧ2 a 2 Φ = 4πGδP
Clustering dark energy Perturbations in scalar field dark energy Φ + 4ȧ a Φ + 2ä a Φ + ȧ2 [ ] a 2 Φ = 4πG ϕ δϕ Φ ϕ 2 V (ϕ)δϕ δϕ + 3ȧ δϕ a + k 2 δϕ a 2 + 2ΦV (ϕ) 4 Φ ϕ + V (ϕ)δϕ = 0 Initial conditions: Φ/Φ in = 1 Φ = 0 δϕ = 0
Clustering dark energy Perturbations in scalar field dark energy V (ϕ) = V o exp [ λ ϕ M P ] S. Unnikrishnan, HKJ, T. R. Seshadri, Phys. Rev. D 08
Clustering dark energy Fluid model Φ + 4ȧ a Φ = 4πGρ DE (c 2 S δ DE + 2wΦ) [ δ DE +(2+3cS 2 6w)ȧ a δ 9ȧ2 ] k 2 DE 3ä a a 2 a 2 (c2 S w)δ DE +cs 2 k 2 a 2 δ DE = (1+w) [ k 2 ] Φ + 3(2 3w)ȧ a2 a Φ + 3 Φ
Clustering dark energy The speed of propagation of perturbations cs 2 δp/δρ is negative As soon as dark energy begins to dominate, pressure assists gravity and the gravitational potential begins to grow Assumption: cs 2 is positive (for quintessence scalar field it is unity)
Clustering dark energy Reconstructed Potential For a constant w: dϕ da = V (a) = 3 2 (1 w)h2 0 M2 P Ω DE a 3(1+w) 3(1 + w)ω DE M 2 P a (5+3w)/2 Ω NR a 3 + Ω DE a 3(1+w) In the dark energy dominated era, the potential is exponential.
Clustering dark energy A comparison HKJ, Phys. Rev. D 09
Clustering dark energy Fluid Vs scalar field Background cosmology modeled well by fluid as well as scalar fields This is not the case if dark energy is perturbed Scalar field: Enhanced matter perturbations as compared to the cosmological constant and suppressed for fluid model Pressure gradients evolve differently in these two cases
Clustering dark energy Distinct classes of scalar field models What about scalar fields in the same class (say canonical scalar fields) For thawing behavior: w = 1 at early time and w > 1 at late times V (ϕ) = V 0 exp( λϕ/m P ). V (ϕ) = V 0 ϕ n. For freezing behavior: w > 1 at early time and w = 1 at late times V (ϕ) = V 0 ϕ n V (ϕ) = V 0 ϕ n exp(αϕ 2 /M 2 P ) Linder, 06
Clustering dark energy Scalar field potentials : A comparison Length scale: λ = 10 5 Mpc HKJ, Phys. Rev. D 10
Integrated Sachs Wolfe effect Integrated Sachs Wolfe Effect Results from the late time decay of gravitational potential fluctuations T dφ (ˆn) = 2 T ISW dz (ˆn)dz Angular correlation : C l = 2 π C l Vs l C(θ) = C l P l [cos(θ)] l=2 dφ(k) k 2 dkp(k)iq(k), 2 I q (k) = 2 dz j l[kχ(z)]dz
Integrated Sachs Wolfe effect Integrated Sachs Wolfe effect HKJ, Phys. Rev. D 12
Integrated Sachs Wolfe effect Integrated Sachs Wolfe Effect HKJ, Phys. Rev. D 12
Summary Understanding the nature of dark energy is amongst the most important questions, one needs to distinguish between a plethora of models using observations Combination of different observations required to constrain dark energy parameters Pure distance measurements do not suffice
Summary Dark energy clustering affects growth rate of dark matter perturbations The effect is more pronounced at large scales affect ISW (for canonical scalar fields) Scalar tensor gravtity models shows large effects even at small scales. Growth of perturbations at large scales depends on the details of the model. More observations needed to constrain models and rule out some of them.
Cosmological Constraints Back to
Back to Cosmological Constraints Back to ISW
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Observations give us the apparent magnitude of a supernova. [ ] d obs L (z) m obs = M + 5.0 log 10pc dl th (z) = ch 1 0 f (Ω i; z) m th = M+log ( ) ch 1 0 +25+log (f (Ω i ; z)) = M+25+log (f (Ω i ; z)) Mpc with M and Ω i as unknown parameters.
Goodness of fit is measured using χ 2 : χ 2 (M, Ω i ) = N (m obs m th ) 2 σ 2 j=1 m The model with the least χ 2 is the best fit. Models with χ 2 χ 2 min + m are allowed. m depends on the number of unknown parameters and the confidence level. Back to SN
Cosmological constant Cosmological constant : = 8πG 3 (ρ NR + ρ RAD ) + Λ 3 K a 2 ȧ 2 a 2 ä a = 4πG 3 (ρ + 3P) + Λ 3 Arises as energy density of the vacuum Fine tuning problem Theoretical value of vacuum energy density 10 121 orders of magnitude larger than what observations require For cosmological constant: w = 1 Back to DE Models
Barotropic Fluid Barotropic fluid : P = wρ If w is a constant ρ fluid = ρ 0 a 3(1+w fluid ) Acceleration for w < 1/3 In general, w can be a function of redshift (time) Back to DE Models
Quintessence Scalar field minimally coupled to gravity S = d 4 x [ ] 1 g 2 iϕ i ϕ + V (ϕ) An example: V (ϕ) = V 0 ϕ n Scalar field equation : ϕ + 3H ϕ + V,ϕ = 0 ρ = 1 2 ϕ2 + V (ϕ) P = 1 2 ϕ2 V (ϕ) Acceleration for ϕ 2 < V (ϕ) Back to DE Models
K-essence Non canonical scalar field Pressure density as action S = d 4 x gp(ϕ, i ϕ i ϕ) Motivation: Low energy effective action of string theory Equation of state governed by kinetic energy Constraint on kinetic energy if scalar field is to drive acceleration Back to DE Models
Tachyon Arise as decay modes of D-branes Action S = d 4 xv (ϕ) 1 i ϕ i ϕ Pressure P = V (ϕ) 1 ϕ 2 and energy density ρ = V (ϕ) ; w = ϕ2 1 1 ϕ 2 Tachyon behaves as dark energy at low redshifts and is dustlike at higher redshifts J. S. Bagla, HKJ, T. Padmanabhan, PRD 03 (used tachyon as dark energy) Back to DE Models
Phantom Equation of state w < 1 Scalar field with a negative kinetic energy Can also arise in braneworld models and in Brans-Dicke type gravity Action: [ ] g S = d 4 1 x 2 iϕ i φ V (ϕ) Pressure P = 1 2 ϕ2 + V (ϕ) and energy density ρ = 1 2 ϕ2 V (ϕ) Big Rip in finite time in phantom dominated universe Back to DE Models
Chaplygin Gas Fluid as dark energy P = A ρ α Same as tachyon if potential is a constant Behaves as pressureless gas at early times and as cosmological constant at late times Back to DE Models
Parameterisations Equation of state w P/ρ Taylor series expansion around redshift w(z) = w 0 + w (z = 0)z Taylor series expansion around scale factor w(z) = w 0 + w (z = 0) z 1+z Modified parameterisation w(z) = w 0 + w z (z = 0) (1 + z) 2 HKJ, J. S. Bagla and T. Padmanabhan, MNRAS Letters 05 Kink w(a) = w 0 + (w m w 0 )Γ(a, a t, ) Logarithmic expansion w(z) = w 0 + w n [log(1 + z)] n Back to w(z)