chapter 12 supernovae of type ia - leiden...

Chapter 12

Supernovae of Type Ia

12.1 Standard candles and distances

12.1.1 The principle

• Before starting with the details of supernovae, their type Ia andtheir cosmological relevance, let us set the stage with a few illus-trative considerations.

• Suppose we had a standard candle whose luminosity, L, we knowprecisely. Then, according to the definition of the luminosity dis-tance in (2.16), the distance can be inferred from the measuredflux, f , through

Dlum =

L

4π f. (12.1)

• Besides the redshift z, the luminosity distance will depend on thecosmological parameters,

Dlum = Dlum(z;Ωm0,ΩΛ0,H0, . . .) , (12.2)

which can be used in principle to determine cosmological param-eters from a set of distance measurements from a class of standardcandles.

• For this to work, the standard candles must be at a suitably highredshift for the luminosity distance to depend on the cosmologicalmodel. As we have seen in (2.17), all distance measures tend to

D ≈ cz

H0(12.3)

at low redshift and lose their sensitivity to all cosmological pa-rameters except H0.

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CHAPTER 12. SUPERNOVAE OF TYPE IA 126

• In reality, we rarely know the absolute luminosity L even of cos-mological standard candles. The problem is that they need to becalibrated first, which is only possible from a flux measurementonce the distance is known by other means, such as from paral-laxes in case of the Cepheids.

• Supernovae, however, which are the subject of this chapter, aretypically found at distances which are way too large to allow di-rect distance measurements. Therefore, the only way out is tocombine distant supernovae with local ones, for which the ap-proximate distance relation (12.3) holds.

• Any measurement of flux fi and redshift zi of the i-th standardcandle in a sample then yields an estimate for the luminosity L interms of the squared inverse Hubble constant,

L = 4π fi

czi

H0

2. (12.4)

Since all cosmological distance measures are proportional to theHubble length c/H0, the dependences on H0 on both sides of(12.1) cancels, and the determination of cosmological parametersother than the Hubble constant becomes possible. Thus, the firstlesson to learn is that cosmology from distant supernovae requiresa sample of nearby supernovae for calibration.

• Of course, this nearby sample must satisfy the same criterion asthe distance indicators used for the determination of the Hubbleconstant: their redshifts must be high enough for the peculiar ve-locities to be negligible, thus z 0.02. On the other hand, theredshifts must be low enough for the linear appoximation (12.3)to hold.

• It is important to note that it is not necessary to know the abso-lute luminosity L even up to the uncertainty in H0. If L is trulyindependent of redshift, cosmological parameters could still bedetermined through (12.1) from the shape of the measured rela-tion between flux and redshift even though its precise amplitude

may be unknown. It is only important that the objects used are

standard candles, but not how bright they are.

12.1.2 Requirements and degeneracies

• Let us now collect several facts about cosmological inferencefrom standard candles. Since we aim at the determination of cos-mological parameters, say Ωm0, it is important to estimate the ac-curacy that we can achieve from measurements of the luminositydistance.


• Suppose we restrict the attention to spatially flat cosmologicalmodels, for which ΩΛ0 = 1−Ωm0. Then, because the dependenceon the Hubble constant was canceled, Ωm0 is the only remainingrelevant parameter. We estimate the accuracy through first-orderTaylor expansion,

∆Dlum ≈dDlum

dΩm0∆Ωm0 , (12.5)

about a fiducial model, such as a ΛCDM model with Ωm0 = 0.3.

Logarithmic derivative of the lumi-nosity distance with respect to Ωm0.

• At a fiducial redshift of z ≈ 0.8, we find numerically

d ln Dlum

dΩm0≈ −0.5 , (12.6)

which shows that a relative distance accuracy of

∆Dlum

Dlum≈ −0.5∆Ωm0 (12.7)

is required to achieve an absolute accuracy of ∆Ωm0. For ∆Ωm0 ≈0.02, say, distances thus need to be known to ≈ 1%.

• This accuracy requires sufficiently large supernova samples. As-suming Poisson statistics for simplicity and distance measure-ments to N supernovae, the combined accuracy is

|∆Ωm0| ≈2√N

∆Dlum

Dlum. (12.8)

That is, an accuracy of ∆Ωm0 ≈ 0.02 can be achieved from ≈ 100supernovae whose individual distances are known to ≈ 10%.

• Anticipating physical properties of type-Ia supernovae, their in-trinsic peak luminosities in blue light are L ≈ 3.3 × 1043 erg s−1,with a relative scatter of order 10%. (As we shall see later, type-Iasupernovae are standardisable rather than standard candles, andthe standardising procedure is currently not able to reduce thescatter further.)

• Given uncertainties in the luminosity L and in the flux measure-ment, error propagation on (12.1) yields the distance uncertainty

Dlum =

dDlum

dL

2∆L

2 +

dDlum

d f

2∆ f

2

1/2

, (12.9)

or the relative uncertainty

∆Dlum

Dlum=

12

∆L

L

2+

∆ f

f

21/2

. (12.10)

Even if the flux could be measured precisely, the intrinsic lumi-nosity scatter currently forbids distance determinations to betterthan 10%.


• Fluxes have to be inferred from photon counts. For various rea-sons to be clarified later, supernova light curves should be de-termined until ∼ 35 days after the peak, when the luminosityhas typically dropped to ≈ 2.5 × 1042 erg s−1. The luminos-ity distance to z ≈ 0.8 is ≈ 5 Gpc, which implies fluxes f ≈1.1 × 10−14 erg s−1 cm−2 at peak and f ≈ 8.7 × 10−16 erg s−1 cm−2

35 days later.

The luminosity distance in a uni-verse with Ωm0 = 0.3 and ΩΛ0 =

0.7 with Hubble constant h = 0.72.

• Dividing by an average photon energy of 5 × 10−12 erg, multiply-ing with the area of a typical telescope mirror with 4 m diameter,and assuming a total quantum efficiency of 30%, we find detectedphoton fluxes of fγ ≈ 85 s−1 at peak and fγ ≈ 7 s−1 35 days af-terwards. These fluxes are typically distributed over a few CCDpixels.

• Supernovae occur in galaxies, which means that their fluxes needto be measured on the background of the galactic light. On thearea of a distant supernova image, the photon flux from the hostgalaxy is comparable to the flux from the supernova. Therefore,an estimate for the signal-to-noise ratio for the detection is

SN≈ N

2√

N

=

√N

2, (12.11)

where N is the number of photons per pixel detected from super-nova and host galaxy during the exposure time. Signal-to-noiseratios of 10 up to 35 days after the maximum thus requireN ≈ 400 photons per pixel. Assuming that the supernova ap-pears on typically ∼ 4 pixels, this implies exposure times of order4 × 400/7 ≈ 230 s, or a few minutes. Typical exposure times areof order 15 . . . 30 minutes to capture supernovae out to redshiftsz ∼ 1. Then, the photometric error around peak luminosity is cer-tainly less than the remaining scatter in the intrinsic luminosity,and relative distance accuracies of order 10% are within reach.

• However, a major difficulty is the fact that the identification oftype-Ia supernovae requires spectroscopy. Sufficiently accuratespectra typically require long exposures on the world’s largesttelescopes, such as ESO’s Very Large Telescope which consistsof four individual mirrors with 8 m diameter each.

• In order to see what we can hope to constrain by measuringangular-diameter distances, we form the gradient of Dlum in theΩm0-ΩΛ0 plane,

g ≡∂Dlum

∂Ωm0,∂Dlum

∂ΩΛ0

t, (12.12)

at a fiducial ΛCDM model with Ωm0 = 0.3. When normalised to


unit length, it turns out to point into the direction

g =

−0.760.65

. (12.13)

• This vector rotated by 90 then points into the direction in theΩm0-ΩΛ0 plane along which the luminosity distance does not

change. Thus, near the fiducial ΛCDM model, the parametercombination

P ≡ g ·Ωm0

ΩΛ0

= −0.76Ωm0 + 0.65ΩΛ0 (12.14)

is degenerate. The degeneracy direction, characterised bythe vector R(π/2)g = (0.65, 0.76)t, points under an angle ofarctan(0.76/0.65) = 49.5 with the Ωm0 axis, almost along thediagonal from the lower left to the upper right corner of the pa-rameter plane. Thus, it is almost perpendicular to the degeneracydirection obtained from the curvature constraint due to the CMB.This illustrates how parameter degeneracies can very efficientlybe broken by combining suitably different types of measurement.

12.2 Supernovae

12.2.1 Types and classification

Supernova 1994d in its host galaxy.• Supernovae are “eruptively variable” stars. A sudden rise inbrightness is followed by a gentle decline. They are unique eventswhich at peak brightness reach luminosities comparable to thoseof an entire galaxy, or 1010 . . . 1011

L⊙. They reach their maximawithin days and fade within several months.

• Supernovae are traditionally characterised according to their earlyspectra. If hydrogen lines are missing, they are of type I, oth-erwise of type II. Type-Ia supernovae show silicon lines, un-like type-Ib/c supernovae, which are distinguished by the promi-nence of helium lines. Normal type-II supernovae have spectradominated by hydrogen. They are subdivided according to theirlightcurve shape into type-IIL and type-IIP. Type-IIb supernovaspectra are dominated by helium instead.

Lightcurves of supernovae of differ-ent types.• Except for type-Ia, supernovae arise due to the collapse of a mas-

sive stellar core, followed by a thermonuclear explosion whichdisrupts the star by driving a shock wave through it. Core-collapse supernovae of type-I (i.e. types Ib/c) arise from stars withmasses between 8 . . . 30 M⊙, those of type-II from more massivestars.


• Type-Ia supernovae, which we are dealing with here, arise when awhite dwarf is driven over the Chandrasekhar mass limit by massoverflowing from a companion star. In a binary system, the moremassive star evolves faster and can reach its white-dwarf stagebefore its companion leaves the main sequence and becomes ared giant. When this happens, and the stars are close enough,matter will flow from the expanding red giant on the white dwarf.

Early (top) and late spectra of dif-ferent supernova types.

• Electron degeneracy pressure can stabilise white dwarfs up to theChandrasekhar mass limit of ∼ 1.4 M⊙. When the white dwarf isdriven over that limit, it collapses, starts a thermonuclear runawayand explodes. Since this type of explosion involves an approxi-mately fixed amount of mass, it is physically plausible that theexplosion releases a fixed amount of energy. Thus, the Chan-drasekhar mass limit is the main responsible for type-Ia super-novae to be approximate standard candles.

• The thermonuclear runaway in type-Ia supernovae converts thecarbon and oxygen in the core of the white dwarf into 56Ni, whichlater decays through 56Co into the stable 56Fe. According to de-tailed numerical explosion models, the nuclear fusion is started atrandom points near the centre of the white dwarf.

Supernova classification.

Type-Ia supernovae occur whenwhite dwarfs are driven over theChandrasekhar mass limit by massflowing from a companion star.

• Since the core material is degenerate, its pressure is independentof its temperature. The mass accreted from the companion star in-creases the pressure. Once it exceeds the Fermi pressure, inversebeta decay sets in,

p + e− → n + νe + γ , (12.15)

which suddenly removes the degenerate electrons. Under the highpressure, the temperature rises dramatically and ignites the fu-sion. The neutrinos carry away much of the explosion energy un-noticed because they can leave the supernova essentially withoutfurther interaction.

• The presence of silicon lines in the type-Ia spectra indicates thatnot all of the white dwarf’s material is converted into 56Ni. Thisshows that there is no explosion, but a deflagration, in which theflame front propagates at velocities below the sound speed. Thedeflagration can burn the material fast enough if it is turbulent,because the turbulence dramatically increases the surface of theflame front and thus the amount of material burnt per unit time.Typically, ∼ 0.5 M⊙ of 56Ni is produced in theoretical models.

• The peak brightness is reached when the deflagration frontreaches the former white dwarf’s surface and drives it as a rapidlyexpanding envelope into the surrounding space. The γ photons re-leased in the nuclear fusion processes are redshifted by scattering


with the expanding material and finally leave the explosion site asX-ray, UV, optical and infrared photons.

• Once the thermonuclear fusion has ended, additional energy isreleased by the β decay of 56Co into 56Fe with a half life of 77.12days. The exponential nature of the radioactive decay causes thetypical exponential decline phase in supernova light curves.

• Since the supernova light has to propagate through the expandingenvelope before we can see it, the opacity of the envelope and thusits metallicity are important for the appearance of the supernova.

12.2.2 Observations

• Since supernovae are transient phenomena, they can only be de-tected by sufficiently frequent monitoring of selected areas in thesky. Typically, fields are selected by their accessibility for thetelescope to be used and the least degree of absorption by theGalaxy. Since a type-Ia supernova event lasts for about a month,monitoring is required every few days.

• Supernovae are then detected by differential photometry, in whichthe average of all preceding images is subtracted from the lastimage taken. Since the seeing varies, the images appear con-volved with point-spread functions of variable width even if theyare taken with identical optics, thus the objects on them appearmore or less blurred. Before they can be meaningfully subtracted,they therefore have to be convolved with the same effective point-spread function. This causes several complications in the lateranalysis procedure, in particular with the photometry.

• Of course, this detection procedure returns many variable starsand supernovae of other types, which are not standard candlesand have to be removed from the sample. Pre-selection of type-Ia candidates is done by colour and the light-curve shape, butthe identification of type-Ia supernovae requires spectroscopy inorder to identify the decisive silicon lines at 6347 Å and 6371 Å.Since these lines move out of the optical spectrum for redshiftsz 0.5, near-infrared observations are crucially important for thehigh-redshift supernovae relevant for cosmology.

• Nearby supernovae, which we have seen need to be observed forcalibration, show that type-Ia supernovae are not standard can-dles but show a substantial scatter in luminosity. It turned out thatthere is an empirical relation between the duration of the super-nova event and its peak brightness in that brighter supernovae lastlonger.


• This relation between the light-curve shape and the brightness canbe used to standardise type-Ia supernovae. It was seen as a majorproblem for their cosmological interpretation that the origin forthis relation was unknown, and that its application to high-redshiftsupernovae was based on the untested assumption that the relationfound and calibrated with local supernovae would also hold there.Recent simulations indicate that the relation is an opacity effect:brighter supernovae produce more 56Ni and thus have a highermetallicity, which causes the envelope to be more opaque, the en-ergy transport through it to be slower, and therefore the supernovato last longer.

Lightcurves of type-Ia supernovaebefore (top) and after correction.

• Thus, before a type-Ia supernova can be used as a standard candle,its duration must be determined, which requires the light-curve tobe observed over sufficiently long time. It has to be taken intoaccount here that the cosmic expansion leads to a time dilation,due to which supernovae at redshift z appear longer by a factorof (1 + z). We note in passing that the confirmation of this timedilation effect indirectly confirms the cosmic expansion. After thestandardisation, the scatter in the peak brightnesses of nearby su-pernovae is substantially reduced. This encourages (and justifies)their use as standardisable candles for cosmology.

• The remaining relative uncertainty is now typically between10 . . . 15% for individual supernovae. Since, as we have seen fol-lowing (12.7), we require relative distance uncertainties at the percent level, of order a hundred distant supernovae are required be-fore meaningful cosmological constraints can be placed, whichjustifies the remark after (12.8).

• An example for the several currently ongoing supernova surveysis the Supernova Legacy Survey (SNLS) in the framework of theCanada-France-Hawaii Legacy Survey (CFHTLS), which is car-ried out with the 4-m Canada-France-Hawaii telescope on MaunaKea. It monitors four fields of one square degree each five timesduring the 18 days of dark time between two full moons (luna-tions).

• Differential photometry is performed to find out variables, andcandidate type-Ia supernovae are selected by light-curve fittingafter removing known variable stars. Spectroscopy on the largesttelescopes (mostly ESO’s VLT, but also the Keck and Geminitelescopes) is then needed to identify type-Ia supernovae. To givea few characteristic numbers, the SNLS has taken 142 spectra oftype-Ia candidates during its first year of operation, of which 91were identified as type-Ia supernovae.

• The light curves of these objects are observed in several differentfilter bands. This is important to correct for interstellar absorp-


tion. Any dimming by intervening material makes supernovaeappear fainter, and thus more distant, and will bias the cosmolog-ical results towards faster expansion. Since the intrinsic coloursof type-Ia supernovae are characteristic, any deviation betweenthe observed and the intrinsic colours signals interstellar absorp-tion which is corrected by adapting the amount of absorption suchthat the observed is transformed back into the intrinsic colour.

• This correction procedure is expected to work well unless there ismaterial on the way which absorbs equally at all wavelengths, so-called “grey dust”. This could happen if the absorbing dust grainsare large compared to the wavelength. Currently, it is quite diffi-cult to concusively rule out grey dust, although it is implausiblebased on the interstellar absorption observed in the Galaxy.

Distances to type-Ia supernovae (inlogarithmic units) as a function oftheir redshift, as measured by theSupernova Legacy Survey.

• After applying the corrections for absorption and duration, eachsupernova yields an estimate for the luminosity distance to its red-shift. Together, the supernovae in the observed sample constrainthe evolution of the luminosity distance with redshift, whichis then fit varying the cosmological parameters except for H0,i.e. typically Ωm0 and ΩΛ0. This yields an “allowed” region inthe Ωm0-ΩΛ0 plane compatible with the measurements which isdegenerate in the direction calculated before.

Cosmological parameter constraintsderived from the same data.

• More information or further assumptions are necessary to breakthe degeneracy. The most common assumption, justified by theCMB measurements, is that the Universe is spatially flat. Basedupon it, the SNLS data yield a matter density parameter of

Ωm0 = 0.263 ± 0.037 . (12.16)

This is a remarkable result. First of all, it confirms the other in-dependent measurements we have already discussed, which werebased on kinematics, cluster evolution and the CMB. Second, itshows that, in the assumed spatially flat universe, the dominantcontribution to the total energy density must come from some-thing else than matter, possibly the cosmological constant.

• It is important for the later discussion to realise in what way theparameter constraints from supernovae differ from those from theCMB. The fluctuations in the latter show that the Universe is atleast nearly spatially flat, and the density parameters in dark andbaryonic matter are near 0.25 and 0.045, respectively. The restmust be the cosmological constant, or the dark energy. Arisingearly in the cosmic history, the CMB itself is almost insensitiveto the cosmological constant, and thus it can only constrain itindirectly.

• Type-Ia supernovae, however, measure the angular-diameter dis-tance during the late cosmic evolution, when the cosmological


constant is much more important. As (12.14) shows, the luminos-ity distance constrains the difference between the two parameters,

ΩΛ0 = 1.17Ωm0 + P , (12.17)

where the degenerate parameter P is determined by the measure-ment. Assuming ΩΛ0 = 1 − Ωm0 as in a spatially-flat universeyields

P = 1 − 2.17Ωm0 ≈ 0.43 (12.18)

from the SNLS first-year result (12.16), illustrating that the sur-vey has constrained the density parameters to follow the relation

ΩΛ0 ≈ 1.17Ωm0 + 0.43 . (12.19)

• The relative acceleration of the universe, a/a, is given by theequation

a

a= H

20

ΩΛ0 −

Ωm0

2a3

(12.20)

if matter is pressure-less, which follows directly from Einstein’sfield equations. Thus, the expansion of the universe acceleratestoday (a = 1) if a = H

20(ΩΛ0 − Ωm0/2) > 0, or ΩΛ0 > Ωm0/2.

Given the measurement (12.19), the conclusion seems inevitablethat the Universe’s expansion does indeed accelerate today.

Above redshift z ≈ 1, the cosmic ac-celeration seems to turn into decel-eration.

• If the Universe is indeed spatially flat, then the transition betweendecelerated and accelerated expansion happened at

1 − 0.263 ≈ 0.2632a3 ⇒ a = 0.56 , (12.21)

or at redshift z ≈ 0.78. Luminosity distances to supernovae atlarger redshifts should show this transition, and in fact they do.

12.2.3 Potential problems

• The main observational issues is dust in the SN host galaxy. If oneis careful enough (an observational issue), then one can estimatethe reddening of the supernova accurately. Yet, there is a terri-ble conceptual issue that is currently one of the major stumbling-blocks of the field - we do not know with certainty what the re-lationship between reddening and the extinction (the amount oflight lost - this affects the inferred absolute magnitude).

• There are two ways around this issue - trying to better estimatethe attenuation curve shape (the relationship between reddeningand extinction, this is not straightforward and requires excellentphotometry over a long wavelength range; see Wood-Vasey et al.


2007), and going to systems that one a priori expects to be free ofdust (elliptical galaxies; this involves a considerable reduction insample size; see, e.g., Sullivan et al. 2003, MNRAS, 340, 1057).Both approaches have been taken.

• The problem with possible grey dust has already been mentioned:While the typical colours of type-Ia supernovae allow the detec-tion and correction of the reddening coming with typical inter-stellar absorption, grey dust would leave no trace in the couloursand remain undetectable. However, grey dust would re-emit theabsorbed radiation in the infrared and add to the infrared back-ground, which is quite well constrained. It thus seems that greydust is not an important contaminant, if it exists.

• Gravitational lensing is inevitable for distant supernovae. De-pending on the line-of-sight, they are either magnified or demag-nified. Since, due to nonlinear structures, high magnificationscan occasionally happen, the magnification distribution must beskewed towards demagnification to keep the mean of zero mag-nification. Thus, the most probable magnification experienced bysupernova is below unity. In other words, lensing may lead toa slight demagnification if lines-of-sight towards type-Ia super-novae are random with respect to the matter distribution. In anycase, the rms cosmic magnification adds to the intrinsic scatterof the supernova luminosities. It may become significant for red-shifts z 1.

• It is a difficult and debated question whether supernovae at highredshifts are intrinsically the same as at low redshifts where theyare calibrated. Should there be undetected systematic differences,cosmological inferences could be wrong. In particular, it may benatural to assume that metallicties at high redshifts are lower thanat low redshifts. Since supernovae last longer if their atmospheresare more opaque, lower metallicity may imply shorter supernovaevents, leading to underestimated luminosities and overestimateddistances. Simulations of type-Ia supernovae, however, seem toshow that such an effect is probably not significant.

• It was also speculated that distant supernovae may be intrinsicallybluer than nearby ones due to their lower metallicity. Should thisbe so, the extinction correction, which is derived from redden-ing, would be underestimated, causing intrinsic luminosities tobe under- and luminosity distances to be overestimated. Thus,this effect would lead to an underestimate of the expansion rateand counteract the cosmological constant. There is currently noindication of such a colour effect.

• Supernovae of types Ib/c may be mistaken for those of type Iaif the identification of the characteristic silicon lines fails for


some reason. Since they are typically fainter than type-Ia su-pernovae, they would contaminate the sample and bias resultstowards higher luminosity distances, and thus towards a highercosmological constant. It seems, however, that the possible con-tamination by non-type-Ia supernovae is so small that it has nonoticeable effect.

• Several more potential problems exist. It has been argued for awhile that, if the evidence for a cosmological constant was basedexclusively on type-Ia supernovae, it would probably not be con-sidered entirely convincing. However, since the supernova ob-servations come to conclusions compatible with virtually all in-dependent cosmological measurements, they add substantially tothe persuasiveness of the cosmological standard model. More-over, recent supernova simulations reveal good physical reasonswhy they should in fact be reliable, standardisable candles.

Chapter 2

The cosmological standardmodel

2.1 Introduction

• One of the landmark achievements of the last decade of astro-nomical study is the establishment of a ‘cosmological standardmodel’. By this term, we mean a consistent theoretical back-ground which is at the same time simple and broad enough tooffer coherent explanations for the vast majority of cosmoogicalphenomena.

• This lecture will explain and discuss the empirical evidence towhich this cosmological standard model owes its convincingpower. The construction of homogeneous and isotropic cosmolo-gies from general relativity, and the study of their physical prop-erties and evolution, is treated elsewhere (see, e.g. the separatelecture scripts on general relativity and on cosmology).

• We will start with a brief overview of a few relevant observationalfacts about the Universe which played a critical role in motivatingour current cosmological picture, continue with a short timelineof how it is currently imagined that the Universe evolved, and wewill review a few key aspects of the cosmological model. Thebulk of the course will discuss in depth a number of observa-tions/methods, and how they fit in to our current cosmologicalpicture.

2.2 Observational overview; the basics

• Galaxies exist - the Universe is filled with hundreds of billionsof galaxies, with a wide range in properties. In their inner parts,

5

CHAPTER 2. THE COSMOLOGICAL STANDARD MODEL 6

their mass is dominated by cold gas (mostly neutral or molecularhydrogen) and stars (really dense hydrogen!).

• Exapanding Universe - Slipher and Hubble demonstrated con-vincingly in the 20s and 30s that the Universe is expanding, andthat the redshift is proportional to distance (at least locally). Thetruly astounding result, initially determined in the 1990s usingSupernova Ia, is that the expansion of the Universe is accelerating

at the present time. Naıvely, this is counterintuitive, inasmuch asthe matter content of the Universe should be decelerating its ownexpansion.

• Dark Matter - both inside galaxies and in galaxy clusters, mo-tions of gas and stars imply gravitational masses dramatically inexcess of any mass plausibly associated with stars and gas, i.e.,v √GMvisible/R. This discrepancy was first noted in the 1940sby Fritz Zwicky, and became really obvious in the 1970s in thestudy of spiral galaxy rotation curves.

• Cosmic Microwave Background - There is a cosmic microwavebackground with T ∼ 2.73K. Its main defining feature is its as-tonishing level of homogeneity – it is flat to 1 part in 105.

• Helium - There is way too much Helium in the Universe to havebeen made in just stars (or, put differently, the ratio between He-lium and the other, heavier elements is just too high to be ex-plained by being made in stars alone).

There are a large number of other important observations, but these arethe key aspects which are useful to bear in mind when trying to parsethe elements of our current cosmological picture.

2.3 A brief history of time

Our cosmological picture unifies ‘known physics’ with a few novel in-gredients motivated entirely by astronomical observation.

• Inflation - it is currently postulated that the Universe had very

early in its evolution a phase of exponential expansion (roughly60 e-foldings) which took microscopically small parts of the Uni-verse and boosted them in scale to giant, macroscopic scales (∼galaxy scales and larger). This has two main advantages. It solvesthe flatness problem (the Universe would recollapse or expandvery rapidly in the case of Ωinitial > 1 or Ωinitial < 1. Ωinitial needsto be within 1 part in 1060 of unity to ensure that Ω ∼ 1 today (as


|Ωtotal−1| ∝ t, and given that the age of the Universe is ∼ 1017s to-

day and evaluating the Ωinitial at the Planck time tPlanck ∼ 10−43s).

There is the advantage also that it solves the horizon problem, andblows up tiny quantum fluctuations in the initial density field intodensity perturbations that later seed structure formation. Inflation

is completely motivated by cosmology, although physics theoristsare happy about it.

• Radiation Dominated Era - At this time T ∝ 1/a; i.e., as theUniverse expands it cools also. Protons and neutrons freeze outof the mix as T 1 GeV. The ratio of p to n is determined bytheir mass difference coupled with the decay half life of neutrons,ending up in a n/p number ratio of 1/7. When temperatures arelow enough T ∼ 1 MeV nucleosynthesis can proceed, forming2H, 3

H, 4He, 7

Li, etc.

• Recombination - At T ∼ 3000K, the hydrogen can no longer stayionised, so it recombines. There are no more electrons to Thomp-son scatter off of, so the Universe suddenly becomes transparentand photons can freely propagate. The CMB is a relic of this tran-sition, and offers a direct view of the structure of the Universe atthis time.

• Matter Dominated Era - structure formation, galaxy formation,expansion of the Universe. A ‘boring’ time for cosmologists (whowant to measure the formation of structure and the expansion his-tory to measure cosmological parameters), and the most excitingtime for astrophysicists (galaxy formation, star formation, planetformation, life, etc - minor details!) The main astrophysically-motivated ingredients here are dark matter and dark energy / cos-

mological constant.

2.4 Friedmann models

2.4.1 The metric

• Cosmology deals with the physical properties of the Universe asa whole. The only of the four known interactions which can playa role on cosmic length scales is gravity. Electromagnetism, theonly other interaction with infinite range, has sources of oppo-site charge which tend to shield each other on comparatively verysmall scales. Cosmic magnetic fields can perhaps reach coher-ence lengths on the order of 10 Mpc, but their strengths arefar too low for them to be important for the cosmic evolution.The weak and the strong interaction, of course, have microscopicrange and must thus be unimportant for cosmology as a whole.


• The best current theory of gravity is Einstein’s theory of gen-eral relativity, which relates the geometry of a four-dimensionalspace-time manifold to its material and energy content. Cosmo-logical models must thus be constructed as solutions of Einstein’sfield equations.

• Symmetry assumptions greatly simplify this process. Guided byobservations to be specified later, we assume that the Universe ap-pears approximately identically in all directions of observation,in other words, it is assumed to be isotropic on average. Whilethis assumption is obviously incorrect in our cosmological neigh-bourhood, it holds with increasing precision if observations areaveraged on increasingly large scales.

• Strictly speaking, the assumption of isotropy can only be validin a prefered reference frame which is at rest with respect to themean cosmic motion. The motion of the Earth within this restframe must be subtracted before any observation can be expectedto appear isotropic.

• The second assumption holds that the Universe should appearequally isotropic about any of its points. Then, it is homoge-neous. Searching for isotropic and homogeneous solutions forEinstein’s field equations leads uniquely to line element of theRobertson-Walker metric,

ds2 = −c

2dt2 + a

2(t)

dr2

1 − kr2 + r2dθ2 + sin2 θdφ2

, (2.1)

in which r is a radial coordinate, k is a parameter quantifyingthe curvature, and the scale factor a(t) isotropically stretchesor shrinks the three-dimensional spatial sections of the four-dimensional space-time; the scale factor is commonly normalisedsuch that a0 = 1 at the present time;

• as usual, the line element ds gives the proper time measured by anobserver who moves by (dr, rdθ, r sin θdφ) within the coordinate

time interval dt; for light, in particular, ds = 0;

• coordinates can always be scaled such that the curvature parame-ter k is either zero or ±1;

• by a suitable transformation of the radial coordinate r, we canrewrite the metric in the form

ds2 = −c

2dt2 + a

2(t)dw2 + f

2k

(w)dθ2 + sin2 θdφ2

, (2.2)

where the radial function fk(w) is given by

fk(w) =

sin(w) (k = 1)w (k = 0)sinh(w) (k = −1)

; (2.3)


sometimes one or the other form of the metric is more convenient;

2.4.2 Redshift and expansion

• the changing scale of the Universe gives rise to the cosmologicalredshift z; the wavelength of light from a distant source seen by anobserver changes by the same amount as the Universe changes itsscale while the light is travelling; thus, if λe and λo are the emittedand observed wavelengths, respectively, they are given by

λo

λe

=a0

a=

1a, (2.4)

where a is the scale factor at the time of emission and a0 is nor-malised to unity; the relative wavelength change is the redshift,

z ≡ λo − λe

λe

=1a− 1 , (2.5)

and thus1 + z =

1a, a =

11 + z

; (2.6)

• when inserted into Einstein’s field equations, two ordinary differ-ential equations for the scale factor a(t) result; when combined,they can be brought into the form

H(a)2 = H20

Ωm,0

a3 +Ωr,0

a4 +ΩΛ +1 −Ωm,0 −Ωr,0 −ΩΛ

a2

≡ H20 E

2(a) ; (2.7)

this is Friedmann’s equation, in which the relative expansion ratea/a ≡ H(a) is replaced by the Hubble function whose presentvalue is the Hubble constant, and the matter-energy content isdescribed by the three density parameters Ωr,0, Ωm,0 and ΩΛ,0;

• the dimension-less parametersΩm,0 andΩr,0 describe the densitiesof matter and radiation in units of the critical density

ρcr,0 ≡3H

20

8πG; (2.8)

matter and radiation are distinguished by their pressure; for mat-ter, the pressure p is neglected because it is very small com-pared to the energy density ρc2, while radiation is characterisedby p = ρc2/3;

• a Robertson-Walker metric whose scale factor satisfies Fried-mann’s equation is called a Friedmann-Lemaıtre-Robertson-Walker metric; the cosmological standard model asserts that theUniverse at large is described by such a metric, and is thus char-acterised by the four parameters Ωm,0, Ωr,0, ΩΛ and H0;


• since the critical density evolves in time, so do the density param-eters; their evolution is given by

Ωm(a) =Ωm,0

a +Ωm,0(1 − a) +ΩΛ,0(a3 − a)(2.9)

for the matter-density parameter and

ΩΛ(a) =ΩΛ,0a

3

a +Ωm,0(1 − a) +ΩΛ,0(a3 − a)(2.10)

for the cosmological constant; in particular, these two equationsshow that Ωm(a) → 1 and ΩΛ(a) → 0 for a → 0, independent oftheir present values, and thatΩm(a)+ΩΛ(a) = 1 ifΩm,0+ΩΛ,0 = 1today;

• this lecture is devoted to answering two essential questions: (1)What are the values of the parameters defining characterisingFriedmann’s equation? (2) How can we understand the deviations

of the real universe from a purely homogeneous and isotropicspace-time?

2.4.3 Age and distances

• since Friedmann’s equation gives the relative expansion rate a/a,we can use it to infer the age of the Universe,

t =

t

0dt =

1

0

da

a=

1

0

da

aH(a)=

1H0

1

0

da

aE(a), (2.11)

which illustrates that the age scale is the inverse Hubble constantH−10 ; a simple example is given by the Einstein-de Sitter model,

which (unrealistically, as we shall see later) assumes Ωm,0 = 1,Ωr,0 = 0 and ΩΛ = 0; then, E(a) = a

−3/2 and

t =1

H0

1

0

√ada =

23H0

; (2.12)

• distances can be defined in many ways which typically lead to dif-ferent expressions; we summarise the most common definitionshere; the proper distance Dprop is the distance measured by thelight-travel time, thus

dDprop = cdt ⇒ Dprop =c

H0

da

aE(a), (2.13)

where the integral has to be evaluated between the scale factorsof emission and observation of the light signal;


• the comoving distance Dcom is simply defined as the distance mea-sured along a radial light ray ignoring changes in the scale factor,thus dDcom = dw; since light rays propagate with zero propertime, ds = 0, which gives

dDcom = dw =cdt

a=

c

H0

da

aa=

c

H0

da

a2E(a); (2.14)

• the angular-diameter distance Dang is defined such that the samerelation as in Euclidean space holds between the physical size ofan object and its angular size; it turns out to be

Dang(a) = a fK[w(a)] = a fK[Dcom(a)] , (2.15)

where fK(w) is given by (2.3);

• the luminosity distance Dlum is analogously defined to reproducethe Euclidean relation between the luminosity of an object and itsobserved flux; this gives

Dlum(a) =Dang(a)

a2 =fK[w(a)]

a=

fK[Dcom(a)]a

, (2.16)

• these distance measures can vastly differ at scale factors a 1;for small distances, i.e. for a ∼ 1, they all reproduce the linearrelation

D(z) =cz

H0. (2.17)

• since time is finite in a universe with Big Bang, any particle canonly be influenced by, and can only influence, events within afinite region; such regions are called horizons; several differentdefinitions of horizons exist; they are typically characterised bysome speed, e.g. the light speed, times the inverse Hubble func-tion which sets the time scale;

2.4.4 The radiation-dominated phase

• it is an empirical fact that the Universe is expanding; earlier intime, therefore, the scale factor must have been smaller than to-day, a < 1; in principle, it is possible for Friedmann models thatthey had a finite minimum size at a finite time in the past and thusnever reached a vanishing radius, a = 0; however, it turns outthat a few crucial observational results rule out such “bouncing”models; this implies that a Unniverse like ours which is expand-ing today must have started from a = 0 a finite time ago, in otherwords, there must have been a Big Bang;


• equation (2.7) shows that the radiation density increases like a−4

as the scale factor decreases, while the matter density increaseswith one power of a less; even though the radiation density is verymuch smaller today than the matter density, this means that therehas been a period in the early evolution of the Universe in whichradiation dominated the energy density; this radiation-dominated

era is very important for several observational aspects of the cos-mological standard model;

• since the radiation retains the Planckian spectrum which it ac-quired in the very early Universe in the intense interactions withcharged particles, its energy density is fully characterised by itstemperature T ; since the energy density is both proportional to T

4

and a−4, its temperature falls like T ∝ a

−1;

2.5 Structures

2.5.1 Structure growth

• the hierarchy of cosmic structures is assumed to have grown fromprimordial seed fluctuations in the process of gravitational col-lapse: overdense regions attract material and grow; they are de-scribed by the density contrast δ, which is the density fluctuationrelative to the mean density ρ,

δ ≡ ρ − ρρ

; (2.18)

• linear perturbation theory shows that the density contrast δ is de-scribed by the second-order differential equation

δ + 2Hδ − 4πGρδ = 0 (2.19)

if the dark matter is cold, i.e. if its constituens move with negli-gible velocities; notice that this is an oscillator equation with animaginary frequency and a characteristic time scale (4πGρ)−1/2,and a damping term 2Hδ which shows that the cosmic expansionslows down the gravitational instability;

• equation (2.19) has two solutions, a growing and a decayingmode; while the latter is irrelevant for structure growth, the grow-ing mode is described by the growth factor D+(a), defined suchthat the density contrast at the scale factor a is related to an initialdensity contrast δi by δ(a) = D+(a)δi; in most cases of practicalrelevance, the growth factor is accurately described by the fitting


formula

D+(a) =5a

2Ωm

Ω4/7

m −ΩΛ +1 +

12Ωm

1 +

170ΩΛ

−1

,

(2.20)where the density parameters have to be evaluated at the scalefactor a;

• a very important length scale for cosmic structure growth is setby the horizon size at the end of the radiation-dominated phase;structures smaller than that became causally connected while ra-diation was still dominating; the fast expansion due to the radia-tion density inhibited further growth of such structures until thematter density became dominant; small structures are thereforesuppressed compared to large structures which became causallyconnected only after radiation domination; the horizon size at theend of the radiation-dominated era thus divides between largerstructures which could grow without inhibition, and smaller struc-tures which were suppressed during radiation domination; it turnsout to be

req =c

H0

a3/2eq

2Ωm,0; (2.21)

2.5.2 The power spectrum

• it is physically plausible that the density contrast in the Universeis a Gaussian random field, i.e. that the probability for finding avalue between δ and δ + dδ is given by a Gaussian distribution;the principal reason for this is the central limit theorem, whichholds that the distribution of a quantity which is obtained by su-perposition of random contributions which are all drawn from thesame probability distribution (with finite variance) turns into aGaussian in the limit of infinitely many contribtions;

• a Gaussian random process is characterised by two numbers, themean and the variance; by construction, the mean of the densitycontrast vanishes, such that the variance defines it completely;

• in linear approximation, density perturbations grow in place, aseq. (2.19) shows because the density contrast at one position xdoes not depend on the density contrast at another; as long asstructures evolve linearly, their scale will be preserved, which im-plies that it is advantageous to study structure growth in Fourierrather than in configuration space;

• the variance of the density contrast δ(k) in Fourier space is calledthe power spectrum

δ(k)δ∗(k)

≡ (2π)3

Pδ(k) δD(k − k) , (2.22)


where the Dirac δ function ensures that modes with different wavevectors are independent;

• once the power spectrum is known, the statistical properties of thelinear density contrast are completely specified; it is a remarkablefact that two simple assumptions about the nature of the cosmicstructures and the dark matter constrain the shape of Pδ(k) com-pletely; if the mass contained in fluctuations of horizon size isindependent of time, and if the dark matter is cold, the powerspectrum will behave as

Pδ(k) ∝

k (k k0)k−3 (k k0)

, (2.23)

where k0 = 2πreq is the wave number of the horizon size at theend of radiation domination. The rise of the amount of structuretowards larger k (for small k) comes from the fact that the horizonsize is very small at early times and becomes larger with time,allowing a head-start for growth of smaller scale modes; the steepdecline for structures smaller than req reflects the suppression ofstructure growth during radiation domination;

The linear CDM power spectrumwith its characteristic shape (red),and the deformation by non-linearevolution at the small-scale end(green).

2.5.3 Non-linear evolution

• as the density contrast approaches unity, its evolution becomesnon-linear; the onset of non-linear evolution can be described bythe so-called Zel’dovich approximation, which gives an approxi-mate description of particle trajectories;

• although the Zel’dovich approximation breaks down as the non-linear evolution proceeds, it is remarkable for two applications;first, it allows a computation of the shapes of collapsing dark-matter structures and arrives at the conclusion that the collapsemust be anisotropic, leading to the formation of sheets and fil-aments; second, it provides an explanation for the origin of theangular momentum of cosmic structures; filamentary structuresthus appear as a natural consequence of gravitational collapse ina Gaussian random field;

• in the course of non-linear evolution, overdensities contract,which implies that matter is transported from larger to smallerscales; the linear result that density fluctuations grow in placetherefore becomes invalid, and power in the density fluctuationfield is transported towards smaller modes, or towards larger wavenumber k; this mode coupling process deforms the power spec-trum on small scales, i.e. for large k;


• detailed studies of the non-linear evolution of cosmic structuresrequire numerical simulations, which need to cover large scalesand to resolve small scales well at the same time; much progresshas been achieved in this field within the last two decades dueto the fortunate combination of increasing computer power withhighly sophisticated numerical algorithms, such as particle-meshand tree codes, and adaptive mesh refinement techniques;

chapter 12 supernovae of type ia - leiden...

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