The Cosmic Microwave Background Radiation
Magnus Axelsson
November 11, 2005
Abstract
Predicted in the mid-1940s and discovered in 1964, the cosmic microwave background (CMB)
radiation has become a valuable tool for both cosmologers and particle physicists. This report
attempts to summarize the results from observations, and place them in the framework of current
theoretical models. As this is an active and quite extensive field, the focus will be towards the
thermal nature and anisotropies (particularly the anisotropy power spectrum) of the CMB, and the
implications for cosmological parameters and models.
1
Introduction
In 1946, the physicist George Gamow and collaborators published a paper where they discussed the
possibility of the elements being formed in the early Universe. Reversing the expansion of the Universe
discovered by Edwin Hubble in 1929, Gamow reasoned that the young Universe must have been dense
and hot enough for nuclear reactions to occur. While it would later be shown that no element heavier
than 42 He could form this way, the idea of a hot, dense early universe lived on. As the photon mean
free path would have been very short, such a universe would be extremely close to thermodynamical
equilibrium. As the Universe expanded, this blackbody radiation would have cooled as the energy density
dropped. Using the Wien displacement law:
λT = const.
giving
λ(R)T (R) = λ0 T0
(1)
Where where T (R) and λ(R) are the temperature and peak wavelength at scale factor R, and T0 and
λ0 the present temperature and peak wavelength (when R = 1). Substituting with λ0 = Rλ(R), the
current temperature of the blackbody radiation may now be expressed as:
T0 = RT (R)
(2)
At the time of helium synthesis, the Universe had to have a temperature of T ' 109 K, and a scale
factor of R ' 3.11 × 10−9 [3]. Inserting these values into Eq. 2 gives a current blackbody temperature of
∼ 3.1 K.
In 1964, Arno Penzias and Robert Wilson were working at Bell Laboratories, trying to measure weak
radio emission from the galactic halo. While making calibration measurements in preparation for the
study, they encountered a problem. They were consistently detecting an extra noise component of ∼ 3 K.
After trying to eliminate all sources of noise from their antenna (they even “..evicted [a pair of] pigeons
and cleaned up their mess” [8]), the resolution of their problem came from an unexpected source. By
chance they learned of theoretical work being done by P.J.E. Peebles and his group at Princeton, and
their prediction of relic blackbody radiation in the Universe. The observations and a theoretical paper
were published side by side in a 1965 issue of the Astrophysical Journal.
2
Observational results
Since the detection of the cosmic microwave background (CMB) by Penzias & Wilson, numerous observations have been made, using both balloon-flown instruments and satellites. In recent years, two
satellite missions have been very successful – the COsmic Background Explorer (COBE) and the Wilkinson Microwave Anisotropy Probe (WMAP). While other instruments have been equally important to
the knowledge of the CMB, these two are special in the sense that they are the only instruments yielding
full sky coverage.
1
Figure 1: Results from the COBE satellite. The left panel shows the data points superimposed on the
theoretical 2.728 K blackbody curve, and the right panel shows the temperature variations on the sky,
after correction for the peculiar motion of the Earth and galactic emission [6, 9].
2.1
COBE
The COBE satellite was launched into orbit on November 18, 1989. Its primary mission was to measure
the CMB over the entire sky. The results were to test whether the CMB was truly thermal, as well as
detecting any anisotropy in the radiation.
Results from the Far Infrared Absolute Spectrophotometer and Differential Microwave Radiometer instruments onboard COBE are shown in Fig .1. The left hand panel shows the analytical curve
of a blackbody of 2.728 K, together with the observational results. The right hand panel shows the
temperature fluctuations for the full sky, after correction for the peculiar motion of the Earth.
The COBE data led to two main conclusions. The first is that the CMB is thermal with a peak of
2.728 ± 0.004 K (95 % confidence level) [2]. The other is that while it is isotropic down to very low
−5
levels, there are fluctuations on the scale of ∆T
. As the angular resolution of COBE was about
T ∼ 10
◦
7 , it could only show the power of these fluctuations on large scales. To investigate the anisotropy on
smaller scales became the task of another satellite.
2.2
WMAP and the anisotropy power spectrum
Launched on June 30, 2001, the WMAP mission was ‘designed to advance observational cosmology by
making full sky CMB maps with accuracy, precision, and reliability’ [1]. With an angular resolution of
∼ 1◦ , it is able to map the anisotropies of the full sky down to very small scales.
When studying the temperature anisotropy T (n) of the sky, it is frequently expanded using a basis
of spherical harmonics, Ylm , giving
X
T (n) =
alm Ylm (n) .
(3)
l,m
The angular power spectrum observed for the actual sky is then
Clsky =
1 X
2
|alm |
2l + 1 m
(4)
and with the assumption of random phases, the temperature anisotropy for each dipole moment, ∆Tl ,
can be associated with the observed power spectrum as
∆Tl2 = Clsky l(l + 1)/2π.
(5)
The improved resolution anisotropy map of the sky as observed by WMAP is presented in the left
panel of Fig. 2. The figure also shows the measured anisotropy power spectrum using data from both
WMAP and other recent missions. Unlike the results from COBE, the results from WMAP give the
power for the multipole moments up to l ∼ 800, corresponding to an angular scale of ∼ 0.3◦ . The figure
clearly shows the so-called acoustic peaks, which will be discussed in more detail in Sect. 3.2 below.
As will be seen, the location and strength of these peaks can be used to constrain the cosmological
parameters, and thereby discern between various cosmological models.
Although the WMAP satellite has been gathering data for over four years, the results after the
first year have not yet been made public. The later years have mainly been focused on improving the
polarization measurements, and there are still problems with the calibration and systematic errors.
2
Figure 2: Left panel: The temperature fluctuations of the CMB as measured by WMAP. Note the
higher resolution compared to the COBE image in Fig 1. Right panel: The temperature anisotropy
power spectrum of the CMB as measured by WMAP and other recent missions. The solid line shows
the best fit to a cosmological model including cold dark matter and a non-zero cosmological constant
(Λ − CDM ) [1, 2, 9].
3
Theory of the CMB
In this section, the theoretical models predicting the CMB will be discussed, and tested against the
observations presented above. The discussion takes reference in a model of an expanding universe, with
an early hot, dense phase (i.e., a Big Bang scenario). The redshift, z, is then also a measaure of time;
the higher the redshift, the further back in time.
3.1
Why thermal?
For the CMB to have a thermal spectrum, two requirements must be fulfilled. The Universe must have
been in thermal equilibrium at some point in the past, and maintained its thermal nature throughout
the expansion of the Universe since that time.
In a hot and dense plasma, photons are generated and thermalized by interactions with matter,
e.g. γ + γ ↔ e+ + e− . As long as the temperature remains sufficiently high (in the example above
while kT ∼ me c2 ), it is reasonable to assume that matter and radiation were in thermal equilibrium,
and the photons close to a Planck distribution. As the temperature drops below the limit for creation/annihilation processes, there are three other processes which become important: free-free emission
(thermal bremsstrahlung as e− scatter of protons), the ordinary Compton effect (where photons scatter
of electrons, changing energy) and the radiative Compton effect (where an extra photon is produced in
the scattering). All these mechanisms are more effective when density and temperature are high, and
as it turns out these processes will create a thermal radiation spectrum by zth ' 106 − 107 , where zth is
the redshift, regardless of the shape of the spectrum before that time [5].
The question now becomes to see whether the thermal nature of the radiation can be maintained
in an expanding universe. The first event which could affect the photon distribution is the so-called
recombination, which occurred when the expansion caused the temperature to drop below the limit
required to keep the electrons and protons from combining to hydrogen atoms. At this time, the crosssection for photon interaction with matter dropped significantly, and the Universe essentially became
transparent as the photons decoupled. However, the recombination did not occur instantaneously, giving
this surface of last scattering a finite thickness. In 1972, Steven Weinberg expressed the present CMB
energy density in terms of the probability that a photon at a current frequency ν will survive from an
earlier time t to the present:
¸−1
Z ·
hν(z(t) + 1)
d
8πhν 3 dν t0
exp
−
1
P (t0 , t, ν)dt,
(6)
u0 (ν)dν =
c3
kT
(t)
dt
0
where P is the probability, z the redshift and T the temperature of matter at time t. Here P → 0
for sufficiently early times t, and P → 1 for times much later than the epoch of last scattering. The
right-hand side of Eq. 6 is thus a weighted average of Planck distributions. Furthermore, assuming that
the heat capacity of the radiation was large enough to keep matter and radiation cooling at the same
rate (T ∝ (z + 1)), the quantity in square brackets becomes independent and may be moved outside the
integral. This leads to u(ν) having a Planckian form, even through an extended recombination. In the
hot Big Bang model, the predicted ratio of photon to baryon number density is nγ /nb ∼ 109 , and the
heat capacity of the radiation large enough to justify the assumption above. In this model, the Big Bang
produced an exactly thermal spectrum, and unless heating was added at some later time, the blackbody
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radiation was preserved through the expansion of the Universe until the present time. The detection
of the ∼ 3 K background by Penzias and Wilson, and the extremely close match between a theoretical
blackbody curve and the COBE data in Fig. 1 above therefore lends considerable support to this model.
The close agreement also puts limits on any mechanism of energy emission into the CMB during later
epochs, distorting the spectrum.
The temperature of the CMB is currently the best determined cosmological parameter, known to
an accuracy of about 1%. Together with the knowledge that the spectrum is that of a blackbody, it is
possible to calculate the present number density of photons:
T0 = 2.728 K → nγ,0 = 410 cm−1 .
(7)
The density has implications both for theories on primordial nucleosynthesis (which depends sensitively
on the photon/baryon ratio) as well as interactions between the CMB and cosmic rays. Using the number
density of the CMB photons it is possible to calculate the mean free path of high energy cosmic rays,
perhaps putting constraints on their origin.
3.2
Anisotropy and observational support for inflation
In order to create the inhomogeneity and structures seen in the Universe, the primordial Universe cannot
have been completely homogeneous. Anisotropy in the CMB was therefore a necessary prediction of the
Big Bang model. However, some of the large scale fluctuations observed by COBE were a problem for
the standard Big Bang model. The regions in causal contact at the time of recombination subtend ∼ 1◦
on the sky today, and the model did not agree with isothermal regions many times that size. The answer
to the problem was to invoke a period of accelerated expansion in the early Universe, inflation. In this
scenario, smaller regions in causal contact were quickly expanded, keeping the same temperature and
thereby reconciling the Big Bang model with the large isothermal regions observed by COBE.
As radiation and matter were coupled until the time of recombination, any density fluctuations in the
matter distribution will be imprinted on the CMB. Measuring the fluctuations thus provides information
on the gravitational potential at the time of recombination.
Hu & White [4] liken the fluctuations to standing waves in music, where integer numbers of halfwavelengths are amplified to give a fundamental tone and overtones. At the time of inflation, quantum
fluctuations in the inflation field provide initial disturbances which are then magnified during inflation.
This provides fluctuations in the energy density of the primordial plasma, which are approximately
equal on all scales. The fluctuations then begin to oscillate in time, such that a hotter region will tend
toward average temperature, overshoot and become cooler than average, then reheat. At the time of
recombination, the photons decouple, and thus the regions at maximum deviation at that time will show
the largest temperature differences in the CMB. Since the oscillations all started at the same time, the
regions showing the largest temperature fluctuations will be the ones who have completed an integer
number of half cycles in the time between inflation and recombination, thereby reaching maximum
deviation at recombination. Inflationary cosmology thus predicts clear peaks in the anisotropy power
spectrum of the CMB, where the first appears at the scale of one half-cycle, the second at the scale of
one cycle, and so on.
3.2.1
Large scale fluctuations
The first peak in the CMB power spectrum occurs at a scale of about 1◦ (see Fig. 2). This gives
a measure of the frequency of the wave, and combined with calculations of the sound speed of the
primordial plasma, it is possible to determine the size of the wave at recombination. Together with
knowledge of the distance travelled by the CMB photons until the present time, the location of the peak
may be used to test space curvature, or equivalently the total energy density of the Universe. The result
from the WMAP measurements show that this is very close to the critical density, 10−29 g/cm3 .
On scales larger than a few degrees, the regions are too large to undergo the density fluctuations. The
observations of anisotropy at these scales is therefore solely due to the perturbations in the gravitational
potential at recombination. CMB photons released in denser regions will lose more energy reaching us
compared with photons from less dense regions, as the former must climb out of deeper potential wells.
This is known as the Sachs-Wolfe effect. Measuring the power of large scale temperature anisotropies
thus provides information on the primordial fluctuations of the potential field.
3.2.2
Small scale fluctuations
In the theory of inflation, the amplitude is the same on all scales. This predicts that all peaks in the
power spectrum should be of equal strength, which clearly does not agree with the observations in
Fig. 2. One reason for the dampening of the higher order peaks is dissipation. As the small scale CMB
4
anisotropies correspond to density perturbations, they will be suppressed if the wavelength is shorter
than the mean free path in the plasma, leading to a loss of power for higher frequencies in the power
spectrum.
As the density waves propagate in the early Universe, they are modified by gravity, which compresses
gas in denser regions (thus counteracting the gas pressure). This force can act to either amplify or
counteract the sonic wave. Matter in the early Universe was a mix of ‘ordinary matter’ (i.e. baryons)
and cold dark matter. Baryonic matter interacts with radiation and therefore is affected by both gravity
and the sonic waves. Cold dark matter, on the other hand, only contributes to gravity, enhancing the
potential wells. For the fundamental wave, gravity worked to enhance the sonic compression, but for the
first harmonic (i.e., the second peak) the two effects counteracted. The prediction is therefore that the
second peak in the power spectrum will be weaker than the first. Furthermore, comparing the strengths
of the two peaks gauges the relative strengths of gravity and gas pressure in the early Universe. Current
observations indicate that energy density of baryons and photons were roughly equal at recombination.
By measuring also the strength of the third acoustic peak (where gravity once again enhanced the
compression), it is possible to determine the density of cold dark matter, about five times the baryonic
density.
3.2.3
Other influences
Once the radiation had decoupled from matter, it travelled relatively unimpeded through the Universe.
However, there are later effects which, although small, may be measured and provide additional information.
As noted above, photons will lose energy as they climb out of potential wells. This can be used to
constrain a possible dark energy, which acts to accelerate the expansion of the Universe. It thus weakens
the gravitational wells associated with the clustering of galaxies. A photon travelling through such a
region will increase in energy as it falls into the well, and as the well is less deep by the time it climbs
out, the loss is less than the gain. The mechanism is known as the integrated Sachs-Wolfe effect, and
predicts large scale fluctuations in the CMB.
Another effect on the CMB connected with galaxy clusters is the Sunyaev-Zel’dovich effect. The
intergalactic plasma in these clusters is known to be hot, with X-ray emission leading to an inferred
temperature of 107 − 108 K. Photons from the CMB may thus gain energy from inverse Compton scattering in these regions. Clusters of galaxies are therefore predicted to produce temperature fluctuations
in the CMB, an effect which has been detected [1, 5]. While small, reliable measurements of the temperature changes could e.g. be combined with measurements of the X-ray flux to accurately measure the
distance to the clusters. Together with the redshift, this can then be used to more accurately compute
the current value of the Hubble constant, an important cosmological parameter. Another more easily
obtained result of the detection of the Sunyaev-Zel’dovich effect from distant galactic clusters is the
conclusion that the CMB cannot be a local effect.
3.3
Problems to be resolved
In the wealth of data from both COBE and WMAP, as well as from other missions, there are also results
not predicted by theory. One is the drop in power at the largest angular scales, l < 6 in Fig. 2. This
could be due to undersampling; since the sky is only 360◦ , there are not enough samples of the larger
scales to accurately measure the power. However, the discrepancy has led to some speculation about
inadequacies in the theory of inflation.
Inflationary cosmology is also troubled by another observational finding. Measurements of the large
scale polarization of the CMB by WMAP showed a high degree of polarization. This has been interpreted
as 17 % of the CMB being scattered against a thin layer of ionized gas only a few hundred million years
after the Big Bang. Such a scattering had been predicted to occur around a billion years after the
Big Bang, as the first stars were born and reionized the surrounding medium. These new observations
indicate that star formation occurred earlier than expected. One possible solution to this is that the
initial fluctuations were not scale invariant, but stronger at smaller scales, allowing stars to form faster.
This challenges the prediction of inflationary cosmology that the fluctuations were scale invariant.
Another slightly worrying result is the fact that the l = 2 quadrupole and l = 3 octopole terms are
surprisingly aligned with one another, and the axis of the combined low-l signal is perpendicular to the
ecliptic plane. This three-axis alignment (with a random probability of < 0.1 %) has been given the
nickname the ‘Axis of Evil’. It has been suggested that this is a result of the dipole induced by the flow
of local large scale structures being scattered by gravitational interaction into the higher-order moments
[7]. However, while this mechanism may cause the axes to align, it also predicts increased power at the
lower moments, thereby increasing the discrepancy between observations and predictions at the largest
scales.
5
4
Summary
The CMB provides one of the most important observational tools to probe the Universe at early times,
and determine the cosmological parameters that govern its evolution. At present, the CMB has been
observed to have a temperature of (2.728 ± 0.004) K, giving a photon density of nγ,0 = 410 cm−1 .
−5
level on both large and small scales. This supports
Anisotropies have been detected at the ∆T
T ∼ 10
inflationary Big Bang models including cold dark matter and a non-zero cosmological constant. The
current best-fit values of the cosmological parameters from measurements of the CMB indicate a flat
Ωtot ≈ 1 Universe with an age of 13.7 ± 0.2 Gyr, composed of 4.4 % baryons, 22 % dark matter, and
73 % dark energy [1].
In order to further explore the CMB, the European Space Agency (ESA) is planning to launch the
Planck satellite in 2007. With an angular resolution of 10 arcminutes (∼ 0.2◦ ), it may be able to
determine the cosmological parameters to within 1 %.
References
[1] C.L. Bennet et al., Astrophys. J. Suppl. 148, 1 (2003).
[2] L. Bergström & A. Goobar, Cosmology and Particle Astrophysics, Springer, 2004.
[3] B.W. Carroll & D.A. Ostlie, An Introduction to Modern Astrophysics, Addison Wesley, 1996.
[4] W. Hu & M. White, The Cosmic Symphony, Sci. Am. 290, 44 (2004).
[5] R.B. Partridge, 3K: The Cosmic Microwave Background Radiation, Cambridge, 1995.
[6] G. F. Smoot, ArXiv Astrophysics e-prints, arXiv:astro-ph/9705101 (1997).
[7] C. Vale, ArXiv Astrophysics e-prints, arXiv:astro-ph/0509039 (2005).
[8] R.W. Wilson, The Cosmic Microwave Background Radiation, Nobel lecture, December 8, 1978,
http://www.nobel.se/physics/laureates/1978/wilson-lecture.html
[9] The Legacy Archive for Microwave Background Data (NASA/LAMBDA),
http://lambda.gsfc.nasa.gov/
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