Geometrical reconstruction of dark energy

1
Geometrical reconstruction of dark energy

• Stéphane Fay
• School of Mathematical Science
• Queen Mary, University of London, UK
• steph.fay_at_gmail.com
• and
• LUTH,
• Paris-Meudon Observatory, France

2
Outline

• Model
• Data
• What means "reconstructing dark energy"
• 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 pFwF?F with wFlt-1/3,
  the equation of state (EOS)
• Dark energy is a cosmological constant when
  wF-1, quintessence when wF gt-1 and ghost when wF
  lt-1.
• wF-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
• RGscalar field defined by the Lagrangian
• LRFµFµ-ULm with wF(FµFµU)/(FµFµ-U)
• We recover LCDM when Fµ0
• In this interpretation, either wF , lt or gt-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.
• They consist in 115 supernovae at zlt1.01
• We will also consider the BAO data consisting in
  the dimensionless quantity A(0.35)0.4690.017,
  where A is defined by
• A(z)Dm(z)2 cz H(z)-1 1/3 Om1/2H0 (zc)-1 with
  Dm 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 dl.
• Which cosmological quantities?
• The distance luminosity dl related to the
  magnitude m by dl 10Exp(m-25)/5
• and
• The Hubble function H2H02(Om(1z)3 ODE ?F/?F0)
• The potential U and kinetic term FµFµ of the
  scalar field.
• The deceleration parameter q, qgt0 when the
  expansion decelerates and qlt0 when it accelerates
• The EOS of dark energy
• ? All these quantities can be expressed with dl
  and its derivatives

6
Reconstruction model independent method

• 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 'gt0 true for all expanding Universe
• (b) dl''gt0 means that the deceleration parameter
  qlt1. True for any presently accelerating Universe
  (qlt0) undergoing a transition to an EdS Universe
  (q1/2)
• (c) dl'''lt0 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

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How to define a dl curve?

• 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 ?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.
• 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 (yax)
• Assuming that the 8 points dli are equidistant in
  redshift, the properties (a)-(c) will be
  respected if
• dli1gtdli
• dli2-dli1gtdli1-dli
• dli2-2dli1dliltdli3-2dli2dli1

8
Testing reconstruction with mock data

• 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 1s confidence
  level.
• Best ?2113.41 the reconstruction is efficient

9
Reconstruction of dl with real data
Best ?2113.85 the ?CDM model (?2114) cannot be
ruled out at 1s Note that the best fit
corresponds to a dl slightly below the ?CDM
model slower acceleration than with the ?CDM
model.
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Reconstruction of H
SN data do not constrain H0' Hence to
reconstruct H, we will assume H0'gt-40,
i.e. 9375ltdl0'' lt13333 The last condition is
equivalent to a lower limit for the EOS, i.e.
pF0/?F0gt-2. The LCDM model is well inside the
1s contour.
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Reconstruction of Om

• To reconstruct the other cosmological quantities,
  we need to know Om but the supernovae do not give
  any information about Om because each curve dl is
  degenerated. Why?
• Lets take the curve dl representing a constant
  EOS for dark energy defined by wFG-1 with the
  Hubble function
• H2/H20 Om(1z)3 OF(1z)3G.
• Now rewrite the Hubble function as
• H2/H20 Om1(1z)3 Om2(1z)3 OF(1z)3G with
  Om1Om2Om.
• It represents the same dl but can mimic a new
  DE with the Hubble function
• H2/H20 Om1(1z)3 OF1?F(z)/?F(0)
• and the varying EOS
• wF (A(1z)2G B(1z)3G-1)/(A(1z)2B(1z)3G-1)-1
• Hence
• a same dl can model several DE theories with
  different values of Om.
• A constant EOS can misleadingly becomes time
  dependant if Om is incorrectly chosen
  Shafieloo06

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Reconstruction of Om

• To get some information about the best fitting
  value of Om we use the BAO. Then
• 0.16 lt Om lt 0.41 for the set of theories fitting
  SNBAO
• The best fit is got when Om 0.27
• So we are now assuming that Om 0.27

13
Reconstruction of dF/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 1s, at
least until z0.6
14
Reconstruction of the EOS

• The LCDM model is well within the 1s level but
  some large degeneracy occurs for large redshift.
• Deceleration begins at least at z0.35 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 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 1s.
• The best fitting EOS is closed from -1 today,
  describe a transition at z0.45 from accelerated
  to decelerated expansion. In a general way,
  deceleration begins for zgt0.35
• 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
  Om parameter in particular a constant EOS can
  misleadingly becomes time dependant if Om is
  incorrectly chosen
• Using BAO we get the constrain 0.14ltOmlt0.48,
  with the best fit for Om0.27