Physics Beyond the Standard Model FIRST RIO-SACLAY MEETING ΛCDM And Some of Its Variants: An Overview "A scientist commonly professes to base his beliefs on observations, ons, not theories. Theories, it is said, are useful in suggesting new ideas and new lines of investigation for the experimenter; but "hard facts" are the only proper ground for conclusions. I have never come across anyone who carries this profession into practice -- certainly not the hard-headed headed experimentalist, who is the more swayed by his theories because he is less accustomed to scrutinise them. Observation is not sufficient. We do not believe our eyes unless we are first convinced that what they appear to tell us is credible". Arthur Eddington in the Expanding Universe" Instituto de Física Universidade Federal do Rio de Janeiro Ioav Waga December 4-15, 006
ΛCDM is in excellent accordance with current cosmological observations. What motivates us to look for alternatives?
The cosmological constant problems I-Why is the vacuum energy density so small? Watson - "Is there any other point to which you would wish to draw my attention?" Holmes - "To the curious incident of the dog in the night-time." Watson - But Holmes, the dog did nothing in the night-time! " Holmes - "That was the curious incident." Sherlock Holmes in "The Adventure of Silver Blazes" ( ) 10 ( ) Why ρ obs estim vac 10 ρvac? Theory and observations differ by 10 orders of magnitud!!!
The cosmological constant problem - I vacuum minimum energy state. Gliner 1965 T = Lorentz invariance vac vac ρ g µν µν perfect fluid T = ( + p) u u pg µν µ ν µν ρ We can consider the vacuum as a perfect fluid with the following equation of state: p vac = ρ vac EE take the form: G = 8πGT + 8πGρ g +Λg µν µν µν µν m vac µν µν = 8 πgt +Λ g. Λ m eff eff =Λ+ 8πG ρ vac 4
The cosmological constant problem - I Outside GRT vacuum energy is irrelevant. In quantum mechanics we have zero-point energy (uncertainty principle) Zero-point energy of fields contribute to the dark energy density. See (Adler, Casey & Jacob; Am. J. Phys., 63, july 95 and Zel dovich, Sov. Phys. 4, 16, 81 ) 1 ρvac = dkk k m ultraviolet divergence ρ k 4π + 0 4 19 k max 9 3 74 4 if kmax 10 Gev ρ vac 10 g / cm 10 GeV 16π max 4 vac j ( 1) (j + 1) 0 supersymmetry ρvac = dkk k + m 4π 18 3 if k 1Tev ρ 10 g / cm vac -9 3-47 4 ρ c0 =1.879 h 10 g/cm = 8.099 h 10 GeV Zel dovich 5
The cosmological constant problem - I what local observations can tell us? Λ 0 Advance of the Mercury perihelia precession Λeff ( ) arc sec/ century 4 3 10 cm Λ < 10 Observed precession 43.1±0.1 arcsec/century GRT 4. 98 arcsec/century eff 4 1 eff 10 Λ cm 10 cm 1kpc cm 4 The Newtonian potential would have a significant term in the galactic scale. ρ ρ cluster Λ eff 10 g/ cm cluster 8 3 < ρ Λ < Ω < eff F m 55 10 cm or Λ 10. GM 1 = Λ r 3 cr 6
The cosmological constant problem - I 8 Λ 8πG ρ +Λ 10 eff = vac < < Λ Λ 10 9 Λ Λ QFT eff QFT = 1+ Λ Λ QFT QFT Fine tuning 10 10 (k max = 10 19 Gev) How to deal with this? Λ = Λ QFT = 0 ; (another field act as an effective Λ) Λ = 0 & Λ QFT is small Λ eff = Λ(t) 7
The cosmological constant problems The cosmological constant problem - II Why now? Why do we live at a special time when ρ Λ is comparable to ρ m? nr rad 1 Ω r Ω m curv Λ 0 Ω Λ
The cosmological constant problem - II Cosmic Coincidence Why do we live at a special time when Ω Λ is comparable to Ω m? Λ Dark energy 70% 30% Matter a =1/5 a =1/ a =1 a = a =5 Today For ΛCDM, the why now problem coincides with the initial condition problem (to be discussed).
Some questions we would like to answer What is the nature of the dark energy (DE)? Is it Λ, quintessence or even a more exotic component? How does the DE equation of state evolve? Is w constant or w=w(t)? Is w<-1? Why now? Are dark matter (DM) and DE coupled? Is cosmic acceleration a manifestation of a new gravitation theory? Extra dimensions? Are DM and DE really different substances? Is it possible to have a unified scenario (UDM or quartessence )?
Determining the metric from observations Assuming homogeneity and isotropy ( χ ) k ds = dt a () t d + S dω Sen χ if k =+ 1 Sk = χ if k = 0 Senh χ if k = 1 1 da = () = () ds dt a t dl a t dl H ( a) a ds 1 dz 1+ z H ( z) = dl ( ) Assuming no spatial curvature ( Ω 3 H = H 0 Ω m0(1 + z) + (1 Ωm0) e k =0) and GR. z dz 3 ( 1 + w( z ) ) 1 + z 0
Determining the metric from observations Comoving Distance: dx dr rz H z H( x) dz z 1 ( ) = = ( ) 0 For k = 0: Cosmological Test Luminosity Distance: d (z)= rz ( )(1 + z) L Sne Ia Angular Diameter Distance: d (z)= A rz ( ) (1+ z) CMB, Radio Galaxies, Gravitational Lensing, Alcock-Paczynski Comoving volume element per unit redshift dv r ( z) and solid angle : f = = dzd Ω H( z) Number Counts of quasars and galaxies Age: t 0 = 0 dx (1 + xh ) ( x)
Linder
What is the nature of the dark energy? Is ΛCDM the best candidate to explain the cosmic acceleration? To explain cosmic acceleration, alternatives to ΛCDM modify Einstein equations in two different ways: a) left side (geomet ry) 1 ( m) R Rg + L ( g ) = 8π G T µν µν µν µν µν b)right side (energy-momentum tensor) 1 ( m) R Rg = 8 πg T + T () φ µν µν µ ν µν
What is the nature of the dark energy? Cosmological Constant (w = -1) XCDM (w = constant ; -1 > w > -1/3 ; Phantom w < -1) Quintessence: Dynamical Scalar Field - Ratra & Peebles 1988; Wetterich 1988 ; Frieman, Hill, Watkins 199 ; Sahni, Feldman & Stebbins 199. Frieman, Hill, Stebbins & Waga 1995 ; Ferreira & Joyce 1997 ; Caldwell et al. 1998 ; Brax & Martin 000 ; Albrecht & Skordis 000. Coupled Quintessence: coupling with dark matter (Amendola 000) K-essence: non-canonical scalar fields (Aramendariz-Picon et al. 001, Malquarti et al. 003) Tachyon : A.Sen, 00 ; G. W. Gibbons, 00 ; Padmanabhan & Choudhury 00 Quartessence Makler et al. 00 ; Reis et al 005 ; K-quartessence: Scherrer 004 Chaplygin Gas: Kamenshchik et al. 001, Bento et al. (00). Extended Quintessence: non-minimal coupling to Gravity (Chiba 1999, Uzan 1999, Perrotta et al. 000, Baccigalupi et al. 000, Faraoni 000, Matarrese, Baccigalupi, Perrotta 004, Perivolaropoulos 005) Brane Cosmology : extra dimensions (Dvali, Gabadadze &Porrati 000 ; Sahni & Sthanov 00) 4 Modified Gravity : M 4 P µ 4 S = d x g. (S. M. Carroll, V. Duvvuri, M. Trodden & M. S. Turner, 004) R + d x g LM R
What is the nature of the dark energy? Cosmological Constant ( 1 ) ( 1 ) H H z z = 3 0 Ω m0 + +Ω Λ0 +Ω k0 + XCDM = wρ ; w = constant p X X w = 1 cosmological constant w = 3 domain walls 1 w = 3 cosmic strings w < 1 phantom w+ ( 1 ) ( 1 ) ( 1 ) H H z z z = 3 3( 1) 0 Ω m0 + +Ω x0 + +Ω k0 + a ρm H =, Ω m =, a ρ ρx k Ω x =, Ω k =, ρ ah cr ρ cr 3H = 8π G cr
Phantom Caldwell, PLB 545, 3 (00) Caldwell, Kamionkowski &Weinberg astro-ph/030506 1 w > 3 H -1 grows more rapidly with time than the scale factor a(t) 1 > w > 1 a(t) grows more rapidl 3 bound systems are stable. ρ + -1 y than H, ( 3 ) decreases, and p w< 1 ( ρ + 3 p) increases bound systems will be dissociated The Universe ends in a "Big Rip" (assuming w=const). 500 at () a ( t t ) t t = a w t t w t t /3 m m m /3(1 + w) m[(1 + ) / m ] > m 400 300 M. Tegmark et al a 00 astro-ph/060863 100 1.5.5 3 3.5 4 t tm
Phantom What mechanisms lead to phantom behavior? 1. Braneworld cosmological models. Scalar-tensor gravity 3. Minimal coupled scalar field with a negative kinetic term 4 1 µν S = d x g g µ φ νφ V ( φ) If the potential has a maximum it is possible to avoid Big Rip. Most of this kind of phantom models are plagued with quantum instabilities. See Carrol et al PRD 68,03509 (003) and Cline et al PRD 70,043543, (004).
Phantom How to cross the phantom divide? (no coupling) Two or more dark energy components, at least one is phantom. Modify General Relativity at large scales. Alam et al ; Mon.Not.Roy.Astron.Soc. 354 (004) 75 Ω = 0.4 m Ω = 0.4 m z w( z) = w0 + w1 1+ z z w( z) = w0 + w1 1+ z Nesseris & Perivolaropoulos astro-ph/061009
What is the nature of the dark energy? Quintessence Models with a single scalar field with a canonical kinetic term Dynamical Scalar Field Cosmologies All models assume Λ is zero 4 R 1 µν S = d x g + g µ φ νφ V( φ) + L 16 π G R Ricci scalar V(φ) scalar field potential L lagrangian density of radiation and nr matter Classical action L ρ p = 1 µ φ φ V ( φ ) = φ + V ( φ ) = φ V ( φ ) φ µ φ φ H 3 π dφ 4π 4π + H + V φ + ρ r = pl pl 3 pl dh ( ) 0 dt m dt m m d φ dφ dv H dt dt d + 3 + = 0 φ Field equations 1 da 8π 1 dφ = = V ( φ ) ρ + + m + ρ r a dt 3m pl dt
Quintessence a V M 4 ) ( φ ) = (1+ cos ) b) V ( φ) = M 4 + α α φ 4 f c) V ( φ ) = M e φ 4 + α α f d) V ( φ) = M φ e φ 4 φ α f e) V ( φ ) = M [( B) + A] e φ 4 f φ ) ( φ ) = (1+ cos ) f V M e g )... φ f φ f PNGB potential Power law potential Exponential potential f Frieman, Hill, Watkins PRD 46, 16, 199 Hill, Schramm, Fry, Comm. Nucl. Part. Phys. 19, 5,1989 Frieman, Hill, Stebbins & Waga PRL 75, 077, 1995 Ratra & Peebles PRD 37, 3406, 1988 Caldwell et al. PRL 80, 158, 1988. Sahni, Feldman & Stebbins Ap J 385, 1, 199. Ferreira & Joyce PRL 79, 4740, 1997. SUGRA potential Brax & Martin PRD 60, 10350 (000) Albrecht & Skordis PRL 84, 076, 000. Dodelson, Kaplinghat & Stewart,PRL 85 (000) 576
Quintessence Effective equation of state parameter w φ () t dv ( φ ) φ + 3H φ + = 0 d φ p dφ V dt φ = = ρ φ dφ + dt V ( φ ) ( φ ) V w w i f 1 0 V Φ = F Thawing Φ f 4 ( ) M cos ) V Φ = M Φ n= 4 n n ( ) ; 1,,3 V V V ( ) Φ = M ( ) Φ = Φ 4+ n n Φ 4 f M e w w 0 1 F 01 0 03 04 05 Freezing i f
PNGB Potential = + 4 φ V( φ) M 1 Cos f V(φ) φ Ω 1 today M φ 3 1/ M h ev '' 3 10 KE.. PE.. V ( φ ) 3H i 0 3H 8π 4 0 m pl 19 f mpl 10 GeV m φ M f 10 33 ev φ m
V(φ) ) = M 4 (1+Cos(φ/ φ/f) ) Sne Ia (f /10 18 )GeV lensing M (h 1/ ev) Waga & Frieman
What is the nature of the dark energy? α 4 M ( φ) = M ( α > 0) Tracking Power-law Potential V φ α + Γ= = There are some potentials that present attractor (tracker) solutions for α the field evolution. The two Tracking, conditions with w Ωφ0 1 and V ( φ) 3 H0 (KE< PE) φ <w B, can be achieved if the potential obeys, 10 VV 4 Imply φ0 mpl and Γ = M 10 > + 1 α mpl (V ) and For such the power-law that Γ is potential constant. there are attractor (tracking) solutions such that wb = 0 MDE TR ρ TR φ 3( wb wφ ) 1 a wb = RDE TR ρ 3 B ρφ So 1) grows with ρb TR TR wφ w B wφ = ( wb + 1) > 0 = + α < + α 1 constant >1 ) during MDE 0 (negative pressure)
There are two main ways the field can achieve tracking 1) ρ ρ ρ 0 cr initial φ initial TR Zlatev et al. PRL 8, 846, 98 Steinhardt et al. PRD 59, 13504, 99 In this case the field remains frozen until ρ φ ~ ρ TR and then the field starts to follow the tracking solution. ) ρ ρ ρ initial initial initial TR φ B In this case the field enters a phase of kinetic energy domination (w φ ~1), ρ φ decreases fast, ρ φ ~ a -6 overshooting the tracking solution, after that the field frozen and when ρ φ ~ ρ TR the field starts to follow the tracking solution.
Tracker solution radiation tracker matter undershooting overshooting ρ Λ Steinhardt et al. PRD59 59,, 13504, 1999. For power-law potentials it is difficult o achieve tracking before matter-radiation equality with w 0 ~ -1.
Scalar field with noncanonical kinetic terms R = + + 16 π G 4 S d x g p( φ, X) Lm 1 da 8π p H X p a dt 3m pl X = = + ρ B a 4π G = [ ρ B(1 + 3 wb) + ρk(1 + 3 wk) ] a 3 p p p p p φ + φ + 3H φ + φ = 0 X X X X φ φ a ρk + 3 ( ρk + pk) = 0 a Quintessence: c s =1 K-essence sk p = V( φ) W( X) w c k k k Chiba, Okabe, Yamaguchi, PRD 6, 03511 (00). Armendariz-Picon, Mukhanov, Steinhardt PRL 85,1 (00) ; PRD 63, 103510, (01) X 1 ( µ = µφ φ ) p = V ( φ ) W ( X ) ( W X ) ρ = V( φ) ε( X) = V( φ) XW ( X) ( ) W( X) = XW ( X) W( X) W ( X) W ( X) = = ε ( X) W ( X) + XW ( X)
w = - 0.8 c s =1 Sound Speed drop in power due to c s (z) c s =0 c s 1 z > 5 = 0 z < 5 k-essence models same w(z) different c s (z) DeDeo et al. astro-ph/030184 Sound speed can produce features in the mass power spectrum
What is the nature of the dark energy? Cosmological Constant w = - 1 Fine tuning XCDM dark energy K-essence dark energy w X = const. w = w(t) Fine tuning Self-adjusting (?) today w X = const. radiation =>matter w Λ = -1 K-essence 1+z = a 0 /a Abramo
During RDE y = X During MDE
astro-ph/030477 p(x).01 + (1 + X) 1/ + 3 10 17 X 3 10 4 X 4 K-atractor K-essence may not solve the coincidence problem basin of attraction is small
Quartessence Are DM and DE really different substances? Is it possible to have a unified scenario?
Quartessence finally found in Bolt Castel, Alexandria Bay, New York. Quartessence is related to STRINGS!!!!
K-quartessence p = V( φ) W( X) k k ( ) ρ = V( φ) XW ( X) W( X) Supose now p = W( X) and such that there is an extremum for p at X 0. Consider the case of small perturbations around X. X = X + ε 0 W ' W"( X ) ε w c s 0 W W"( X ) ε X W = O( ε ) 0 0 0 0 1 Scherrer astro-ph/040316 ; PRL 004 w c k sk W( X) = X W ( X) W( X) W ( X) = W ( X) + XW ( X) a ρk + 3 ( ρk + pk) = 0 a ( W ' + XW") X + 6 HW ' X = 0 (*) dx ( W ' + XW") a + 6 W ' X = 0 da α XW' = ( α integration const.) 6 a Expanding Eq (*) up to first order in ε we get: ε = 3Hε Let us analise how one arrives to the extremum state:
Assume that near X, W(X) can be expanded as: W( X) = W + W ( X X ) substituting in α XW' = 6 a we get: 0 0 0 0 α 4W ( X X0) = 6 a ( X X0) 1 X a a 1 3 1 3 X=X0 1 + ε1( ) ; note that ε1( ) 1 a a ρ = W ( X) W( X) ρ = 4 W X( X X ) W W ( X X ) Constraints and fine tunning 0 0 0 substituting the solution for X we get ρ = W + W X ε + W X ε a a 3 3 a1 a1 0 4 0 1 3 0 1 3 a1 0 WX0 ε1 W0 ρ = W + 4 ; < 0 a c X X 1 a = ε 1 0 1 s 1 3X X0 a Ωm0eff 0.3 W a 0 eq ΩΛ eff 0. 7 < 8 ε 1( ) 10 4 WX a 0 0 a1 < aeq 3 10 a 0 3 3 10
Chaplygin Gas Tachyon A.Sen, J. High Energy Phys 04,48 (00) G. W. Gibbons, PLB, 537,1, (00) Padmanabhan & Choudhury PRD, 66, 081301 (00) Motivation strings theory and D-branes 1 1 ( ) φ φ ( φ) µ L= q V q LQ = µ V µ L= m q φ φ φ 1 LT = V( ) 1 µ Born-Infeld action When φ= φ(t) V ( φ) ρt = 1 φ p = V( φ) 1 φ T w = + φ c = w T 1 0 ; T T 0 µ νφ µ µ ν µ φ+ φ φ+ (log V ) = 0 µ 1+ φ φ φ if φ = φ() t φ + 3 H φ + (log V ) = 0 1 φ φ if V( φ) = M p T T 4 µ 8 M = ρ = constant Chaplygin EOS
Generalized Chaplygin Gas The Chaplygin Gas Kamenshchik et al. PLB 511, 65 (001) Initially it was suggested as an alternative to quintessence. Motivation D-branas, Tachyons The Chaplygin gas as a model of dark matter dark energy unification was presented by Bilic et al. PLB 66, 08130 (00) Bento et al. PRD 66, 043507 (00) - discussed theoretical motivation for GCG - Generalized Chaplygin Gas α 1 1+ α 1+ α α + 1 α µ ( µφ φ) L= A 1 Generalized Chaplygin Gas p = M 4( α + 1) α ρ α = 0 ΛCDM-like α = 1 Chaplygin M ~10 3 ev
Solution to the energy conservation equation dρ da = 3( ρ + p ) a q q q ρ q ρ q0 (1 A) 3( α + 1) a 0 = + a A 1 ( α + 1) Limits: a + = >> a a 0 1 z 1 ρq (Dark matter) 3 a 0 << 1 q = ρq = = a 4 p M cte 1< α 1 0 < A 1 (Dark energy - Cosmological constant) w w q q 0 p q = = ρ A = q = A M ρ 4( α + 1) α + 1 q0 M ρ 4( α + 1) α + 1 q Adiabatic sound speed c qs = p αw ρ = q
Combined Analysis Background Tests Makler, Oliveira & Waga Combined Analysis of SNIa, lensing, FR IIb, and X-ray data Tightest constraints from background observables GCG fluid consistent with data, including pure Chaplygin and ΛCDM
Mass Power Spectra Chaplygin (adiabatic) α = -10-4 α =-10-5 α =0 α =10-4 α =10-5 Sandvik et al astro-ph/01114 Reis et al PRD 003, astro-ph/0306004
Silent Quartessence a h L a δ + 3 ( c w ) δ = (1 + w ) kv + 3w Γ a a a c w v c v k k i si i i i i i i si i i + ( 1 3 si) i = δ i + Γi a 1+ wi 1+ wi a h + h = 1+ 3c 8πGρ a δ 4πGa pγ ( ) L L si i i i a i i Γ = α δ ch ch dδ pch δ pch = 0 = 0 dt Amendola, Waga, Finelli JCAP (005).
Mass Power Spectra Chaplygin & Baryons (non-adiabatic) α = 0.6, 0.3,0,0.3,0.6,1 Γ = 0.18 eff Reis et al.- PRD 003
Important other cases Coupled dark energy Modified gravity Brane World
Coupled Dark Energy 1 1 ( ) L = F( ϕ) R ζ( ϕ) ϕ V ( ϕ) L m 8 π G = 1 After a conformal transformation ( ϕ ) g = F g µν µν The action above reduces to the Einstein frame: 1 1 L = R ( φ) V ( φ) Lm ( φ) where 3 F, ϕ φ G( ϕ)d ϕ ; G( ϕ) + F F, ϕ df dϕ ζ F
Coupled Dark Energy Several quantities in the Einstein frame are related to those in Jordan frame via a = F a ; dt = F dt ρ m p m ρ m = ; p m = F F V V = F In the Jordan frame we have: dρm + 3H( ρm + pm ) = 0 dt In Einstein frame the above equation is writen as dρ F, m ϕ + 3H( ρm + pm) = ( ρm 3pm) φ dt FG
Cosmic Acceleration Without Dark Energy Modified Gravity M P 4 4 S = d x g f R + d x glm ( ). S. M. Carroll et al, PRD 70, 04358 (004) 4 M 4 P µ 4 S = d x g R + d x g L R Chiba PLB 575,1, 003 Dolgov & Kawasaki, PLB 575,1, 003 Soussa & Woodard, GRG 36, 855, 004 M. 1 Model with f( R) is not compatible with R solar system experiments.
f( R) = R 1+ α R H 0 β 1 excluded ΛCDM α=-4.38 β=0 SNeIa, CMB, BAO, LSS. Amarzguioui et al astro-ph/0510519
Brane Cosmology - DGP model H H z z 1 Ω rc = 4rH ( ) 3 = 0 Ω k0(1 + ) + Ω rc + Ω rc +Ω m0(1 + ) c 0 { } k m 3 0 Ω 0 + +Ω 0 + if z 1 H H (1 z) (1 z) we have Ω + Ω + Ω +Ω = 1 k0 rc rc m0 1 Ωm0 if k=0 Ω rc = and Ω rc < 1 ; ( Ω rc <Ωx ) Dvali, Gabadadze &Porrati PLB 485,08, (000)
astro-ph/0611775 Berger, Gao and Marfatia
Growing Factor δρ m δ = ; = ρ m δ a 1 1 3 + 4 ( ln H ) 3 ( ln H + + + ) GΩ m ( a) = 0 w = w +w x 0 a z 1+z Q w0 w a (, ) = ( 0.78,0.3) Linder astro-ph/050763