Cosmic microwave background fluctuations

8/17/2019 Cosmic microwave background fluctuations

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PR10 : Cosmic Microwave Background - From the

frequency map to cosmological parameters.

-Final Report-

Aubert Marie, Sharma Nouri

Master Space 2015



Contents

1 Introduction 2

2 Origin of the CMB : History of the discovery. 3

2.1 First Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 First measurement and interpretation . . . . . . . . . . . . . . . . . . . . 42.3 Observations and Instrumentation . . . . . . . . . . . . . . . . . . . . . . 5

3 Calculation of the temperature of the universe at the time of the CMB 7

4 Anisotropies of the CMB 11

5 Data Analysis : Extracting the CMB from Planck frequency map. 13

5.1 Components of the frequency map . . . . . . . . . . . . . . . . . . . . . . 135.2 Extracting the CMB Map. . . . . . . . . . . . . . . . . . . . . . . . . . . 145.3 Qualication of the CMB map . . . . . . . . . . . . . . . . . . . . . . . . 175.4 Trying to compute the power spectrum . . . . . . . . . . . . . . . . . . 20

6 Conclusion 22

7 References 24

1



1 Introduction

Thanks to astronomical observations we found that stars visible during the night dwells10 to 100 light years away and the nearest galaxy, Andromeda, is at 2.3 million light

years away. Thus, the Cosmic Microwave Background which was emitted 13.7 billionyears ago, only a few hundred thousand years after the Big Bang, long before stars orgalaxies ever existed is the oldest signal emitted by photons in the universe.

Indeed, when matter started forming in the plasma that composed the early universe,the photons, that were trapped up until that point, were released from the ionized gas.They then travelled to space and time carrying with them a snapshot of the early universewhich led to a further understanding of the universe as we know it.

Although uniform at a large scale, the radiation presents uctuations in its signalthat correspond to variation of the temperature, and so, density in the early universe.Thus, by studying the detailed physical properties of the radiation, we can learn moreabout the universe and its creation since the radiation we see today has traveled oversuch a large distance. The information provided by the CMB allows us to determine theconditions of the universe at very early times and to understand the repartition of largescale objects like galaxies.

Through this report we are going to present you how the CMB was theorized then af-rmed by empirical observations and measurements, then explain what are the anisotropiesand why they are important and, nally, present you our own extraction of the CMBanisotropies map and its power spectrum.

Figure 1: The photons are released and travel through space (Credits : NASA/Planck

2





Those assumptions permitted Gamow to calculate the temperature of the universe atthe time of the decoupling. He found a value of T = 103 K . This value is very importantfor cosmology because, as the CMB radiation is a blackbody, it allowed the scientist tohave an idea of the magnitude of the CMB radiation at our epoch.

At this time the astronomical community was wary of cosmology as it wasn’t fullyconsidered as a part of physics due to the lack of empiric results but Alpher, Hermanand Gamow’s prediction was rediscovered by Yakov Zel’dovich in the early 1960s, andindependently predicted by Robert Dicke at the same time.

2.2 First measurement and interpretation

Figure 2: Above gures the horn antenna with which Penzias and Wilson received the CMB.(Credit: NASA)

Dicke and his colleagues reasoned that the Big Bang must have scattered not only thematter that condensed into galaxies but also must have released a tremendous blast of radiation. And this radiation should be calculable with very sensitive instrument. They

calculated wavelength of the radiation (at T 0 = 103

K) and assigned some key points fortheir observations:

• This radiation should be isotropic ;

• Due to an important redshift, it should be detectable in the microwave region of the Electromagnetic spectrum.

At the same time in 1965, Arno Penzias and Robert Wilson were experimenting witha 6 meter horn antenna originally built to detect radio waves. While reducing their data,they removed all noises but were surprised to nd an anomaly unpredicted in their the-oretical model : a signal at 3 ,5 K measured at 4080 Mc/ s (MHz). They were very clear

4



that this signal was coming from all direction in space and independent of season andunpolarized. They were adamant that the unknown sound was not coming from galaxiesor other celestial source and were at loss to explain this phenomenon until Penzias andWilson contacted Dicke (and al.) who predicted the possible existence of a radiation relic

of the early universe.

Dicke and al. provided the interpretation to Penzias and Wilson measurement becausethe signal being isotropic and detectable in the microwave region (around the 10 − 2 m of wavelength) t their prediction. They assumed that this signal was that of a black bodyspectrum around 3 .5 K ±1 K.

2.3 Observations and Instrumentation

In order to obtain a more precise measurement of this black body and its temperature,it was necessary to go out of the atmosphere, into space, to get rid of the atmosphericperturbations that altered the signal. The satellite COBE equipped with the FIRASand DMR instrumentations was developed and launched into space in 1989. The dataobtained by COBE provided conrmation of the CMB being a black-body spectrum anda more precise measurement of the temperature of the black body : 2 ,726 ±0,03K .The satellite provided some maps of the CMB and the DMR allowed the discovery of anisotropies in the signal in the range of tens of µ K. These anisotropies being uctuations

of the temperature at the time of decoupling.

Figure 3: COBE satellite

Some years later, the WMAP was sent to space to collect more information aboutthe anisotropies found in the CMB. Measuring the CMB at ve different frequency from

23 to 94 GHz, it provided a map of the pattern of the uctuations in the signal whichallowed to precise the previous calculations, among others, about densities of baryonic

5







By doing partial differentiation we obtain :

−dT dt ×

ddT

(ln T ) = (8π ×G ×σ ×T 4

3c2 )12

−dT dt ×

1T

= T 2

c (

8π ×G ×σ3

)12

Integrating the previous equation,

−dT T 3

= 1c

(8π ×G ×σ

3 )

12 dt

−T − 2

−2

= tc

(8π ×G ×σ

3 )

12

T − 2 = tc ×2(

8π ×G ×σ3

)12

T 2 = ct(

332π ×G ×σ

)12

Knowing that the pressure of the molecules inside a closed container is :

P = 13

nM V

c2

P = 13

ρc2

P ∝ρc2

According to Stefan’s law :P = σT 4

Thus we can nd the density for the radiation :

ρc2

∝σT 4

ρrad ∝σT 4

c2

ρrad ∝σc2 ·

332πGσ ·

c2

t2

ρrad ∝σc2 ·

332πGσ ·

c2

t2

ρrad ∝3

32πG ·t− 2

8



ρrad ∝t − 2

ρrad ∝a − 4

We are now going to nd the density for the matter, the equation of the adiabatic

expansion of the universe is the following :

ddt

(ρa3 c2 ) + P ddt

(a3 ) = 0

For the limiting case of cold matter, the pressure goes like this :

P ρc2

The adiabatic expansion of universe becomes :

ddt

(ρa3 c2 ) = 0

So, after integrating we have :ρmatt a3 c2 = C

with C as a constant.And consequently :

ρmatt

∝

1a − 3

We are now able to calcule the temperature at the time of the decoupling. Using theassumption that ρrad = ρmatt at that time, we get :

ρmatt = ρ0matt (

aa0

)− 3

ρrad = ρ0rad (

aa0

)− 4

ρ0rad

ρ0matt

= ( aa0

)

ρ0rad

ρ0matt

= (T 0T

)+1 = ( T T 0

)− 1

T = ρ0

rad

ρ0matt ×T 0

With the following value, given in Gamow’s paper, at the time of decoupling : T 0 = 109

K ; ρmatt = 106

g ·cm− 1

; ρrad = 1 g ·cm− 1

, we can calculate T :

9



T = 103 K

Knowing T, we can now calculate the age of the universe corresponding to this tem-

perature knowing that T 0

corresponds to t0

= 100s

T ∝1√ t

T T 0

=√ t0

√ t√ t = √ t0 ×

T 0T

t = t0

×T 20

T 2

t = 1014 s

We can now try to calculate the temperature of the present universe with t =13.7109 yrs :

T = 15.2 K

We can see that the value obtained by Gamow is just one order of magnitude higherthan the actual value. We can conclude that this approximation is good enough but tohave a more precise value, maybe one of the hypothesis put forward in the calculationmust be rened.

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4 Anisotropies of the CMB

The discovery of small uctuations in the black body CMB came as a surprise. Thoseanisotropies were temperature uctuations frozen at the decoupling. Indeed, before thedecoupling, the universe was made of a plasma of different particles (baryons, electrons,protons, dark matter) distributed between overdense and underdense area. Photonswere trapped in the plasma. At the decoupling, the plasma is gradually replaced byneutral gas leading the photons to be released and as a result of Compton scattering, thenucleosynthesis process starts and matter is created. This matter is known as Baryonicmatter.

In the overdense area, the ionized gas clouds oscillated between contractions (hot anddense) and dilatation (less hot and less dense). Indeed plasma had its own gravitationaleld , this attracted matter towards it, the heat of photon-matter interactions createda large amount of outward pressure. These counteracting forces of gravity and pressurecreate oscillations analogous to sound waves (created by pressure difference). Theseoscillations, which are periodic uctuations in the density of the baryonic matter of theuniverse, are known as baryon acoustic oscillations (BAO).

Figure 7: Photons in plasma pushes matter inside the potential well and matter pushes back,creating oscillations (Credits : Wayne Hu)

They illustrate temperature uctuations as a peak of temperature when plasma con-tracts and a dip of temperature when it dilates. The ionized clouds, being of differentsizes and oscillating at different rythms, create the different acoustic peaks noticed in thedata analysis of the mission WMAP.

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Figure 8: Power spectrum of the CMB (Credits: APC univ-Paris

Those are peaks and dips corresponding to the maximum contraction and dilatationrespectively. These several stages of contraction and dilatation were frozen in the signalat the time of the decoupling, when photons separated from matter, which released theCMB.

The anisotropies provide a large number of information about the behaviour of matter

at that time and its evolution because they indicate the different densities in the plasma.For example, it seems that the densest areas of the universe are the areas were the rststars or galaxies clusters were born because of gravitation between the different lumps of matter. It is the starting point of the cosmology we know today, indeed, the fact that thedensity uctuations were adiabatic assured the physicist/cosmologist that the universewas itself adiabatic and that the theoretical model that described the early universe wasan ination of a “primeval rebal” : the Hot Big Bang.

The power spectrum of the CMB also conrmed the presence of Cold Dark Matter(CDM) and Dark Energy ( Λ) and allowed the physicists to determine the values of thesix cosmological parameters needed to describe our universe among them : the physicalbaryon density, the physical dark matter density, the age of the universe etc. and deducedthe cosmic expansion rate.

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5 Data Analysis : Extracting the CMB from Planck

frequency map.

5.1 Components of the frequency map

When reading the map, we can see that they contain much more than the CMB signal.Indeed, a frequency map is composed of the radiation of the CMB but also other radiationscoming from galaxies and matter disseminated in the universe. Those emissions, whichare concentrated on the galactic plane of the map, are the foreground components of theCMB. They are composed of the free-free emission, the synchrotron emission and theinterstellar dust emission.

Free free interaction : The free-free emission comes from the scattering of free elec-trons off ions (when electrons collide with ions and are redirected) in interstellarplasma when electrons collide with ions, and the electrons remain free after thecollision. Therefore they are free before and after the interaction with the ions.

Synchrotron : The synchrotron emission comes from interaction between electrons andthe magnetic elds in the galaxy. Indeed, the electrons that are parts of the cosmicrays are accelerated by the magnetic els, resulting in the emission of this signal.It is generally emitted at low frequency.

Interstellar dust : The interstellar dust is the dust created by dying stars. It is foundin all the universe inside or outside of the galaxies. With gravitation, the dust grainsgather in clouds. The dust in the clouds heats emitting a blackbody radiation thatbecomes more important on the frequency maps as the frequency grows.

Figure 9: Foreground components : Antenna temperature vs frequencies, (Credits: Bennet etal.)

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As we can see on the gure 9, each kind of emission is more or less present at a certainfrequency. That is to say that the intensity of the emission depends on the frequency.This behavior allowed us to select the frequency map used in the determination of ourCMB maps because they all hold a different relevance. Indeed, as we settled on only four

maps, we had to select the one were the CMB was more present and were the foregroundwas dominated by one kind of emission. That is why we chose the maps from 100 GHz,where the brightness temperature of the CMB is more important than the foreground,to 353 GHz where the map is mainly dominated by the dust emission.

5.2 Extracting the CMB Map.

To extract the CMB Map, we used four different maps retrieved from the Planck LegacyArchive. Each map is a full-sky frequency map obtained with the mission Planck and hasa certain angular resolution. We used the 100, 143, 217 and 353 GHz maps which havean angular resolution of 9 .68, 7.3, 5.02, 4.94 arcmin respectively.

The maps are all in a K CM B units, because as the CMB is a blackbody we can con-vert the measurement made by Planck into a brightness temperature and in this case, abrightness temperature obtained through the derivative of the blackbody of the CMB atT = 2.73 K. This brightness temperature of a blackbody is the same at all wavelengthand thus frequencies which means that the signal of the CMB is the same in our four

maps and thus that the anisotropies of the CMB are the same in the four maps.

Before extracting the CMB signal from the maps, we had to prepare the map so thatthe treatment is more convenient. Indeed, the maps retrieved from the Planck LegacyArchive were to big and contained to many values, which would have made the manipu-lation through Python quite long and difficult with our computers. The maps comes ina Fits le that requires the Healpy module in order to be manipulated in Python. Eachmap is an array of values - brightness temperature - associated to a point of the sky.

The maps are all in galactic coordinates with an initial pixel resolution (NSIDE) of 2048.As they all have a different angular resolution, the CMB can not be retrieved directly.Therefore the angular resolution had to be changed. We converted the higher angularresolution into the lowest angular resolution : the maps were smoothed to the lowestresolution : 9.68 arcmin. This smoothing was done by doing a gaussian convolution of the map with a full width half maximum (fwhm) of : FWHM = √ 9.682 −θ2 ×

pi180 × 60 rad

with θ the angular resolution of the other maps.

We then changed the NSIDE of all the maps to 128 and obtained four maps of equalangular resolution and NSIDE which allowed us to treat the data more efficiently.

14



Figure 10: Above is the 143 GHz frequency after preparation, we can distinctly see theforeground contamination in red.

Once this preliminary work done, the maps are ready to be combined to extract theCMB map. As the maps are in K CM B units, we know that the signal of the CMB is thesame in all of the maps. The only variation is then the noise created by the foregroundcomponents and by the instrument. Indeed, when we display the map, we can see thatthe CMB signal is masked by the intensity of the emitted by the foreground components,essentially the dust. So, if we remove the noise coming from the foreground and we sumthe maps, we obtain the map of the anisotropies of the CMB.

In order to remove the noise, we did an Internal Linear Combination. By supposingthat there are k CMB map observed at different frequency, we can describe each of themap as follows :

T kmap = T kCM B ( p) + T kforeground ( p) + T knoise ( p)

The T CM B map in each map is independent of the foreground and the noise of theinstrument because it is not dependent on the frequency. This allows us to write thelinear combination :

T =4

i=1

wi ×T (ν i )

and, if we set the condition that4

i=1

wi = 1

.

15



We obtain a map that can be described as :

T = T CM B +4

i=1

wi ×T noise (ν i )

Finally, to extract the CMB map it appears that this linear combination has to becompleted by the minimization of the noise of each map in order to compute the weights.Thus, the variance of the map is minimized through the Lagrange Multipliers method.The expression of the weights wi is the solution of the system ;

wi =4 j =1 C − 1

ij4ij =1 C − 1

ij

with C i,j the map-to-map covariance matrix.

Each weights actually reects the intensity of the noise contained in the correspondingmap. We can see that the 100 GHz map will have a weight closer to 1 than the 353 GHzthat is dominated by thermal dust for example.

Before doing the ILC, we removed the inuence of the galactic plane and its high in-tensity. Indeed, we created a mask removing the highest values of brightness temperature(> 0, 006) that we knew hold no relevance in extracting the CMB signal. This allowedus to obtain more precise values of the weights and thus, to diminish the uncertainties.

We then applied the resulting weights to their corresponding maps and we obtaineda map of the CMB.

Figure 11: Above is a nal result of the ILC with a map of the CMB radiation. We can seethat it is not perfect because there is remnants of the foreground contamination. It was

computed with a mask of 0.006

16



Figure 12: Here is a map of the CMB with a mask of 0.006 hiding the galactic plane

Before doing the ILC, we removed the inuence of the galactic plane and its high in-tensity. Indeed, we created a mask removing the highest values of brightness temperature(> 0, 006) that we knew hold no relevance in extracting the CMB signal (see Fig.12) .This allowed us to obtain more precise values of the weights and thus, to diminish theuncertainties.

5.3 Qualication of the CMB map

After obtaining a CMB map, we have to qualify it. Indeed, even though we obtain a mapwe are not sure if this map is truly representative of the CMB. There is two factors onwhich we have inuence that could modify the map given :

• The value used to make a mask : To suppress the galactic plane, we use a valueof brightness temperature that we know is above the value of the anisotropies fromthe map the most contaminated by the dust, in our case, the 353 GHz map. But,this mask still remain arbitrary and depending cover a more or less important part

of the map.

• The Nside can be changed when we convolve the map but it should not have adirect incidence on the values of the weights used to compute the CMB map.

From this, we can deduce that, to obtain a map which qualies as a CMB map wehave to compare the weights in regard to the value used for the masking of the maps.The second point tells us that the weights should not vary too much from one NSIDE toanother, the contrary would then tell us that the map obtained doesn’t qualify or that

we have a problem with the computing of the map.

17



Finally, we thought that studying the correlation between our map and a CMB mapobtained by the Planck mission, the map obtained by the SMICA method, could help usin the qualication of our map.

We computed the weights for the CMB map for three different NSIDE, 128, 512 and

1024 with different masks, that is to say that we remove the values higher than the maskvalue.

(a) The weights of the maps are computed with different masks for a xed NSIDE.NSIDE Mask Weights (from the 100 GHz map to the 353 GHz map

128

0.008 1.5291357811618715 0.28526562139520928 -0.88784468715324827 0.0734432845961704440.007 2.4327618135712679 -0.11678523350479744 -1.4612788860017343 0.145302305935264970.006 3.243135295179957 -0.56074275106708138 -1.8830920635240622 0.20069951941120270.005 3.6959853540788905 -0.97890300779567652 -1.9309916757022143 0.21390932941899835

(b) The weights of the map computed with a 0.008 mask for three different NSIDE.NSIDE Mask Weights (from W1 to W4)

1024 0.008 2.0481065142136066 -0.093985541006738629 -1.0523127422500691 0.098191769043206797512 0.008 2.097041584039042 -0.095550982995337277 -1.105780162767914 0.10428956172420759128 0.008 1.5291357811618715 0.28526562139520928 -0.88784468715324827 0.073443284596170444

Table 1: Weights computation result :

In the Table 1 above, the panel 2a, we displayed the weights computed with the ILCwhile using a certain mask value. The results below 0.005 do not gure because they

tend to remove some information, contained on the lowest frequency maps, regarding theCMB in addition to the foreground. The values obtained vary when we change the mask.It would seem that the best set of weights to use would be the one obtained with the0.008 Mask because the value of the weight associated to the 100 GHz is the closest to1 and the dustiest map, at 353 GHz, has the lowest weight value. The panel 2b displaysthe weights for a set mask of 0.008 for each NSIDE. The weights are quite close for themaps of NSIDE 512 and 1024 but we notice a bigger difference (of about 0.5) betweenthe weights of the maps with a NSIDE 128 compared to the other weights. This could

be explained by the loss of information due to the lowering of the pixel resolution of themaps.In order to determine if the mask at 0.008 is the best one to get a CMB map, we

decided to compute the correlation between our map and the Planck map with a differentvalue of the mask each time.

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Nside Mask Corrcoeff p-value slope

512

0.008 0.9537487797770 0.0 0.95192853979100.007 0.9465600954880 0.0 0.95473901368000.006 0.9445335421450 0.0 0.9563616544780

0.005 0.9431692737980 0.0 0.9599847324270

Table 3: Correlation and slope between Planck map and our map with a NSIDE of 512 fordifferent mask values

The Table 3 shows that the CMB map masked with a value of 0.008 has a bettercorrelation coefficient compared to the other mask. However its slope is lower. This tableapplies to the NSIDE 512 but, for any value of NSIDE, the correlation coefficient for the0.008 mask is the highest.

Figure 13: CMB map vs Planck mapfor masks 0.007 and 0.008 with a NSIDE

of 512

If we take a look at the Fig-ure 14, the reason for this dif-ference between the slope andthe correlation is clear : an es-tranged point around 0.006 islowering the slope. But, we canactually see that the points aremore concentrated toward theslope for in the graph of theCMB vs Planck CMB for themask 0.008 than for 0.007.

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To conclude, after the analysis of the weight computed. We can affirm that the bestweights to apply to the maps are the one computed with a mask of 0.008. This allow usto present a map that we can call CMB map :

Figure 14: Even though it doesn’t look like the perfect map like the Planck mission, wefound a good approximate.

5.4 Trying to compute the power spectrum

Once we obtain what qualies as a good CMB map, we can attempt to plot its powerspectrum. We use our CMB map at a NSIDE of 1024 and masked at 0.008 in order toplot its power spectrum. Before plotting the power spectrum, two thing must be takeninto account :

• The temperature brightness temperature weredistributed on the map according toa certain pixel window function.

• Our CMB map was obtained by smoothed value in order to create a map of FWHW = 9.68 arcmin .

The two precedent points have an incidence on the data contained in the map. Indeed,if we want to obtain the real power spectrum of the CMB, that is to say, the observedpower spectrum of the anisotropies we have to remove the pixel window function and thegaussian beam :

Clmeas = Clobs

·w2l

but also :

Clmeas = Clobs

·1

σ√ 2πexp −

(l)2

2σ2

20







6 Conclusion

The anisotropies of the CMB are a rich mine of information concerning the universe.Indeed, they can provide information on the density matter, both baryonic and dark,

and on the dark energy which was a huge discovery at that time because it conrmedthe Λ−CDM Hot Big Bang as a good theoretical model of the universe. The mysteryof the universe structure can also be investigated through the anisotropies, that is to say,the densest area of the anisotropy map can be related to the different clusters be theyglobular or galactic.Though these anisotropies are very important, we have to treat the data provided bydifferent satellite to obtain a good representation of what they look like. Indeed, at eachfrequency the satellite measure a temperature for each coordinate of the sky. These dataare then converted to the K CM B unit in order to have a uniform CMB in each of thefrequency maps and thus to compare the maps. The CMB is then extracted from themaps by removing the highest values of the brightness temperature, which corresponds tothe galactic plan and then doing an Internal Linear Combination with the determinationof the weights using the Lagrange Multipliers. Once the weights are applied and if wekeep the mask on, we obtain a representation of the CMB. While that in itself is reallyinteresting because we can already see the comportment of the anisotropies through thecolors variation, it would not allow us to deduce the different cosmological parameters.This is why we then compute the power spectrum from which the parameters are de-duced. Our CMB map not being a perfect map, we were just able to display the acousticpeak of the CMB.

The project allowed us to reproduce the analytical work of an astrophysician fromthe collection of the data to the nearly nal result which is the computation of the powerspectrum (and in normal circumstances the cosmological parameters). We were ableto try our hands on extracting a CMB map and its power spectrum but also to nallyunderstand why the CMB is a really important signal to the physicist. Though we focused

on the acoustic peaks because it is the only noticeable feature on our computed powerspectrum, we can now look forward to the next analysis of the anisotropies which willconsist in the analysis of a more precise measure of the polarization of the photons whichwill maybe unravel more information on the early universe and present universe.

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7 References

1. W. Hu, CMB Introduction , http://background.uchicago.edu/ ˜ whu/beginners/

introduction.html .

2 . T. D. E. Survey, Baryon Acoustic Oscillations , https://www.darkenergysurvey.org/science/bao.shtml .

3 . Wikipedia, Baryon Acoustic Oscillations , https://en.wikipedia.org/wiki/

baryon_acoustic_oscillations .

4. Wikipedia, Cosmic microwave background , https://en.wikipedia.org/wiki/

cosmic_microwave_background .

5 . apc.univ paris7.fr, Le Fond Diffus Cosmologique (CMB), http://www.apc.univ-

paris7.fr/APC_CS/content/le-fond-diffus-cosmologique-cmb .6 . E. S. Agency., Planck And the Cosmic Microwave Background. http://www.esa.

int / Our _ Activities / Space _ Science / Planck / Planck _ and _ the _ cosmic _

microwave_backgroundtk1bp .

7 . J. C. Mather et al., The Astrophysical Journal 420 , 439–444 (1994).

8 . K. Ichiki, Progress of Theoretical and Experimental Physics 2014 , 06B109 (2014).

9 . C. Bennett et al., The Astrophysical Journal Supplement Series 148 , 97 (2003).

10 . M. Bucher, International Journal of Modern Physics D 24 , 1530004 (2015).11. R. H. Dicke, P. J. E. Peebles, P. G. Roll, D. T. Wilkinson, The Astrophysical Journal

142 , 414–419 (1965).

12 . A. A. Penzias, R. W. Wilson, The Astrophysical Journal 142 , 419–421 (1965).

13 . H. K. Eriksen, A. J. Banday, K. M. G´orski, P. Lilje, The Astrophysical Journal 612 ,633 (2004).

14. G. Gamow, Physical Review 74 , 505 (1948).

15 . W. Hu, S. Dodelson, arXiv preprint astro-ph/0110414 (2001).16 . H. Mo, F. Van den Bosch, S. White, Galaxy formation and evolution (Cambridge

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Cosmic microwave background fluctuations