e
Test de la param´trisation CPL pour contraindre
l’energie noire
Sebastian Linden
(en collaboration avec Jean-Marc Virey)
e
Centre de Physique Th´orique
November 20, 2007
Sebastian Linden (en collaboration avec Jean-Marc Virey) (Centre de Physique Th´orique)
e e
Test de la param´trisation CPL pour contraindre l’energie noire November 20, 2007 1 / 14
Content
1 Parametrisation of Dark Energy’s Equation of State
Model Independence
Test of CPL
2 Approach
Fitting Procedure
3 Results
Two examples
Confusion with ΛCDM possible?
Sebastian Linden (en collaboration avec Jean-Marc Virey) (Centre de Physique Th´orique)
e e
Test de la param´trisation CPL pour contraindre l’energie noire November 20, 2007 2 / 14
Parametrisation of Dark Energy’s Equation of State Model Independence
Model Independence
Open landscape of theories proposed to explain the present accelerated
expansion of the universe. Modify Friedmann equation or add a new
component.
Direct reconstruction of dark energy’s Hubble function and/or it’s equation of
state is stymied by statistical and systematic errors of the obervations.
p
→ Parametrize w (z) := ρ (z) to extract maximal information about it’s
present value w0 and, where possible, it’s time evolution w .
Using statistics of a SNAP-like mission Linder and Huterer (2005) showed
that maximally two parameters can be usefully constrained.
Sebastian Linden (en collaboration avec Jean-Marc Virey) (Centre de Physique Th´orique)
e e
Test de la param´trisation CPL pour contraindre l’energie noire November 20, 2007 3 / 14
Parametrisation of Dark Energy’s Equation of State Model Independence
Model Independence
Open landscape of theories proposed to explain the present accelerated
expansion of the universe. Modify Friedmann equation or add a new
component.
Direct reconstruction of dark energy’s Hubble function and/or it’s equation of
state is stymied by statistical and systematic errors of the obervations.
p
→ Parametrize w (z) := ρ (z) to extract maximal information about it’s
present value w0 and, where possible, it’s time evolution w .
Using statistics of a SNAP-like mission Linder and Huterer (2005) showed
that maximally two parameters can be usefully constrained.
Sebastian Linden (en collaboration avec Jean-Marc Virey) (Centre de Physique Th´orique)
e e
Test de la param´trisation CPL pour contraindre l’energie noire November 20, 2007 3 / 14
Parametrisation of Dark Energy’s Equation of State Model Independence
Model Independence
Open landscape of theories proposed to explain the present accelerated
expansion of the universe. Modify Friedmann equation or add a new
component.
Direct reconstruction of dark energy’s Hubble function and/or it’s equation of
state is stymied by statistical and systematic errors of the obervations.
p
→ Parametrize w (z) := ρ (z) to extract maximal information about it’s
present value w0 and, where possible, it’s time evolution w .
Using statistics of a SNAP-like mission Linder and Huterer (2005) showed
that maximally two parameters can be usefully constrained.
Sebastian Linden (en collaboration avec Jean-Marc Virey) (Centre de Physique Th´orique)
e e
Test de la param´trisation CPL pour contraindre l’energie noire November 20, 2007 3 / 14
Parametrisation of Dark Energy’s Equation of State Model Independence
Model Independence
Open landscape of theories proposed to explain the present accelerated
expansion of the universe. Modify Friedmann equation or add a new
component.
Direct reconstruction of dark energy’s Hubble function and/or it’s equation of
state is stymied by statistical and systematic errors of the obervations.
p
→ Parametrize w (z) := ρ (z) to extract maximal information about it’s
present value w0 and, where possible, it’s time evolution w .
Using statistics of a SNAP-like mission Linder and Huterer (2005) showed
that maximally two parameters can be usefully constrained.
Sebastian Linden (en collaboration avec Jean-Marc Virey) (Centre de Physique Th´orique)
e e
Test de la param´trisation CPL pour contraindre l’energie noire November 20, 2007 3 / 14
Parametrisation of Dark Energy’s Equation of State Model Independence
CPL-Parametrization
Chevallier, Polarski and Linder’s two-parametric parametrization of dark
energy’s equation of state,
z
w (z) = w0 + wa 1+z ,
shows bounded behaviour, covers slow time evolution and has a simple
physical interpretation.
It’s logarithmic derivate is
proportional to w (a):
dw (a)
d log a = −(w0 + wa ) + w (a).
Sebastian Linden (en collaboration avec Jean-Marc Virey) (Centre de Physique Th´orique)
e e
Test de la param´trisation CPL pour contraindre l’energie noire November 20, 2007 4 / 14
Parametrisation of Dark Energy’s Equation of State Model Independence
CPL-Parametrization
Chevallier, Polarski and Linder’s two-parametric parametrization of dark
energy’s equation of state,
z
w (z) = w0 + wa 1+z ,
shows bounded behaviour, covers slow time evolution and has a simple
physical interpretation.
It’s logarithmic derivate is
proportional to w (a):
dw (a)
d log a = −(w0 + wa ) + w (a).
Sebastian Linden (en collaboration avec Jean-Marc Virey) (Centre de Physique Th´orique)
e e
Test de la param´trisation CPL pour contraindre l’energie noire November 20, 2007 4 / 14
Parametrisation of Dark Energy’s Equation of State Model Independence
But what happens, if dark energy’s equation of state is not at all like CPL?
Neither slowly evolving, nor linear in (w -w )–space?
Two extreme cases are principally possible:
1 Oscillating w (z)
2 Sharp, abrupt transition in w (z)
We study the second case: we test CPL with a step-like fiducial equation of
state.
Sebastian Linden (en collaboration avec Jean-Marc Virey) (Centre de Physique Th´orique)
e e
Test de la param´trisation CPL pour contraindre l’energie noire November 20, 2007 5 / 14
Parametrisation of Dark Energy’s Equation of State Model Independence
But what happens, if dark energy’s equation of state is not at all like CPL?
Neither slowly evolving, nor linear in (w -w )–space?
Two extreme cases are principally possible:
1 Oscillating w (z)
2 Sharp, abrupt transition in w (z)
We study the second case: we test CPL with a step-like fiducial equation of
state.
Sebastian Linden (en collaboration avec Jean-Marc Virey) (Centre de Physique Th´orique)
e e
Test de la param´trisation CPL pour contraindre l’energie noire November 20, 2007 5 / 14
Parametrisation of Dark Energy’s Equation of State Model Independence
But what happens, if dark energy’s equation of state is not at all like CPL?
Neither slowly evolving, nor linear in (w -w )–space?
Two extreme cases are principally possible:
1 Oscillating w (z)
2 Sharp, abrupt transition in w (z)
We study the second case: we test CPL with a step-like fiducial equation of
state.
Sebastian Linden (en collaboration avec Jean-Marc Virey) (Centre de Physique Th´orique)
e e
Test de la param´trisation CPL pour contraindre l’energie noire November 20, 2007 5 / 14
Parametrisation of Dark Energy’s Equation of State Test of CPL
Test of CPL(1): Why steplike?
A subclass of quintessence models
predict transition from some initial
value wi to wf = −1.
Sebastian Linden (en collaboration avec Jean-Marc Virey) (Centre de Physique Th´orique)
e e
Test de la param´trisation CPL pour contraindre l’energie noire November 20, 2007 6 / 14
Parametrisation of Dark Energy’s Equation of State Test of CPL
Test of CPL(2): tanh-like step.
We choose an integrable step-like functional form:
Steplike EoS
1+zt
w (z) = 1 (wi + wf ) − 1 (wi − wf ) tanh Γ log
2 2 1+z
dw (a) wf w 2 (a)
One also quickly calculates d log a = 2Γ 2 + ∆w − w (a) .
Sebastian Linden (en collaboration avec Jean-Marc Virey) (Centre de Physique Th´orique)
e e
Test de la param´trisation CPL pour contraindre l’energie noire November 20, 2007 7 / 14
Parametrisation of Dark Energy’s Equation of State Test of CPL
Test of CPL(2): tanh-like step.
We choose an integrable step-like functional form:
Steplike EoS
1+zt
w (z) = 1 (wi + wf ) − 1 (wi − wf ) tanh Γ log
2 2 1+z
dw (a) wf w 2 (a)
One also quickly calculates d log a = 2Γ 2 + ∆w − w (a) .
Sebastian Linden (en collaboration avec Jean-Marc Virey) (Centre de Physique Th´orique)
e e
Test de la param´trisation CPL pour contraindre l’energie noire November 20, 2007 7 / 14
Approach Fitting Procedure
Fitting Procedure (1)
We fit a fiducial steplike model with CPL’s parametrization.
What does that mean?
We act as if dark energy’s equation of state would be step-like, simulate the
Supernova Distribution (Luminosity-Distance-Relation) that would result, and
especially assume the systematics of a SNAP-like mission, designed to
measure these SNe.
We combine with the CMB shift-parameter R = ΩM × dA (z = 1089) and the
2
1 1
BAO reduced variable A = 1 0.35 dA (z = 0.35) 3
that would follow.
H(z=0.35) 3
Then we try to fit our so obtained data with CPL and take a look at possible
biases introduced by this procedure.
Sebastian Linden (en collaboration avec Jean-Marc Virey) (Centre de Physique Th´orique)
e e
Test de la param´trisation CPL pour contraindre l’energie noire November 20, 2007 8 / 14
Approach Fitting Procedure
Fitting Procedure (1)
We fit a fiducial steplike model with CPL’s parametrization.
What does that mean?
We act as if dark energy’s equation of state would be step-like, simulate the
Supernova Distribution (Luminosity-Distance-Relation) that would result, and
especially assume the systematics of a SNAP-like mission, designed to
measure these SNe.
We combine with the CMB shift-parameter R = ΩM × dA (z = 1089) and the
2
1 1
BAO reduced variable A = 1 0.35 dA (z = 0.35) 3
that would follow.
H(z=0.35) 3
Then we try to fit our so obtained data with CPL and take a look at possible
biases introduced by this procedure.
Sebastian Linden (en collaboration avec Jean-Marc Virey) (Centre de Physique Th´orique)
e e
Test de la param´trisation CPL pour contraindre l’energie noire November 20, 2007 8 / 14
Approach Fitting Procedure
Fitting Procedure (1)
We fit a fiducial steplike model with CPL’s parametrization.
What does that mean?
We act as if dark energy’s equation of state would be step-like, simulate the
Supernova Distribution (Luminosity-Distance-Relation) that would result, and
especially assume the systematics of a SNAP-like mission, designed to
measure these SNe.
We combine with the CMB shift-parameter R = ΩM × dA (z = 1089) and the
2
1 1
BAO reduced variable A = 1 0.35 dA (z = 0.35) 3
that would follow.
H(z=0.35) 3
Then we try to fit our so obtained data with CPL and take a look at possible
biases introduced by this procedure.
Sebastian Linden (en collaboration avec Jean-Marc Virey) (Centre de Physique Th´orique)
e e
Test de la param´trisation CPL pour contraindre l’energie noire November 20, 2007 8 / 14
Approach Fitting Procedure
Fitting Procedure (1)
We fit a fiducial steplike model with CPL’s parametrization.
What does that mean?
We act as if dark energy’s equation of state would be step-like, simulate the
Supernova Distribution (Luminosity-Distance-Relation) that would result, and
especially assume the systematics of a SNAP-like mission, designed to
measure these SNe.
We combine with the CMB shift-parameter R = ΩM × dA (z = 1089) and the
2
1 1
BAO reduced variable A = 1 0.35 dA (z = 0.35) 3
that would follow.
H(z=0.35) 3
Then we try to fit our so obtained data with CPL and take a look at possible
biases introduced by this procedure.
Sebastian Linden (en collaboration avec Jean-Marc Virey) (Centre de Physique Th´orique)
e e
Test de la param´trisation CPL pour contraindre l’energie noire November 20, 2007 8 / 14
Approach Fitting Procedure
Fitting Procedure (1)
We fit a fiducial steplike model with CPL’s parametrization.
What does that mean?
We act as if dark energy’s equation of state would be step-like, simulate the
Supernova Distribution (Luminosity-Distance-Relation) that would result, and
especially assume the systematics of a SNAP-like mission, designed to
measure these SNe.
We combine with the CMB shift-parameter R = ΩM × dA (z = 1089) and the
2
1 1
BAO reduced variable A = 1 0.35 dA (z = 0.35) 3
that would follow.
H(z=0.35) 3
Then we try to fit our so obtained data with CPL and take a look at possible
biases introduced by this procedure.
Sebastian Linden (en collaboration avec Jean-Marc Virey) (Centre de Physique Th´orique)
e e
Test de la param´trisation CPL pour contraindre l’energie noire November 20, 2007 8 / 14
Approach Fitting Procedure
Fitting Procedure (2)
We consider the reconstruction of:
Fitted value Fiducial value /
w0 w f (0) Present value
w0 + wa wif Inital value
w (zp ) w f (zp ) Pivot value (at z = zp )
f
weff weff Effective value
CMB
R
wx (z)Ωx (z)dz
0
weff := CMB
R
Ωx (z)dz
0
σw 0
zp = −Cw0 wa σwa +Cw0 wa σw0 is the redshift with minimal errors on w fit (z).
Sebastian Linden (en collaboration avec Jean-Marc Virey) (Centre de Physique Th´orique)
e e
Test de la param´trisation CPL pour contraindre l’energie noire November 20, 2007 9 / 14
Results Two examples
Two examples (1)
SNAP(+Nearby)+R+A SNAP(+Nearby)+R+A
0,2 zt=0,5 0,2 zt=0,5
Γ=0,5 Γ=10
0,0 0,0
wi=-0,8 wi=0
-0,2 wf=-1 -0,2 wf=-1
-0,4 -0,4
w(z)
w(z)
-0,6 -0,6
-0,8 -0,8
-1,0 -1,0
-1,2 -1,2
zp zp
-1,4 -1,4
0 1 2 3 4 5 6 0 1 2 3 4 5 6
z redshift z
Fit-result (dashed red lines) for two different fiducial cases of a step in dark energy’s equation of state time evolution (black dotted lines), both centered
around zt = 0.5.
Left-hand side: w (z) starts at wi = −0, 8 in the far foretime of the universe and tends to wf = −1 after transition phase. The parameter Γ is set to
Γ = 0.5 which corresponds to a transition width ∆z ≈ 81.
Right-hand side: Γ is set to Γ = 10 which corresponds to a transition width ∆z ≈ 0.6.
Sebastian Linden (en collaboration avec Jean-Marc Virey) (Centre de Physique Th´orique)
e e
Test de la param´trisation CPL pour contraindre l’energie noire November 20, 2007 10 / 14
Results Two examples
Two examples (2): Choose the strongly biased one
We fix zt and Γ to 0.5 and 10, respectivley, (biased case from above) and do
an evolution on fiducial parameters (wi , wf ) in the range wfi = −6...6 .
w
−3...3
We will obtain a (wi , wf )–plane where each fiducial model is charaterised by
a point.
We define a Zone of Confusion for each parameter’s
pifit = (w0 , w0 + wa , w (zp ), weff , ΩM , ...) possible confusion wih some value
pito test we want to test:
Definition:
|pifit −pitested |
Zone of Confusion CZ := {(wi , wf ) : σp i 2,
2 and slow transitions Γ 7(1 + zt )) for models
with zt < 2.
Sebastian Linden (en collaboration avec Jean-Marc Virey) (Centre de Physique Th´orique)
e e
Test de la param´trisation CPL pour contraindre l’energie noire November 20, 2007 13 / 14
Results Confusion with ΛCDM possible?
Conclusions
We presented some results of our tests of the CPL parametrization of dark
energy’s equation of state. Two results of our work are:
1 There isn’t much fear of confusing a step-like behaviour with a Cosmological
Constant model, unless in the deep phantom region or for very early
transition-redshifts zt (that especially lie outside the range of our SN-data).
2 CPL is able to well fit a wide range of step-like models, but biases are
introduced for recent rapid transitions.
Sebastian Linden (en collaboration avec Jean-Marc Virey) (Centre de Physique Th´orique)
e e
Test de la param´trisation CPL pour contraindre l’energie noire November 20, 2007 14 / 14