Geometrical reconstruction
of dark energy
Stéphane Fay
School of Mathematical Science
Queen Mary, University of London, UK
[email protected]
and
LUTH,
Paris-Meudon Observatory, France
Outline
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Model
Data
What means "reconstructing dark energy"
How to reconstruct dark energy from the
data independently on any cosmological
models
The model
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Flat Universe + baryons + CDM + Dark energy
modeled by a perfect fluid pΦ=wΦρΦ with wΦ<-1/3, the
equation of state (EOS)
Dark energy is a cosmological constant when wΦ=-1,
quintessence when wΦ >-1 and ghost when wΦ <-1.
wΦ=-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
• RG+scalar field defined by the Lagrangian
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L=R±ΦμΦμ-U+Lm with wΦ=(ΦμΦμ+U)/(ΦμΦμ-U)
We recover LCDM when Φμ=0
In this interpretation, either wΦ =, < or >-1 but it cannot cross the line
-1 [Vikman 05]
Data
The supernovae data we will consider have been published in 2006 by
SNLS.
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They consist in 115 supernovae at z<1.01
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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)=[Dm(z)2 cz H(z)-1 ]1/3 Ωm1/2H0 (zc)-1 with Dm the angular diameter
distance
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Reconstruction: a model
independent method
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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 dl.
Which cosmological quantities?
• The distance luminosity dl related to the magnitude m by dl
=10Exp[(m-25)/5]
and
• The Hubble function H2=H02(Ωm(1+z)3+ ΩDE ρΦ/ρΦ0)
• The potential U and kinetic term ΦμΦμ of the scalar field.
• The deceleration parameter q, q>0 when the expansion
decelerates and q<0 when it accelerates
• The EOS of dark energy
 All these quantities can be expressed with dl and its derivatives
Reconstruction: model independent
method
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Which general properties?
• We proceed by looking for all the dl curves respecting some
reasonable geometrical properties and fitting the magnitudes
given by SNLS. The properties are as follow:
 (a) dl '>0: true for all expanding Universe
 (b) dl''>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) dl'''<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
How to define a dl curve?
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A dl 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 dl curve of the LCDM model needs at
least 5 points to be described with enough precision to recover the
same χ2 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.
• “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!
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“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 dli are equidistant in redshift, the
properties (a)-(c) will be respected if
• dli+1>dli
• dli+2-dli+1>dli+1-dli
• dli+2-2dli+1+dli<dli+3-2dli+2+dli+1
Testing reconstruction with mock data
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We take the same
distribution and
error bars as SNLS
but we replace the
dl 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.
Best χ2=113.41: the
reconstruction is
efficient
Reconstruction of dl with real data
Best χ2=113.85: the
ΛCDM model
(χ2=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.
Reconstruction of H
SN data do not constrain H0'
Hence to reconstruct H, we
will assume:
H0'>-40, i.e.
9375<dl0'' <13333
The last condition is
equivalent to a lower limit
for the EOS, i.e. pΦ0/ρΦ0>-2.
The LCDM model is well
inside the 1σ contour.
Reconstruction of Ωm
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To reconstruct the other cosmological quantities, we
need to know Ωm but the supernovae do not give any
information about Ωm because each curve dl is
degenerated. Why?
• Let’s take the curve dl representing a constant EOS for dark
energy defined by wΦ=Γ-1 with the Hubble function
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H2/H20= Ωm(1+z)3+ ΩΦ(1+z)3Γ.
• Now rewrite the Hubble function as
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H2/H20= Ωm1(1+z)3+ Ωm2(1+z)3 + ΩΦ(1+z)3Γ with Ωm1+Ωm2=Ωm.
It represents the same dl but can mimic a new DE with the
Hubble function
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H2/H20= Ωm1(1+z)3+ ΩΦ1ρΦ(z)/ρΦ(0)
and the varying EOS
wΦ =(A(1+z)2+Γ B(1+z)3Γ-1)/(A(1+z)2+B(1+z)3Γ-1)-1
Hence
- a same dl can model several DE theories with different values
of Ωm.
- A constant EOS can misleadingly becomes time dependant if
Ωm is incorrectly chosen [Shafieloo06]
Reconstruction of Ωm
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To get some information
about the best fitting
value of Ωm we use the
BAO. Then:
• 0.16 < Ωm < 0.41 for the
set of theories fitting
SN+BAO
• The best fit is got when
Ωm = 0.27
• So we are now assuming
that Ωm = 0.27
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
Reconstruction of the EOS
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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.
To conclude
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We reconstruct some dark energy properties by
imposing some geometrical constraints on dl
The best fitting EOS is a varying one
• Universe expands slower than with a LCDM model
• The LCDM model cannot be ruled out at 1σ.
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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 Malmquist bias.
SN data alone do not provide constraints on the Ωm
parameter: in particular a constant EOS can
misleadingly becomes time dependant if Ωm is
incorrectly chosen
Using BAO we get the constrain: 0.14<Ωm<0.48,
with the best fit for Ωm=0.27