Cosmological Parameters from WMAP and SDSS

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Cosmological Parameters from WMAP and SDSS

• Greg Thompson
• Astronomy Bag Lunch
• February 22, 2005

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Outline

• In the beginning
• The Cosmological Parameters
• The CMB and Temperature Fluctuations
• The WMAP Results
• Large Scale Structure (LSS)
• The SDSS Power Spectrum
• Combining the WMAP and SDSS results

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In the beginning

• There was something and then something happened
  and our universe was born. This birth happened
  in an explosion of space called the big bang.
• The very early universe was in an ultra-hot and
  dense state and radiated as a blackbody. As the
  universe expanded, it cooled and the once very
  hot blackbody radiation was redshifted into the
  microwave regime at T 2.7 K.
• While in its hot, dense state the universe
  consisted of photons and baryons coupled together
  in a photon-baryon fluid.
• At z 3600 or tmr 50,000 yrs the radiation and
  matter densities were equal. Prior to that time,
  the universe was radiation dominated. After that
  time, the universe became matter dominated.
• At z 1400 or trec 250,000 yrs electrons
  combined with baryons to form neutral matter for
  the first time.

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• At z 1100 or tdec 350,000 yrs the photons are
  decoupled from the matter and are free to travel
  to us to be observed. These will be the CMB
  photons.
• The sphere from which we observe these photons is
  called the last scattering surface. Any
  inhomogeneities in the universe at the time of
  last scattering will leave their imprint in the
  CMB.
• What are the cosmological parameters?
• a(t) - the scale factor - a dimensionless
  function that describes how distances in a
  homogeneous, isotropic universe expand or
  contract with time.
• Normalized so that at the current time, t0, a(t0)
  1.
• H(t) - the Hubble parameter - characterizes the
  rate of expansion and has units of km s-1 Mpc-1.
• Defined as

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• Evaluated at current time it is called the Hubble
  constant, H0
• The Hubble constant is often parameterized as
• Typical cosmological scales are set by the Hubble
  length,
• and the Hubble time,
• ?i - the density parameter - for each species i
• ?T???m ????????r
• ?T lt 1 ? ?? -1 ? open universe
• ?T 1 ? ?? 0 ? flat universe
• ?T gt 1 ? ?? 1 ? closed universe

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• Curvature density parameter
• The Friedmann equation in terms of the density
  parameters
• Equation of state
• Define equation of state pi wi ?i with pT
  ?wi ?i .
• The equation of state parameter is wi. Assuming
  no interaction between components, the following
  equation holds,
• The horizon distance
• is the farthest distance from which light has
  had time to reach you.

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The CMB and Temperature Fluctuations

• The mean temperature of the CMB
• Define temperature fluctuations at a given point
  on the sky
• The angular size ?? of a fluctuation in T is
  related to a physical size l on the surface of
  last scattering
• At large redshifts

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• For typical cosmological models,
• For COBE, with ???gt???,?l gt 1.6 Mpc.
• Corrected for the Hubble expansion,?l gt 1700 Mpc,
  much bigger than superclusters.
• Modern measurements see scales corresponding to
  the sizes of superclusters.
• Since the fluctuations are measured on the sky,
  it is useful to do an expansion in spherical
  harmonics,
• Cosmologists are interested in the statistical
  properties of the fluctuations. The most
  important statistic is the correlation function,
• This can be written,

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• It is common to present CMB results as
• which gives the contribution per logarithmic
  interval in l to the total fluctuation ?T in the
  CMB. This is the angular power spectrum.

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What causes the fluctuations?

• Suppose the matter density at the time of last
  scattering was not perfectly homogeneous. In the
  Newtonian limit, the fluctuation in density ??
  will cause a variation in the gravitational
  potential ??
• A full relativistic calculation finds
• This is called the Sachs-Wolfe effect. Photons
  falling into or climbing out of a potential well
  at the time of last scattering will be
  blueshifted or redshifted causing a warm spot or
  cool spot in the CMB. This effect is important
  for potential wells with angular sizes ?? gt 1?.

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• On scales where ?? lt 1?, the motion of the
  photon-baryon fluid becomes important. As the
  fluid falls to the bottom of a potential well it
  will be compressed and heat up. As it is
  compressed its pressure rises and will balance
  with and then overcome gravity and the fluid will
  expand and cool. This cycle of compression and
  expansion causes standing acoustic oscillations
  in the photon-baryon fluid.
• At the time of photon decoupling, the photons are
  released and carry the imprint of the acoustic
  waves with them.
• If the potential well happens to be expanding or
  contracting at the time of decoupling, the
  photons will be blueshifted or redshifted.
• In general, the highest peak in the CMB
  represents the potential wells within which the
  photon-baryon fluid had just reach maximum
  compression at the time of last scattering.

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What can the first peak in ?T tell us?

• The location and amplitude of the first acoustic
  peak is a useful cosmological tool.
• The location of the highest peak is related to
  the curvature of the universe.
• In a negatively curved universe, the angular size
  of an object with known proper size and distance
  is smaller than it is in a positively curved
  universe.
• In general, ?? 180?/l.??For ?? -1, the peak
  would be at ??lt 1? or l gt 180 for ?? 1, the
  peak would be located at ??gt 1? or l gt 180.
• The amplitude of the first peak is dependent on
  the sound speed of the photon-baryon fluid,
• The equation of state parameter wpb is dependent
  on the photon-to-baryon ratio. Thus, the
  amplitude of the peak can be used to constrain
  the photon and baryon densities.

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The WMAP Results

• WMAP vs. COBE

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• Foreground Analyses
• Regions of bright foreground emission are masked
  out. The masks are based on the 13mm K-band
  temperature levels, these being the best tracers
  of foreground contamination.
• Form a linear combination of the multi-frequency
  WMAP data that retains unity response only for
  the emission component of the CMB. The
  statistics of this type of map are too complex
  for use in CMB analyses.
• Remove any point sources found in the WMAP data.
• The Sunyaev-Zeldovich effect a negligible
  contaminant
• Kinematic SZ effect - if a cluster is moving
  toward us, free electrons in the intra-cluster
  gas will Thomson scatter the CMB photons causing
  a Doppler blueshift in the direction of the
  cluster.
• Thermal SZ effect - high temperature of the free
  electrons imparts energy to the CMB photons
  depleting the Rayleigh-Jeans tail and
  overpopulating the Wien tail.

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Deriving the cosmological parameters

• The shape of the angular power spectrum, in
  particular the ratio of the heights of the first
  two peaks gives the baryon density ?bh2. (?bh2
  0.024 /- 0.001)
• The first acoustic peak represents a known
  acoustic size (rs 147 Mpc) and known redshift
  (zdec 1089). One can compute the CMB photon
  travel time over the distance determined from the
  decoupling surface (dA 14 Gpc) and the geometry
  of the universe (flat). (t0 13.7 /- 0.2 Gyr)
• The matter density ?mh2 affects the shape and
  height of the acoustic peaks. The
  baryon-to-matter ratio determines the amplitude
  of the acoustic peaks and the matter-to-radiation
  ratio determines the epoch of equality zeq. The
  Sachs-Wolfe effect also constrains zeq. (?mh2
  0.14 /- 0.02)
• Measurements of the age and ?mh2 yield a value
  for the Hubble constant H0. (H0 72 /- 5 km s-1
  Mpc-1)

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Large Scale Structure

• The basic mechanism for structure formation is
  gravitational instability due to matter density
  perturbations
• The structure seen in galaxy surveys has grown
  from the density perturbations present when the
  universe became matter dominated.
• Consider density fluctuations at a time when
  their amplitude is small (? ltlt 1). The expansion
  of the universe is still nearly homogeneous and
  isotropic as long as this is true. Begin by
  labeling each bit of matter with its comoving
  coordinate position

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• To study large scale structure cosmologists are
  interested in the statistical properties of the
  distribution . A map of density
  perturbations can be Fourier expanded
• where the Fourier components
• This Fourier transform decomposes the function
  into an infinite number of sine waves with
  comoving wavenumber and comoving
  wavelength
• Each component is a complex number
• with the phase remaining constant for small
  amplitude fluctuations.

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• The power spectrum is defined by the mean square
  amplitude of the Fourier components
• with the average being taken over all possible
  orientations of the wavenumber.
• Most inflation scenarios predict the creation of
  isotropic, homogeneous Gaussian density
  fluctuation fields. Further, the power spectrum
  is expected to have a scale invariant power-law
  form
• The favored power-law index is n 1.
• The power spectrum is expected to have an index n
  1 right after inflation, but this index will
  change between the end of inflation and the time
  of matter-radiation equality. How the power
  spectrum changes shape depends on the nature of
  the dark matter.

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• Cold dark matter consists of particles that
  were non-relativistic at the time they decoupled
  from other components of the universe.
• Hot dark matter consists of particles that were
  relativistic at the time of decoupling and
  remained relativistic until the universe becomes
  matter dominated.
• Example Neutrinos decoupled at t 1 s when kT
  1 MeV and would have been relativistic at that
  time (m?c2 ltlt 1 MeV). Neutrinos with m?c2 2 eV
  would have remained relativistic until after the
  time of matter-radiation equality and are thus
  hot dark matter candidates.
• Consider a universe filled with this type of hot
  dark matter. The relativistic particles cool
  along with the expanding universe until their
  velocities become non-relativistic.
• This occurs at a temperature and time,

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• Prior to this time, the particles are moving in
  random directions at close to the speed of light.
  This will effectively remove any density
  fluctuations within the hot dark matter smaller
  than cth.
• If hot dark matter were the dominant constituent
  of dark matter, the oldest structures in the
  universe would be superclusters. Observations
  seem to indicate that superclusters are still in
  the process of collapsing.
• Cold dark matter provides the best fit to
  observations but a least some hot dark matter may
  be present.

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The SDSS Power Spectrum

• Since galaxy redshift surveys only sample finite
  volume, the power spectrum found by the
  traditional Fourier technique is difficult to
  interpret. Thus, the SDSS team uses a matrix
  based method to estimate the power spectrum.
• Their basic approach is
• Prior to analysis, they remove redshift
  distortions known as the Finger of God effect.
• They actually measure three power spectra, the
  galaxy power spectrum, the velocity spectrum and
  the galaxy-velocity spectrum.
• They analyze mock galaxy catalogs to help them
  quantify the accuracy of their results and study
  non-linear efffects.

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Combining the WMAP and SDSS Data
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References

• Bennett, C.L., et al. 2003, ApJS, 148, 1 (for
  WMAP)
• Spergel, D.N., et al. 2003, ApJS, 148, 175 (for
  WMAP)
• Tegmark, Max, "Doppler peaks and all that CMB
  anisotropies and what they can tell us", in Proc.
  Enrico Fermi, Course CXXXII, Varenna, 1995,
  available from http//www.hep.upenn.edu/max/sdss.
  html (for CMB info.)
• Tegmark, Max, et al. 2004, ApJ, 606, 702 (for
  SDSS)
• Tegmark, Max, et al. 2004, Phys. Rev. D, 69,
  103501 (for WMAP and SDSS)
• Ryden, Barbara, Introduction to Cosmology, (San
  Francisco Addison Wesley), 2003 (for CMB and
  LSS information)