Cosmological constraints from baryonic acoustic oscillation measurements

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Jean-Marc Le Goff and Vanina Ruhlmann-Kleider (2015), Scholarpedia, 10(9):32149. doi:10.4249/scholarpedia.32149 revision #152278 [link to/cite this article]
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Curator: Vanina Ruhlmann-Kleider


Baryonic Acoustic Oscillations (BAO) refer to acoustic waves propagating in the ionised matter in the early Universe, before electrons and protons combined to form neutral matter. They are responsible for the acoustic peaks observed in the power spectrum of the cosmic-microwave-background anisotropies and left an observable imprint on the distribution of matter. BAO surveys measure this imprint in the distributions of galaxies, quasars and intergalactic hydrogen clouds at different moments of the history of the Universe. These observations probe the Universe expansion and thus dark energy, which models the late acceleration of this expansion. Good precision requires large surveys observing volumes of several cubic gigaparsecs.


Describing the Universe

Cosmology aims at studying the Universe at large scales. Our understanding of the Universe, the Big Bang model, is based on Einstein’s General Relativity (GR) and several observations. These include the expansion of the Universe, the measurements of primordial abundances of light elements and that of the relic cosmic-microwave-background CMB) radiation. In this framework, the Universe can be thought to be composed of radiation, ordinary matter and dark matter. The latter is non relativistic since the early times and therefore referred to as cold dark matter (CDM).

But, in 1998, measurements of the apparent flux of distant type Ia supernovae (SNe Ia) revealed that the expansion rate of the Universe has been accelerating in the late billion years. BAO measurements provide a way of studying this acceleration, complementary to that of the CMB and distant SNe Ia.

The exact reason for the acceleration of the Universe expansion is still unknown. Phenomenologically, what causes this acceleration can be modelled as a dark energy component added to the content of the Universe. On fundamental grounds, the simplest way to account for it is to add a cosmological constant to GR equations. Theories of modified GR are also possible, as well as dark energy models motivated by high energy physics.

The BAO peak

The origin of BAO goes back to the far past of the Universe. Right after the nucleosynthesis of light elements, the Universe contains ordinary matter, radiation and dark matter. Ordinary matter is dubbed as "baryonic matter" because most of its mass is due to protons and neutrons, which are baryons. In the far past of the Universe, ordinary matter is ionized. Electrons and protons form a plasma, tightly coupled with photons due to continuous Thomson scattering of photons by free electrons. Dark matter interacts only gravitationally with the plasma and the radiation. The distribution of these components exhibits tiny density inhomogeneities. The CMB data indicate that the relative inhomogeneities are initially the same, whatever the component species are (up to a \(\textstyle 4/3\) factor between relativistic and non-relativistic species).

Figure 1: Evolution in time of a single overdensity located at the origin at initial time (see text). The plot shows \(\textstyle r^2\delta(r)\) for the various species, dark matter (black), photons (blue) and baryonic plasma (red), so that the area under each curve is proportional to the mass in the overdensity. Courtesy of Ch. Magneville and J. Rich.

Matter is attracted by gravity towards the regions that are denser in plasma and dark matter than their surroundings. The density of these regions would therefore tend to grow. But, in the tightly coupled plasma-photon fluid, gravity is counteracted by pressure so that the density in the overdense regions cannot grow. Moreover, as overdensities correspond to overpressure, they initiate spherical sound waves that propagate in the surrounding fluid. As the Universe expands, its temperature decreases. After \(\textstyle \approx\) 380,000 years, the Universe has cooled enough so that electrons and protons can combine to form neutral hydrogen, which decouples from radiation. This step is known as "recombination" or "decoupling". The photons then travel freely through space, forming the relic CMB radiation. At decoupling, the acoustic wave propagation stops abruptly, leaving an imprint both in the CMB and in the matter distribution. BAO measurements deal with the latter.

A more precise picture of this process requires to look at the relative density inhomogeneity at a given position \(\textstyle x\), defined as \(\textstyle \delta(x)=(\rho(x)-\bar\rho)/\bar\rho\) where \(\textstyle \bar\rho\) is the mean density. The top panel of Figure 1 depicts a single overdensity at \(\textstyle t = 0\). The following panels show what happens later in time. The dark matter component of the overdensity grows in place, while the high pressure in the fluid generates a spherical sound wave that rapidly drives the baryon-photon component outwards. At recombination (redshift \(\textstyle z \approx 1100\)), the photons decouple from matter and propagate freely. Once decoupled from photons, baryons do no longer feel the radiation pressure and the wave propagation ends up.

The distribution of the baryon inhomogeneity is frozen, leaving an excess of matter at a distance \(\textstyle r_s\) from the initial overdensity. This distance is known as the "sound horizon at decoupling". Its value is about 450,000 light-years. If this distance is rescaled to today, taking into account the expansion of the Universe since the decoupling, it becomes about 500 million light-years or 150 megaparsecs (Mpc), where 1 pc = 3.26 light-years. Distances scaled to today are called comoving distances. While the physical distance between two objects increases with the expansion, the comoving distance stays constant.

What happens then is dictated by gravity. The total gravity potential includes a large well due to the dominant dark matter at the initial overdensity location and a small well at \(\textstyle r_s\). Both baryonic and dark matters fall in these wells. At the end, the distribution of the two species is essentially the same, as depicted in the bottom panel.

Figure 2: The BAO peak in the galaxy correlation function in the first publication of SDSS-II (Eisenstein et al., 2005; Cole et al., 2005).

In fact initial conditions do not correspond to a single overdense region but to a correlated distribution of overdense and underdense regions. As a result of sound wave propagation, the initial overdense regions end up surrounded by overlapping overdense spherical shells of radius \(\textstyle r_s\), and similarly for the underdense regions. More precisely, \(\textstyle \delta(x)\) is a Gaussian random variable and its distribution over \(\textstyle x\) constitutes a correlated field, characterized by its correlation function or its power spectrum. This description in terms of overlapping overdense or underdense shells is only valid as long as the density inhomogeneities remain small, i.e. \(\textstyle |\delta\rho|/\rho \ll 1\).

The overdense shells cannot be individually detected, but overdensities are statistically more likely to be separated by a distance \(\textstyle r_s\) than a bit larger or smaller distances. A peak should then appear in the correlation function \(\textstyle \xi(s)\) of \(\textstyle \delta(x)\) at a separation of \(\textstyle r_s\), as illustrated in Figure 2, where \(\textstyle \xi(s)\) is defined as the average over \(\textstyle x\) of the product of \(\textstyle \delta(x)\) and \(\textstyle \delta(x+s)\), namely \(\textstyle \xi(s) = \langle\delta(x)\delta(x+s) \rangle_x\).

Note that distances in this figure are expressed in \(\textstyle h^{-1}\) Mpc. The dimensionless \(\textstyle h\) parameter is defined as \(\textstyle h = H_0 / [100\) (km/s)/Mpc], where \(\textstyle H_0\) is the Hubble constant that appears in the Hubble's law relating galaxy recession velocities to their distances, \(\textstyle v = H_0 d\). The value of \(\textstyle h\) is about 0.7, so that \(\textstyle r_s \approx 150\) Mpc \(\textstyle \approx\) 105 \(\textstyle h^{-1}\)Mpc.

Summarising, the sound horizon at decoupling, \(\textstyle r_s\), appears as a peak in the correlation function \(\textstyle \xi(s)\) and the peak position, once expressed in comoving coordinates, corresponds to \(\textstyle \approx\)150 Mpc. This is why the BAO peak position is often called a "standard ruler", that is a distance of known intrinsic value. BAO can also be discussed in the Fourier space, in terms of the matter "power spectrum", \(\textstyle P(k)\), which is the Fourier transform of \(\textstyle \xi(s)\). The BAO peak results into oscillations in \(\textstyle P(k)\), as illustrated in Figure 3. As explained in the next section, the BAO peak provides a direct probe of the expansion history of the Universe, and thus a powerful test of cosmological model predictions. This test is, in particular, sensitive to the total matter content of the Universe and to the properties of the dark energy component assumed by these models.

BAO observables \(\textstyle D_A(z)\) and \(\textstyle H(z)\)

BAO studies imply measuring the correlation function, \(\textstyle \xi(s)\), of the distribution of matter, or equivalently its power spectrum, \(\textstyle P(k)\). This is done through a three-dimensional survey of the positions of astrophysical objects of a given type, whose distribution traces the total matter distribution. These objects are therefore called "tracers".

At a given redshift, the correlation function can be measured perpendicular to and along the line of sight. The position of the BAO peak observed in the correlation function in the two cases provides important and complementary information on the expansion history of the Universe:

  • Along the line of sight, the position of the BAO peak in the correlation function corresponds to a separation in redshift \(\textstyle \Delta z\):

\[ \Delta z = r_s H(z) /c\;, \]

where \(\textstyle H(z)\) is the expansion rate of the Universe, also known as the Hubble parameter.
  • In the perpendicular direction, the BAO peak position in the correlation function corresponds to an angle, \(\textstyle \Delta \theta\), which is the angle subtended by \(\textstyle r_s\):

\[ \Delta\theta(z)=\frac{r_s}{(1+z)D_A(z)}\;, \]

where \(\textstyle D_A(z)\), called the angular-diameter distance, is also related to \(\textstyle H(z)\) by the following relation:

\[ D_A(z)=\frac{1}{1+z}\int_0^z \frac{cdz'}{H(z')}\;. \]

By measuring longitudinal and radial separations, \(\textstyle \Delta z\) and \(\textstyle \Delta \theta\), BAO surveys provide measurements of two cosmological quantities, \(\textstyle D_A(z) / r_s\) and \(\textstyle r_s \times H(z)\), related to the expansion history. However, when there is not enough data to achieve separate analyses along and transverse to the line of sight, an isotropic analysis is performed. This provides a measurement of \(\textstyle D_V(z) / r_s\), where \(\textstyle D_V(z)\) is the following combination:

\[ D_V(z) = \left[(1+z)D_A(z) \right]^{2/3} \left[cz/H(z) \right]^{1/3} \;.\]

\(\textstyle D_V\) is sometimes called the volume-averaged distance.

Tracers of matter distribution

A 3D survey implies measuring redshifts in addition to angular positions (right ascension and declination). A photometric survey that records images in several different wavelength bands provides an estimate of the redshift, typically at a few percent accuracy level. This uncertainty on the redshift erases the BAO peak along the line of sight. Only spectroscopic surveys provide accurate measurements in this direction. To do so, a photometric survey first provides the angular positions of candidate tracers. Then a spectroscopic survey measures the spectra of the selected candidates, to confirm their nature and provide accurate redshifts.

Many tracers can be considered. The choice is made so as to minimise the statistical error on \(\textstyle P(k)\) or \(\textstyle \xi(s)\) in a given range in \(\textstyle z\). This is best discussed in terms of the power spectrum \(\textstyle P(k)\), which measures the average strength of modes with wavenumber \(\textstyle k\), i.e. \(\textstyle \sin(2\pi \boldsymbol k \boldsymbol r+\phi)\) functions with \(\textstyle |\boldsymbol k|=k\,\). Since \(\textstyle r\) is in \(\textstyle h^{-1}\) Mpc, \(\textstyle k\) is in Mpc\(\textstyle ^{-1}\) \(\textstyle h\) or \(\textstyle (\)Mpc\(\textstyle /h)^{-1}\). When \(\textstyle \xi(s)\) is Fourier transformed, the BAO peak results into oscillations in \(\textstyle P(k)\). The oscillations are strongly damped after the third peak, as seen in Figure 3, so that measuring up to \(\textstyle k \approx 0.2\) (Mpc\(\textstyle /h)^{-1}\) collects most of the detectable information.

Figure 3: Measurements of \(\textstyle \xi(s)\) (left) and \(\textstyle P(k)\) (right) from the BOSS survey (Anderson et al., 2014). \(\textstyle P(k)\) is actually divided by a smooth function to exhibit the peaks more clearly.

The statistical error on \(\textstyle P(k)\) involves two contributions, the "sample variance" or "cosmic variance" and the "shot noise". The cosmic variance refers to the fact that what is observed is just one realisation of the fluctuations of the large scale structures. This error is proportional to \(\textstyle P(k)\), which is expressed in Mpc\(\textstyle ^3\). The shot noise results from sampling the survey volume with a finite number of objects. This noise is proportional to the inverse of the density, \(\textstyle n\), which is expressed in Mpc\(\textstyle ^{-3}\). Given that the interesting mode range goes up to \(\textstyle k=0.2 \) (Mpc\(\textstyle /h)^{-1}\) and that \(\textstyle P(k)\) decreases with \(\textstyle k\), once the tracer density for a fixed volume reaches \(\textstyle nP(k=0.2)=1\), the cosmic variance becomes larger than the shot noise over all the interesting range in modes. There is not much gain in further increasing the density and the survey is said to be "cosmic variance limited".

The power spectrum of the fluctuations of the tracer density is related to the total matter power-spectrum through a "bias factor" \(\textstyle b\), defined as \(\textstyle P(k) = b^2 P_{matter} (k)\). Tracers with a large bias are preferred because their larger value of \(\textstyle P(k)\) reduces the number density required to reach \(\textstyle nP \approx 1\). A survey is said to be dense if this limit is reached and to be sparse otherwise.

The criteria to choose a tracer include the bias, the density and how easily the tracer can be measured, which results in different choices depending on the redshift, as summarised in Table 1 and described below:

  • The most usual tracers are galaxies, of which different types can be used. Luminous red galaxies (LRG) have a large bias, are easy to select and provide the best tracers at redshifts up to 1. Above \(\textstyle z=1\), LRG become faint in the optical band since their spectra are shifted to the red. On the other hand, at these redshifts, galaxies formed much more stars than they do now, which results in narrow emission lines, such as the oxygen line doublet, known as [OII] in spectroscopic line nomenclature. These emission-line galaxies (ELG), in spite of a lower bias, can thus be used as tracers at redshifts up to 1.6. Some distant galaxies have a large broad Lyman-\(\textstyle \alpha\) emission line. Such Lyman-\(\textstyle \alpha\) emitters (LAE) could be used as tracers at even larger redshifts.
  • Quasars (QSO) are rare objects, whose density does not reach \(\textstyle nP = 1\) in spite of a large bias. However, they are very luminous objects, which can thus be measured at large distances. They are used to make sparse surveys in the redshift range between 1 and 2, where the quasar density is the largest.
  • Quasars can also be used as backlights to illuminate the intergalactic medium. In the "Lyman-\(\textstyle \alpha\) forest" part of the quasar spectrum, the absorption is a tracer of the density of hydrogen along the line of sight and then of the total matter density. The bias is smaller than unity and quasars are very few. Nevertheless, each of them provides hundreds of measurements along the line of sight. Lyman-\(\textstyle \alpha\) forests then allow for sparse surveys using the intergalactic medium as a tracer. Redshifts must be above 2 to have the Lyman-\(\textstyle \alpha\) forest in the optical window observable from the ground. They must also be below 3.5 to detect enough quasars.
  • Galaxies may also be detected with radio telescopes through the hydrogen 21 cm emission line, which results from the hyperfine level transition of neutral hydrogen atoms. This can be done without detecting individual galaxies but summing over all the 21 cm intensity seen within the main lobe of the radio telescope. This intensity mapping technique may potentially provide dense surveys up to redshift 3. The challenge is to deal with foregrounds, which are several orders of magnitude larger than the signal, but which are smooth functions of frequency in contrast with the signal.
  • The next generation of radio interferometers will be able to detect individual galaxies. Foreground subtraction will not be an issue for these surveys, which should indeed allow for dense surveys in this interval of redshifts.

Table 1: Characteristics of various matter distribution tracers for BAO surveys. The acronyms are explained in the text.
LRG ELG LAE QSO Ly\(\textstyle \alpha\) 21 cm
bias \(\textstyle \approx\)2 \(\textstyle \approx\)1 unknown \(\textstyle \approx\)3 \(\textstyle \approx\)0.2 unknown
\(\textstyle nP\) dense dense dense sparse sparse dense
\(\textstyle z \) range < 0.8 < 2 2 - 3.5 1 - 2 2 - 3.5 < 3

Determination of \(\textstyle D_A(z)\) and \(\textstyle H(z)\)

As illustrated in Figure 2, the measured correlation function \(\textstyle \xi(s)\) exhibits a peak on top of a continuum. The first step in the derivation of \(\textstyle D_A(z)\) and \(\textstyle H(z)\) is to determine the position of this peak. This determination involves a model of \(\textstyle \xi(s)\) that includes a theoretical description of the peak and effective parameters to describe the shape of the continuum. This provides enough freedom so that the fitted position of the peak does not depend on the shape of the continuum. As observational effects are expected to vary smoothly with \(\textstyle s\), the measurement of the position of the peak is rather robust.

The second step is to derive values of \(\textstyle D_A(z)\) and \(\textstyle H(z)\) from the peak position measurement. This is also quite robust for two reasons. First, the physics of the baryonic acoustic waves in the early Universe that created the BAO peak involves well-understood physics in the linear regime. Then, in the late stage of structure formation, non-linear effects occur. They displace the tracers and enlarge the BAO peak but do not change significantly its position. BAO measurements are therefore dominated by statistical errors, at least for the current survey generation.

Note that the enlargement of the BAO peak due to tracer displacements can be partially recovered. Indeed, the tracer displacement map can be estimated from that of the measured tracer density and corrected out (Eisenstein et al., 2007). Applying this correction sharpens the BAO peak in \(\textstyle \xi(s)\), significantly improves the statistical accuracy of the measurement and also further reduces possible changes in the peak position (Anderson et al., 2014).

BAO spectroscopic surveys

The BAO peak was first detected in 2005 (Eisenstein et al., 2005; Cole et al., 2005) as illustrated in Figure 2. This was achieved by the Sloan Digital Sky Survey (SDSS) operating at the Apache Point Observatory (APO) in New Mexico and by the 2dF Galaxy Redshift Survey (2dFGRS) at the Anglo-Australian Telescope (AAT) of the Siding Spring Observatory in Australia. Those were the first surveys with large enough volume and tracer density. Since then, larger and/or denser surveys have been performed with upgraded spectrographs. A few characteristics of the different spectroscopic surveys are listed in Table 2.

Table 2: Basic characteristics of spectroscopic surveys for BAO measurements up to 2014. 2dFGRS and 6dFGRS refer to the 2dF (2 degree field) and 6dF Galaxy Redshift Survey operated at the UK Schmidt Telescope (UKST) of the Siding Spring Observatory. BOSS stands for the Baryon Oscillation Spectroscopic Survey of the SDSS-III program and eBOSS for the extended Baryon Oscillation Spectroscopic Survey of the SDSS-IV program. HETDEX is the Hobby-Eberly Telescope (HET) dark energy experiment located in Texas. SDSS-II target numbers include SDSS-I results. "Galaxies" in the target number column refers to galaxies of any type.
survey telescope dates area
target number redshift range
2dFGRS AAT 1997 - 2002 1,800 galaxies: 250,000 < 0.3
SDSS-I APO 2000 - 2005 5,700 LRGs: 78,000
galaxies: 445,000
0.15 - 0.5
< 0.3
6dFGRS UKST 2001 - 2006 17,000 galaxies: 125,000 < 0.1
SDSS-II APO 2005 - 2008 8,000 LRGs: 110,000
galaxies: 670,000
0.15 - 0.5
< 0.3
WiggleZ AAT 2006 - 2011 1,000 ELGs: 238,000 0.1 - 1.0
BOSS APO 2009 - 2014 10,000 LRGs: 1,500,000
Ly\(\alpha\) QSOs: 160,000
0.2 - 0.7
2.1 - 3.5
eBOSS APO 2014 - 2020 7,500
LRGs: 375,000
ELGs: 270,000
QSOs: 675,000
Ly\(\alpha\) QSOs: 150,000
0.6 - 0.8
0.6 - 1.0
1.0 - 2.2
2.2 - 3.5
HETDEX HET 2014 - 2017 60 LAEs: 700,000 1.9 - 3.5

Measurements of cosmological distances from BAO

Figure 4: Left: volume-averaged distance \(\textstyle D_V(z)\) measured by different BAO galaxy surveys compared with the prediction from the Planck 2013 results, assuming flat \(\textstyle \Lambda\)CDM (solid line). The displayed quantity is \(\textstyle D_V(z)\) \(\textstyle r_{s}^{fid} /r_s\) where \(\textstyle r_{s}^{fid}\) is the value of \(\textstyle r_s\) for some reference values of the cosmological parameters in the \(\textstyle \Lambda\)CDM model. This quantity is very close to \(\textstyle D_V(z)\). Right: the same measurements divided by the flat \(\textstyle \Lambda\)CDM prediction from the Planck data. The dashed line shows the flat \(\textstyle \Lambda\)CDM prediction from earlier CMB results. In both cases, the grey region shows the 1\(\textstyle \sigma\) uncertainty in the prediction (Anderson et al., 2014).

Isotropic analyses of galaxy surveys (Beutler et al., 2011; Anderson et al., 2014; Kazin et al., 2014) provide measurements of the BAO volume-averaged distance \(\textstyle D_V(z)\) as illustrated in Figure 4. The measurements are in good agreement with predictions derived from the measurement of the power spectrum of the CMB anisotropies, in the framework of a flat \(\textstyle \Lambda\)CDM cosmological model. This model assumes a Universe with a flat geometry, made of ordinary and cold dark matter, and governed by General Relativity with a cosmological constant \(\textstyle \Lambda\).

Figure 5: 68 and 95% confidence contours in the \(\textstyle H(z)\) vs \(\textstyle D_A(z)\) plane from anisotropic (orange) and isotropic (grey) BAO measurements at \(\textstyle z=0.57\) derived from galaxies (Anderson et al., 2014). Results are compared with \(\textstyle \Lambda\)CDM predictions from CMB data (WMAP and Planck 2013). As in , the quantities displayed are \(\textstyle H(z) (r_d / r_d^{fid})\) and \(\textstyle D_A(z) (r_d^{fid} / r_d)\) where \(\textstyle r_d\) is another notation for \(\textstyle r_s\).

BOSS was the first galaxy survey to reach the statistical power to perform an anisotropic analysis, i.e. a separate measurements of the expansion rate, \(\textstyle H(z)\), and the angular distance, \(\textstyle D_A(z)\). This is illustrated in Figure 5, where these measurements are compared with the isotropic measurement of the volume-averaged distance \(\textstyle D_V(z)\). The gain brought by separate measurements is clear.

Figure 6: Expansion rate as a function of the amount of time elapsed since the Big Bang, compared with predictions from Planck 2013 results assuming flat \(\textstyle \Lambda\)CDM cosmology. The measurement from the Ly-\(\textstyle \alpha\) forest of BOSS quasars is the red point (adapted from Delubac et al., 2015). Other points come from distant galaxy BAO measurements and direct local measurements of \(\textstyle H_0\).

The BAO imprint in the Ly-\(\textstyle \alpha\) forest of quasars was first detected in 2013 (Busca et al., 2013; Slosar et al., 2013). This allowed the expansion rate to be measured for the first time in the far past of the Universe (11 billion years ago) showing that at that time the expansion was still decelerating. An update of this result is given in Figure 6. Since then, the BAO peak in the Ly-\(\textstyle \alpha\) forest was measured with greater accuracy and separate measurements of \(\textstyle H(z)\) and \(\textstyle D_A(z)\) were obtained (Delubac et al., 2015) .

Testing cosmological models

BAO measurements of \(\textstyle D_A(z)\) and \(\textstyle H(z)\) alone do not provide measurements of cosmological parameters. But, they can be combined with the power spectrum of CMB anisotropies and the luminosity distances measured from SNe Ia, to provide measurements of cosmological parameters in the framework of different cosmological scenarios of the Big Bang model. These scenario-dependent measurements are usually called "constraints". The most common scenarios are based on General Relativity and the hypothesis that the Universe is homogeneous and isotropic at large scale. They differ in the content and/or geometry of the Universe.

The \(\textstyle \Lambda\)CDM model assumes a universe made of ordinary and cold dark matter, and supplements GR equations with a cosmological constant \(\textstyle \Lambda\). As a first alternative, the wCDM model replaces \(\textstyle \Lambda\) by a new component in the content of the Universe, called dark energy. This component is described as a fluid whose pressure and density have a ratio, \(\textstyle w=p/\rho\), constant with time. The cosmological constant case corresponds to \(\textstyle w = -1\) in this formulation. The second alternative, the w\(\textstyle _z\)CDM model, has the same content but assumes that \(\textstyle w\) varies with time as \(\textstyle w(a)=w_0+(1-a)w_a\), where \(\textstyle a(z)=(1+z)^{-1}\) is the scale factor of the Universe (that is, at a given redshift \(\textstyle z_1\), \(\textstyle a(z_1)\) measures the increase of the size of the Universe between \(\textstyle z=z_1\) and today, when \(\textstyle z=0\)). The cosmological constant case corresponds to \(\textstyle w_0 = -1\) and \(\textstyle w_a= 0\). The geometry of the Universe in the above models can be free (any curvature possible) or flat (no curvature). These wCDM and w\(\textstyle _z\)CDM models do not imply any assumption about the physical nature of dark energy. They are a way to express the experimental constraints and to compare them to definite models with a given \(\textstyle w(z)\) dependence.

GR equations allow the expansion rate at a given redshift, \(\textstyle H(z)\), to be related to the content of the Universe. This is known as the Friedmann-Lemaître equation, which in the case of the wCDM model stands as:

\[ \frac{H^2(z)}{H^2_0}=\Omega_m(1+z)^3 +\Omega_r(1+z)^4 +\Omega_k(1+z)^2 +\Omega_\Lambda(1+z)^{3(1+w)} \; .\]

Here \(\textstyle H_0 = H( z = 0 )\) is the Hubble constant. The \(\textstyle \Omega_i\) parameters represent present-day energy densities normalised to the critical density, \(\textstyle \rho_c=3H_0^2/8\pi G\). More precisely, \(\textstyle \Omega_m\), \(\textstyle \Omega_r\) and \(\textstyle \Omega_\Lambda\) are related to non-relativistic matter, radiation and dark energy, respectively, and \(\textstyle \Omega_k \equiv 1 -\Omega_m -\Omega_r -\Omega_\Lambda\) characterises the curvature. The values of the various \(\textstyle \Omega_i\)'s in the above equation cannot be predicted but must be constrained from observations. As an example, the mean temperature of the CMB provides the value of \(\textstyle \Omega_r\) and indicates that today the energy density associated to radiation is negligible. It is clear from the Friedmann-Lemaître equation that measuring \(\textstyle H(z)\) constrains the energy content of the Universe, including the dark energy component and its parameter \(\textstyle w\).


The accuracy of present measurements of the CMB, SNe Ia and BAO results in per-cent precision constraints on the content of the Universe and its expansion history.

Figure 7: 68, 95 and 99% \(\textstyle \Lambda\)CDM constraints in the \(\textstyle \Omega_\Lambda - \Omega_m \) plane from CMB data, SNe Ia and the first BAO results. The line corresponds to a flat geometry, \(\textstyle \Omega_k=0\) (Suzuki et al., 2012).

As an example, Figure 7 presents the constraints on \(\textstyle \Omega_m\) and \(\textstyle \Omega_\Lambda\) in the \(\textstyle \Lambda\)CDM model. The results from CMB, SNe Ia and BAO are shown separately, illustrating their complementary role and their remarkable consistency. Combined data are in agreement with a flat universe (\(\textstyle \Omega_k = 1 -\Omega_m -\Omega_\Lambda \approx 0\)) and show that, today, non-relativistic matter represents only \(\textstyle \approx\)30% of the energy balance of the Universe. The Universe is thus dominated by its dark energy component (under the form of a cosmological constant). Anderson et al. (2014), Planck collaboration (2014, 2015) and Betoule et al. (2014) present more recent \(\textstyle \Lambda\)CDM results.

Figure 8: 68 and 95% wCDM constraints in the \(\textstyle w-\Omega_k\) plane. The different contours are from CMB combined with SNe Ia (green), with BAO data (latest results only in grey and all BAO results in red), and from all data (blue) (Anderson et al., 2014).

Figure 8 gives the results in the wCDM model. Combining all measurements provides tight constraints, especially on curvature, which is found to be compatible with zero even in this more general model. Note also that the combined contour encompasses the particular case of a pure cosmological constant (\(\textstyle w=-1\)). In this more general model, data still agree with the hypothesis of a pure cosmological constant to explain the late acceleration of the Universe expansion.

The same conclusions are drawn with the even more general w\(\textstyle _z\)CDM model (Anderson et al., 2014; Planck collaboration, 2014; Planck collaboration, 2015; Betoule et al., 2014).


Measurements of the angular diameter distance \(\textstyle D_A(z)\) and the expansion parameter \(\textstyle H(z)\) have been performed at a few redshift values, with a precision of 1% at best. Future surveys (Weinberg et al., 2013) will cover many thousands of square degrees over a large range of redshifts. Some of them will be dense enough to be cosmic variance limited, i.e. to reach \(\textstyle nP > 1\). The aim is to measure \(\textstyle D_A(z)\) and \(\textstyle H(z)\) with sub-percent accuracy at multiple redshift values (typically, in \(\textstyle \Delta z=0.2\) intervals). Some projects will be dedicated to BAO (and more generally to the clustering of ordinary matter) but most will include other cosmological probes.

Some surveys are purely photometric. They include DES, which started in 2013 and covers \(\textstyle z<1.4\), PanSTARSS, whose second telescope (among four planned) is under commissioning, and LSST (2023, \(\textstyle 0.2<z<3\)). These surveys are not optimal for BAO measurements because redshifts determined from photometric data are less accurate than spectroscopic ones. However, the several billion galaxies measured by LSST should provide \(\textstyle D_A\) measurements with a precision better than 1%.

Most projects are spectroscopic surveys. SuMire should start in 2018 and cover \(\textstyle 0.6 <z< 1.6\), with a limited precision (2%), while DESI (2020, \(\textstyle 0.2<z<4\)) should reach sub-percent accuracies. The Euclid (2020, \(\textstyle 0.7<z<2.1\)) and WFIRST (2023, \(\textstyle 1<z<3\)) satellite surveys will have similar performance.

Finally, radio surveys will aim at cosmic variance limit. Chime (2016, \(\textstyle 0.8<z<2.5\)) will use intensity mapping, while SKA (2023) will measure spectra of billions of galaxies, to cover a similar range in redshift.


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