1. Geometrical reconstruction of dark energy Stéphane Fay School of Mathematical Science Queen Mary, University of London, UK steph.fay@gmail.comandLUTH, Paris-Meudon Observatory, France

2. Outline Model Model Data Data What means "reconstructing dark energy" What means "reconstructing dark energy" How to reconstruct dark energy from the data independently on any cosmological models How to reconstruct dark energy from the data independently on any cosmological models

3. The model Flat Universe + baryons + CDM + Dark energy modeled by a perfect fluid p Φ =w Φ ρ Φ with w Φ <-1/3, the equation of state (EOS) Flat Universe + baryons + CDM + Dark energy modeled by a perfect fluid p Φ =w Φ ρ Φ with w Φ <-1/3, the equation of state (EOS) Dark energy is a cosmological constant when w Φ =-1, quintessence when w Φ >-1 and ghost when w Φ -1 and ghost when w Φ <-1. w Φ =-1 is called the LCDM model which is one of the simplest dark energy model fitting the observations. One possible interpretation of such a dark energy One possible interpretation of such a dark energy RG+scalar field defined by the LagrangianRG+scalar field defined by the Lagrangian L=R±Φ μ Φ μ -U+L m with w Φ =(Φ μ Φ μ +U )/(Φ μ Φ μ -U ) L=R±Φ μ Φ μ -U+L m with w Φ =(Φ μ Φ μ +U )/(Φ μ Φ μ -U ) We recover LCDM when Φ μ =0 We recover LCDM when Φ μ =0 In this interpretation, either w Φ =, -1 but it cannot cross the line -1 [Vikman 05] In this interpretation, either w Φ =, -1 but it cannot cross the line -1 [Vikman 05]

4. Data The supernovae data we will consider have been published in 2006 by SNLS. The supernovae data we will consider have been published in 2006 by SNLS. They consist in 115 supernovae at z<1.01 They consist in 115 supernovae at z<1.01 We will also consider the BAO data consisting in the dimensionless quantity A(0.35)=0.469±0.017, where A is defined by We will also consider the BAO data consisting in the dimensionless quantity A(0.35)=0.469±0.017, where A is defined by A(z)=[D m (z) 2 cz H(z) -1 ] 1/3 Ω m 1/2 H 0 (zc) -1 with D m the angular diameter distance

5. Reconstruction: a model independent method One wants to reconstruct the time evolution of some cosmological quantities without specifying any particular EOS but by assuming some very general properties for the data, here the supernovae d l. One wants to reconstruct the time evolution of some cosmological quantities without specifying any particular EOS but by assuming some very general properties for the data, here the supernovae d l. Which cosmological quantities? Which cosmological quantities? The distance luminosity d l related to the magnitude m by d l =10Exp[(m-25)/5]The distance luminosity d l related to the magnitude m by d l =10Exp[(m-25)/5]and The Hubble function H 2 =H 0 2 (Ω m (1+z) 3 + Ω DE ρ Φ /ρ Φ0 )The Hubble function H 2 =H 0 2 (Ω m (1+z) 3 + Ω DE ρ Φ /ρ Φ0 ) The potential U and kinetic term Φ μ Φ μ of the scalar field.The potential U and kinetic term Φ μ Φ μ of the scalar field. The deceleration parameter q, q>0 when the expansion decelerates and q 0 when the expansion decelerates and q<0 when it accelerates The EOS of dark energyThe EOS of dark energy  All these quantities can be expressed with d l and its derivatives

6. Reconstruction: model independent method Which general properties? Which general properties? We proceed by looking for all the d l curves respecting some reasonable geometrical properties and fitting the magnitudes given by SNLS. The properties are as follow:We proceed by looking for all the d l curves respecting some reasonable geometrical properties and fitting the magnitudes given by SNLS. The properties are as follow: (a) d l '>0: true for all expanding Universe (a) d l '>0: true for all expanding Universe (b) d l ''>0: means that the deceleration parameter q 0: means that the deceleration parameter q<1. True for any presently accelerating Universe (q<0) undergoing a transition to an EdS Universe (q=1/2) (c) d l '''<0: true for LCDM and EdS Universe at all redshift. Most of times, these models are considered as describing late and early dynamics of our Universe (c) d l '''<0: true for LCDM and EdS Universe at all redshift. Most of times, these models are considered as describing late and early dynamics of our Universe

7. How to define a d l curve? A d l curve is defined by the interpolation of 8 points: Why 8? A d l curve is defined by the interpolation of 8 points: Why 8? On one hand, if you do not consider enough points, you cannot fit the data with enough precision: the d l curve of the LCDM model needs at least 5 points to be described with enough precision to recover the same as with its analytical form.On one hand, if you do not consider enough points, you cannot fit the data with enough precision: the d l curve of the LCDM model needs at least 5 points to be described with enough precision to recover the same χ 2 as with its analytical form. On the other hand, considering too many points could lead to overfitting. This is not the case here because of the assumptions (a-c) but it would increase the computing time.On the other hand, considering too many points could lead to overfitting. This is not the case here because of the assumptions (a-c) but it would increase the computing time. “8” is a good compromise between the precision required to reconstruct all the curves respecting the assumptions (a-c) and the necessity of a finite time for the calculations!“8” is a good compromise between the precision required to reconstruct all the curves respecting the assumptions (a-c) and the necessity of a finite time for the calculations! “8” does not correspond to the degrees of freedom of the theory thus reconstructed: a straight line may be defined by 8 points although 2 are sufficient and 1 DOF is necessary (y=ax) “8” does not correspond to the degrees of freedom of the theory thus reconstructed: a straight line may be defined by 8 points although 2 are sufficient and 1 DOF is necessary (y=ax) Assuming that the 8 points d li are equidistant in redshift, the properties (a)-(c) will be respected if Assuming that the 8 points d li are equidistant in redshift, the properties (a)-(c) will be respected if d li+1 >d lid li+1 >d li d li+2 -d li+1 >d li+1 -d lid li+2 -d li+1 >d li+1 -d li d li+2 -2d li+1 +d li <d li+3 -2d li+2 +d li+1d li+2 -2d li+1 +d li <d li+3 -2d li+2 +d li+1

8. Testing reconstruction with mock data We take the same distribution and error bars as SNLS but we replace the d l values by the exact values got with a LCDM model. We also add a noise whose level is comparable to the noise of real data. We take the same distribution and error bars as SNLS but we replace the d l values by the exact values got with a LCDM model. We also add a noise whose level is comparable to the noise of real data. If the reconstruction is efficient, we must recover the LCDM model in the 1σ confidence level. If the reconstruction is efficient, we must recover the LCDM model in the 1σ confidence level. Best χ 2 =113.41: the reconstruction is efficient Best χ 2 =113.41: the reconstruction is efficient

9. Reconstruction of d l with real data Best χ 2 =113.85: the ΛCDM model (χ 2 =114) cannot be ruled out at 1σ Note that the best fit corresponds to a dl slightly below the ΛCDM model: slower acceleration than with the ΛCDM model.

10. Reconstruction of H SN data do not constrain H 0 ' Hence to reconstruct H, we will assume: H 0 '>-40, i.e. 9375<d l0 '' <13333 The last condition is equivalent to a lower limit for the EOS, i.e. p Φ0 /ρ Φ0 >-2. The LCDM model is well inside the 1σ contour.

11. Reconstruction of Ω m To reconstruct the other cosmological quantities, we need to know Ω m but the supernovae do not give any information about Ω m because each curve d l is degenerated. Why? To reconstruct the other cosmological quantities, we need to know Ω m but the supernovae do not give any information about Ω m because each curve d l is degenerated. Why? Let’s take the curve d l representing a constant EOS for dark energy defined by w Φ =Γ-1 with the Hubble functionLet’s take the curve d l representing a constant EOS for dark energy defined by w Φ =Γ-1 with the Hubble function H 2 /H 2 0 = Ω m (1+z) 3 + Ω Φ (1+z) 3Γ. H 2 /H 2 0 = Ω m (1+z) 3 + Ω Φ (1+z) 3Γ. Now rewrite the Hubble function asNow rewrite the Hubble function as H 2 /H 2 0 = Ω m1 (1+z) 3 + Ω m2 (1+z) 3 + Ω Φ (1+z) 3Γ with Ω m1 +Ω m2 =Ω m. H 2 /H 2 0 = Ω m1 (1+z) 3 + Ω m2 (1+z) 3 + Ω Φ (1+z) 3Γ with Ω m1 +Ω m2 =Ω m. It represents the same d l but can mimic a new DE with the Hubble function It represents the same d l but can mimic a new DE with the Hubble function H 2 /H 2 0 = Ω m1 (1+z) 3 + Ω Φ1 ρ Φ (z)/ρ Φ (0) H 2 /H 2 0 = Ω m1 (1+z) 3 + Ω Φ1 ρ Φ (z)/ρ Φ (0) and the varying EOS w Φ =(A(1+z) 2 +Γ B(1+z) 3Γ-1 )/(A(1+z) 2 +B(1+z) 3Γ-1 )-1 Hence -a same d l can model several DE theories with different values of Ω m. -A constant EOS can misleadingly becomes time dependant if Ω m is incorrectly chosen [Shafieloo06]

12. Reconstruction of Ω m To get some information about the best fitting value of Ω m we use the BAO. Then: To get some information about the best fitting value of Ω m we use the BAO. Then: 0.16 < Ω m < 0.41 for the set of theories fitting SN+BAO0.16 < Ω m < 0.41 for the set of theories fitting SN+BAO The best fit is got when Ω m = 0.27The best fit is got when Ω m = 0.27 So we are now assuming that Ω m = 0.27So we are now assuming that Ω m = 0.27

13. Reconstruction of dΦ/dt and U We assume a positive potential and kinetic term, i.e. a quintessent dark energy. In the context of a potitive potential and kinetic term, the reconstructed dark energy is very very closed from a LCDM model at 1σ, at least until z=0.6

14. Reconstruction of the EOS The LCDM model is well within the 1σ level but some large degeneracy occurs for large redshift. The LCDM model is well within the 1σ level but some large degeneracy occurs for large redshift. Deceleration begins at least at z=0.35 but some models with no transition to a decelerated Universe also fit the data. Deceleration begins at least at z=0.35 but some models with no transition to a decelerated Universe also fit the data.

15. To conclude We reconstruct some dark energy properties by imposing some geometrical constraints on d l We reconstruct some dark energy properties by imposing some geometrical constraints on d l The best fitting EOS is a varying one The best fitting EOS is a varying one Universe expands slower than with a LCDM modelUniverse expands slower than with a LCDM model The LCDM model cannot be ruled out at 1σ.The LCDM model cannot be ruled out at 1σ. The best fitting EOS is closed from -1 today, describe a transition at z=0.45 from accelerated to decelerated expansion. In a general way, deceleration begins for z>0.35 The best fitting EOS is closed from -1 today, describe a transition at z=0.45 from accelerated to decelerated expansion. In a general way, deceleration begins for z>0.35 The differences between the best fitting model and the LCDM model could be due to systematic errors in the data such as the The differences between the best fitting model and the LCDM model could be due to systematic errors in the data such as the Malmquist bias. SN data alone do not provide constraints on the Ω m parameter: in particular a constant EOS can misleadingly becomes time dependant if Ω m is incorrectly chosen SN data alone do not provide constraints on the Ω m parameter: in particular a constant EOS can misleadingly becomes time dependant if Ω m is incorrectly chosen Using BAO we get the constrain: 0.14<Ω m <0.48, with the best fit for Ω m =0.27 Using BAO we get the constrain: 0.14<Ω m <0.48, with the best fit for Ω m =0.27