Modern Cosmology -
Part III.2: The Lambda-CDM Universe
Jacobs University Bremen


In 1998, a totally unexpected result from astronomy caused a dramatic rethink of the Big Bang model. Measurements of the light emitted by a certain supernova suggested that it was further away than predicted by the Hubble constant. In other words, the exploding star did not lie on the straight line of the Hubble graph! This is a startling discovery as it implies that the expansion of the universe is not constant - instead the expansion is currently accelerating.

Skeptics at first suggested the result might arise from an error in the measurement of stellar distance – however, a similar observation was reported by a different group within two years. Further, independent support for the result soon emerged from measurements of the cosmic Microwave Background (CMB). In 2002, precision measurements of the CMB by the WMAP satellite suggested a Universe with geometry that is flat to within 1%. This result is completely inexplicable in the context of the known density of the matter of the Universe (both ordinary and dark). The known density of matter points to a Universe with Omega_m = 0.3, a long way from flatness (Omega_m =1). Hence the CMB measurements suggest that there is a great deal of matter/energy in the Universe unaccounted for. These measurements have been fully confirmed in the last ten years.

The Concordance Model of the Universe
1. The Lambda-CDM Universe
Lambda-CDM is an abbreviation for Lambda-Cold Dark Matter, which is also known as the cold dark matter model with Dark Energy. It is frequently referred to as the standard model of Big Bang Cosmology, since it attempts to explain the existence and structure of the cosmic microwave background, the large scale structure of galaxy clusters and the distribution of hydrogen, helium, lithium, oxygen and also the accelerating expansion of the universe observed in the light from distant galaxies and supernovae. It is the simplest model that is in general agreement with observed phenomena.
2. Midterm Review
Midterm Review Key Knowledge
3. Luminosity Distance and Angular Distance

Caption: Distance measures are used in physical cosmology to give a natural notion of the distance between two objects or events in the universe. They are often used to tie some observable quantity (such as the luminosity of a distant quasar, the redshift of a distant galaxy, or the angular size of the acoustic peaks in the CMB power spectrum) to another quantity that is not directly observable, but is more convenient for calculations (such as the comoving coordinates of the quasar, galaxy, etc). The distance measures discussed here all reduce to the naive notion of Euclidean distance at low redshift, z < 0.1.

Types of Distance Measures in an expanding Friedmann Universe (LCDM):

  • Comoving distance: The distance between two points measured along a path defined at the present cosmological time.
  • Cosmological proper distance: The distance between two points measured along a path defined at a constant cosmological time. The cosmological proper distance should not be confused with the more general proper length or proper distance.
  • Luminosity distance: The luminosity distance d_L is defined in terms of the relationship between the absolute magnitude M and apparent magnitude m of an astronomical object at redshift z, M = m - 5*log(d_L) + 25, where the distance d_L is measured in Megaparsecs.
  • Angular diameter distance: Angular Diameter Distance is a good indication (especially in a flat universe) of how near an astronomical object was to us when it emitted the light that we now see.
  • Light travel time or lookback time: This is how long ago light left an object of given redshift.
  • Light travel distance (LTD): The light travel time times the speed of light. For values above 2 billion light years, this value does not equal the comoving distance or the angular diameter distance anymore, because of the expansion of the universe (see Figure).
  • Naive Hubble's law: taking z = H_0*d/c, with H0 today's Hubble constant, z the redshift of the object, c the speed of light, and d the "distance."
Fundamental Plane of the Universe

In 1970, Allan Sandage described Cosmology as the quest for two numbers, H_0 and q_0, which were just beyond reach. Today's cosmological model is described by anywhere from at least 4 to 20 parameters, and the quantity and quality of cosmological data enables precise constraints to be placed upon all of them. However, the results depend on which set of parameters are chosen to describe the Universe as well as the mix of data used.

For definiteness, we refer to the "consensus cosmological model" (or LambdaCDM) as one in which H_0, Ω_k, Ω_B, Ω_M, Ω_Λ, t_0, σ_8, and n_S are free parameters, and dark energy is assumed to be given by an EoS w_0 and w_a (for a cosmological constant: w_0 = -1 and w_a = 0). For given Hubble constant, the density parameters are constrained by their sum to be equal unity:

Ω_k + Ω_M + Ω_Λ = 1.

Each cosmological model is given by a point in the (Ω_M,Ω_Λ)-plane, which is called the Fundamental Plane of Cosmology (see Figure). Also the age t_0 of the Universe can be discussed in terms of this Fundamental Plane.

Exercises IV
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Exercises IV - Solutions
Lecture Notes: Part III.2
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