University of LjubljanaFaculty of Mathematics and Physics

Oddelek za fiziko

Seminar - 4.letnik

Peculiar velocities andType Ia supernovae

Avtor: Vid Irsic

Mentor: dr. Anze Slosar

Ljubljana, December 2009

Abstract

Since the earliest study of supernovae, it has been suggested that these luminousevents might be used as standard candles for cosmological measurements. In the sub-sequent years many studies using Type Ia supernovae has revealed that the expansionof our universe is accelerating due to a new and unexpected gravitational phenomenaoften regarded as the ’dark energy’. From SNe Ia observational data one could mea-sure cosmological parameters of interest through the Hubble rate (H0,Ωm,ΩΛ,w). Inthe most recent years it has become apparent that the correlations between peculiarvelocities of the supernovae are significantly contributing to the results and are pre-senting an oppurtunity to measure underlying matter power spectrum and thus othercosmological parameters (Ωb,ns,σ8).

Contents

1 Introduction 1

2 Type Ia supernovae 22.1 Standard candles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Data corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Distances in cosmology 3

4 Cosmological parameters from SNe Ia 74.1 Peculiar velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5 Conclusion 11

References 12

1 Introduction

Supernovae (SNe) are classified primarily on the basis of their optical spectra. Those thatshow obvious hydrogen lines are called Type II. Those that lack obvious hydrogen linesbut that do develop obvious He I (He I is neutral and He II singly ionized helium) linesare Type Ib. Neither hydrogen nor He I lines are conspicuous in supernovae of Type Ic.Most if not all events of Types II, Ib, and Ic are thought to result from core collapse inmassive stars. The envelopes are ejected as the cores form neutron stars or black holes.The appearance of the optical spectra depends on the extent to which the progenitor starsretain (Type II) or lose their hydrogen layers (Type Ib), or even lose their helium layers(Type Ic), by the time of core collapse. The observed tendency for events of these threetypes to occur only in star-forming regions of galaxies is strong evidence that they comefrom massive stars that exist so briefly (≤ 3 × 108 years) that they die where they wereborn [1].

(a) (b)

Figure 1 — A picutre with Hubble Space Telescope (HST) of a SNe Ia in 1994 (left). The brightdot in the lower left corner is the exploding SN. It has a brightness of the whole host galaxy. Pictureon the right illustrates the possible scenario of close binary system as a most probable candidate forSNe Ia explosions. [2]

The spectra of Type Ia supernovae (SNe Ia) lack hydrogen and He I lines but unlikeType Ic events they do include among other distinguishing characteristics a deep absorp-tion feature near 6100 A that is produced by blueshifted Si II λ 6347 A, 6371 A [1] (e.q.

1

(a) (b)

Figure 2 — A typical light-curve (left) and specter (right) of supernovae of different type (SNLS-3yr)[4].

Figure (2)). Given a good observed spectrum that extends as far as 6100 A in the super-nova rest frame, deciding whether it is or is not a spectrum of a Type Ia is practicallyalways straightforward. Supernovae for which spectra aren’t available are sometimes clas-sified as Type Ia on the basis of their photometry (broad–band light curves and colors),but this must be done with caution. Supernovae that appear in elliptical galaxies are com-monly assumed to be of Type Ia because so far not a single event in an elliptical galaxy hasbeen found to be otherwise. Type Ia supernovae exhibit small magnitude scatter aroundthe peak magnitude.

Although SNe Ia occur in elliptical galaxies, which contain only old (∼ 1010 years)stellar populations, they also occur in spiral galaxies, at a rate that is correlated withgalaxy color: the bluer the galaxy – and by inference the higher the star formation rateduring the last 109 years – the higher the SN Ia rate. This means that most SNe Ia areproduced by stars that were born moderately massive (but ≤ 8 M ) and last ∼ 109 years[1, 3].

2 Type Ia supernovae

As mentioned above, observations indicate that SNe Ia are produced by stars that are bornwith less than 8 M. However, single stars (and effectively single stars in wide binaries)in this mass range lose their envelopes non-explosively and settle down as stable carbon-oxygen white dwarfs that never explode. The standard SN Ia model appeals to a carbon-oxygen white dwarf in a close binary system (Figure (1)) that accretes matter from itsmain-sequence or red-giant companion until it is provoked to explode. As the white-dwarfmass approaches the Chandrasekhar limit of 1.4 M, its slow contraction causes its centraltemperature and density to rise enough to ignite carbon fusion. Within a few seconds, asubstantial fraction of the matter in the white dwarf undergoes nuclear fusion, releasingenough energy (∼ 1044 J) to unbind the star in a supernova explosion. An outwardlyexpanding shock wave is generated, with matter reaching velocities of 10000 km/s, orroughly 3% of the speed of light [1, 3].

The inner half of the white dwarf’s mass is incinerated primarily to unstable 56Ni, whilethe outer half is partially burned to intermediate-mass elements such as silicon, sulfur,and calcium, and partially ejected as unburned carbon-oxygen. The internal energy of theinitially hot ejected matter is rapidly lost by adiabatic expansion, so if it were not for thedelayed energy input provided by the gamma–ray and positron products of the radioactivedecay of 56Ni (6-day half-life) through 56Co (77 days) to stable 56Fe, the explosion wouldbe invisible in the optical. The peak luminosity of an SN Ia therefore depends primarilyon the mass of 56Ni that is ejected, about 0.6 M.

2

The peak luminosity of the light curve was believed to be consistent across Type Iasupernovae, having a maximum absolute magnitude of about −19.3. This would allowthem to be used as a standard candle to measure the distance to their host galaxies [1, 3].

2.1 Standard candles

Making SNe Ia valuable as standard candle estimators of distances to their parent galaxies,began to be discussed on empirical grounds during the late 1960s by Pskovskii and Kowaland it slowly gained further observational support and credence during the following twodecades [1]. The situation as of the early 1990s was reviewed by Branch & Tammann(1992) [5], who strongly emphasized the observational homogenity of SNe Ia. The in-trinsic dispersion in the blue and visual absolute magnitudes, σ(MB) and σ(MV ), aftercorrection for extinction by intervening interstellar dust in our Galaxy and the supernovaparent galaxies, was estimated to be no more than 0.25 mag, which corresponds to a dis-persion in luminosity of 26 percent and therefore a dispersion of only 13 percent in dis-tance when SNe Ia are used as standard candles. This luminosity homogeneity, togetherwith the extremely high luminosity of SNe Ia (approaching 1010 L) that makes themdetectable across the universe, plus the fact that unlike galaxies supernovae are observedas point sources which facilitates accurate photometry, make SNe Ia extremely attractiveas distance indicators for cosmology [1, 3, 6, 7, 5, 8].

Studies of the peak B-band (hereon we shall use standard notation for Johnson’s pho-tometric color filters UBVRI unless stated otherwise) luminosities of high redshift SNeIa led to the surprising discovery by two independent groups (the Supernova CosmologyProject(SCP) [5] and the High-z Supernova Search Team(HSST) [6]), that the expansionof the Universe is accelerating. This acceleration is consistent with some form of ’dark en-ergy’, possibly Einstein’s cosmological constant Λ. The implications of this result for ourunderstanding of the future fate of the Universe and fundamental physics are profound.

2.2 Data corrections

Uncorrected observations of SN Ia absolute magnitudes have an RMS scatter of ∼ 40%.The magnitude scatter is reduced by taking into account correlations between rate ofdecline, colour, and intrinsic luminosity. A number of methods have been developed tomeasure calibrated distances from SN Ia multicolor light curves, with each enjoying asimilar level of success. The first of these was introduced by Phillips (1993) [9], whonoted that the parameter ∆m15(B), the amount by which a SN Ia declined in the B-band during the first fifteen days after maximum light, was well correlated with SN Iaintrinsic luminosity. Phillips et al. (1999) [9] present the current version of this method,which incorporates measurement of the extinction via late-time B–V (blue - visible) colormeasurements (roughly independent of ∆m15(B) ) and B–V and V–I (visible - infrared)measurements at maximum light (for the measured ∆m15(B), determined in an iterativefashion). Other popular methods are SALT(2) [10], Stretch [5], (M)LCS(2k2) [9] andCMAGIC [8]. This current standardization techniques correct for light curve shape andcolor normalize scatter to 15% to 20%, i.e., luminosity distances of SNe Ia can be measuredto 7−10% accuracy. Relatively new method using flux ratios from a single flux-calibratedspectrum per SN is lowering magnitude scatter from 0.161±0.015 mag (SALT2) to 0.125±0.011 mag [11].

3 Distances in cosmology

We have good evidence that the universe is expanding. This means that early in itshistory the distance between us and distant galaxies was smaller than is it today. It isconvinient to describe this effect by introducing the scale factor a, whose present value isset to one. We can picture space as a grid which expands uniformly as time evolvs. Points

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on the grid maintain their coordinates, so the comoving distance between two points –which just measures the difference between coordinates – remains constant. However, thephysical distance is proportinal to the scale factor and the physical distance does evolvewith time. From this we can write relation ~r = a~x, where ~r is physical and ~x comovingdistance [12]. Another important quantity can be defined regarding expanding universe.If the universe is expanding then galaxies should be moving away from each other. Recallthat the wavelegnth of light emitted froma a receding object is stretched out so thatthe observed wavelength is larger than the emitted one. Its is convinient to define thisstretching factor as the redshift:

1 + z ≡ λobsλemit

, (1)

It can be shown that energy scales as 1/a and wavelengths scale as a so we can derivethe relation between the redshift and the scaling factor as 1 + z = 1/a. The redshift tellsus by what factor was the universe smaller than it is today. For instance when z = 1 theuniverse was two times smaller than it is today (1 + z = 1 + 1 = 2).

If the universe is isotropic and homogeneous then we can introduce Friedmann-Robertson-Walker metric of space-time which depends solely on the scale factor a(t). Evolution of thescale factor is governed by the Friedmann equations (derived from Einstein’s field equation[12])

H2(t) ≡(a

a

)2

= 8πG3 ρ− k

a2 , (2)

∂ρ

∂t+ 3

a

a(ρ+ p) = 0. (3)

First equation (Eq. (2)) discribes the evolution of the scale factor and the second equation(Eq. (3)) is simply energy conservation. Here we have introduced density ρ and pressurep of a perfect isotropic fluid and defined the Hubble rate H(t) (throughout this paper weshall use units: ~ = c = 1). Constant k represents the curvature of the universe (k = 0flat, k > 0 closed and for k < 0 open universe) and dot represents derivates over time(a = da/dt). The curvature can also measure the topology of the universe. If we havethree points in the space-time and would create a triangle, such that the line connectingtwo points would be the shortest path between these two points. Than the sum of theinner angles of the triangle would be equal to 180 for flat (or Euclidean) universe, greaterthan 180 for open and less than 180 for open and closed universe respectively. Thecurvature of the two-dimensional surface has analyitical expression.

The distance of importance to our calculations is comoving distance between a distantemitter and us. In that case, the comoving distance out to an object at redshift z can bewritten as

χ(z) =∫ z

0

dz′

H(z′). (4)

It can be shown ([12]) that the flux we observe is

F =L

4πd2L

, (5)

where L is luminosity of the source and dL the luminosity distance defined as

dL(z) = (1 + z)

1√k

sin(χ(z)√k) k > 0

χ(z) k = 01√−k sinh(χ(z)

√−k) k < 0

(6)

4

Following the common lore we introduce critical density as ρcrit = 3H(t)2/8πG anddefine dimensionless energy density Ω = ρ/ρcrit. Usually in the cosmology applicationsusing SNe Ia the most important contributions to energy density come from matter anddark energy (if the redshift is sufficiently small contributions from radiation and neutrinoscan be neglected). Adopting the usual matter density evolution (ρm ∝ a−3) [12] anddividing Eq. (2) with ρcrit we can write Friedmann’s evolution equation in more a familiarform (

H

H0

)2

=Ωm

a3+

Ωk

a2+ ΩΛ. (7)

Here we have introduced constantH0, the Hubble rate today, and today density parametersfor matter (Ωm), dark energy (ΩΛ ∝ Λ, where Λ is the Einstein’s cosmological constant)and curvator (Ωk = −k/H2

0 ). The Hubble constant is often parametrized in the followingway H0 = 100h km/s/Mpc. From energy conservation we know that Ωm+ΩΛ +Ωk = 1 oras it is more commonly parametrised Ωm + ΩΛ = Ωtot = 1− Ωk. First term on the right-hand side of the Eq. (7) stands for both baryonic and dark matter (Ωm = Ωb+Ωdm). In thestandard ΛCDM model matter densities are Ωb ≈ 0.04 and Ωdm ≈ 0.23 [13]. Observationsof cosmic microwave background (CMB) gives value of Ωk consistent with zero and theusual practise in SNe obesrvations is to assume flat universe thus neglecting second termin Eq. (7). The last term is usually asocciated with the cosmological constant that isconsistant with the SNe data, but is by no means the only possibility. We can extendpossible range of solutions by defining equation of state parameter w, where

w =p

ρ. (8)

For cosmological constant w has the value of −1. By letting w vary the cosmologicalconstant effectively changes. Thus we can write Eq. (7) in its more general form(

H

H0

)2

=Ωm

a3+

ΩΛ

a3(1+w). (9)

Moreover, we can let w vary with the scale parameter a which gives us theories knownas quintessence. We can also write associated equation of state parameter for other energydensities. For instance, for matter density we could equivalently write the first term onthe right hand side of Eq. (9) as Ωm/a

3(1+wm). But we know that for matter density theequation of state can be written as p = 1

3ρc2 thus wm = 1/3. Similarly we find that for

radiation p = 14ρc

2 and wr = 1/4.Another important parameter for determining the possible fate of the universe is the

deceleration parameter q (the name and the sign are historical) as

q = − aaa2. (10)

If q < 0 the universe is accelerating and if q > 0 it is deccelerating [5, 15]. So if ourobservations give us value of q at current epoch negative we could say that our modelpredicts accelerating universe.

Figure (4) shows us contour lines in the w−Ωm parameter plane. The innermost linehas the lowest and the outermost has the highest confidence level. We can get betterconstraints on our parameters by combining data from different obeservation techinques.For instance in the Figure (4) the data combined consits of the SNe Ia data, the WMAPobesrvations of the cosmic microwave background and measurements of the primrodalplasma oscillations in the early universe (BAO).

5

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7

a(t)

t [1/H0]

ΩΛ = 0

Ωm = 1; k=0Ωm = 0.5; k < 0Ωm = 2.0; k > 0

(a) ΩΛ = 0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.5 1 1.5 2 2.5 3

a(t)

t [1/H0]

ΩΛ = 0.5

Ωm = 0.5; k=0Ωm = 0.1; k < 0Ωm = 1.0; k > 0

(b) ΩΛ = 0.5

Figure 3 — Examples for scale factor (a(t)) dependance on time for different cosmological parame-ters. On the left we can see the scenarios without the dark energy contribution (ΩΛ = 0) while on theright we have scenarios with non-zero cosmological constant (ΩΛ = 0.5). On both plots there are linesrepresenting different curvator factor (k). Red line represents flat universe (k = 0), green line openuniverse (k < 0) and blue line closed universe (k > 0). We have used the common notation where thescale factor today is equal to 1 (a(t = 1) = 1) and where the time is in the units of 1/H0 which is9.77×109 h−1 years. Todays value of the dimensionless hubble constant h varies between 0.5 and 0.7.

Figure 4 — An example of results on cosmological parameters from the SNe Ia data. Resultsfrom Supernova Legacy Survey (1st year) on w − Ωm plane combining data from SNe Ia + BAO +WMAP-3yr + flat universe (black contours). BAO stands for Baryonic Acoustic Oscillations and refersto observational data of baryonic plasma oscillations in the early universe. WMAP-3yr is the WMAPthird year release data from the satellite WMAP that is observing the cosmic microwave background.[14].

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Figure 5 — The Hubble diagram for SCP low-extinction subsample. The lines represent theoreticalpredictions for different cosmologies (Ωm,ΩΛ): (0.25, 0.75) full, (0.25, 0) dashed and (1, 0) dotted.[16]

4 Cosmological parameters from SNe Ia

As already mentioned the early observational studies of SNe Ia has lead to an astonishingdiscovery that the expansion of the universe is accelerating [5, 6]. Transforming luminosityfrom Eq. (5) to a dimensionless magnitude (m) we can write equation Eq. (5) in the formmore commonly found in literature:

m−M = 5 log10

(dL

Mpc

)+ 25, (11)

where dL is measured in megaparsecs (Mpc). It is a unit of length typically used byastronomers measuring distances between galaxies or galaxy clusters. Figure (5) representstypical plot of measured magnitudes vs redshift (in this case magnitueds are in B-band),while Figure (6) shows confidence levels in the ΩΛ − Ωm parameter plane.

From observations of light curves and SNe spectra we can derive a data set of mag-nitudes (and errors) (mi, σmi) at given redshifts zi. The likelihood for the cosmologicalparameters can be determined from a χ2 statistics, where

χ2(H0,Ωm,ΩΛ, w) =∑i

(mi −m(zi;H0,Ωm,ΩΛ, w))2

σ2mi

+ σ2m + σ2

v

(12)

and σv is the dispersion in galaxy redshift due to peculiar velocities and σm is theabsolute magnitude scatter. Unfortunately the limiting factor in the active SN experimetnsis the systematic uncertainty principally due to the evolution in redshift of SN Ia luminositythat is unknown and usually unmodeled. We can than caluclate probability for a given χ2

in logarithmic scale (dimensionless) as [17]

logL ∝ −12χ2 − 1

2

∑i

log(σ2mi

+ σ2m + σ2

v), (13)

and calculate logL for a wide range of paramter space seeking maximum. Unphysicalregions such as Ωm < 0 are not considered since equation Eq. (5) describes the effect of the

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massive particles on the luminosity distance. Sometimes the region of (Ωm,ΩΛ) parameterspace which gives rise to rebounding universes are also neglected.

Figure 6 — Confidence regions for Ωm − ΩΛ parameter plane. Regions representing differentcosmological scenarios are illustrated. Contours are closed by the intersection with the line Ωm = 0[16]. The innermost contour coresponds to 1σ (68%) confidence level (C.L.), the next to 90% C.L.and the last two to 95% and 99% confidence levels. The line dividing the parameter plane to open,flator closed is the line ΩΛ = 1−Ωm. The acceleration criteria comes from q0 = ΩΛ − Ωm

2 = 0. Whetherthe universe wil expand forever or eventually collapse is given by [6, 15].

Figure (6) shows contour plots in the (Ωm,ΩΛ) parameter space. It can be concludedthat the SNe Ia data most favoured cosmology certainly lies in the part of the diagramthat corresponds to the accelerated expansion. From our analysis we can thus derive themost probable values for cosmological parameters of interest.

4.1 Peculiar velocities

Our derivation of the theoretical framework ( Eqs. (6) and (9)) implied, as stated above,isotropic and homogeneous expansion of the universe. However, this is not the case andwe have sufficient observational evidence (CMB, galaxies, stars) that the universe is per-turbed. Accouting for this in the metric tensor and Einstein’s field equation leads to aperturbation in the luminosity distance [18, 19]. The corrections to the Eq. (6) arisedue to several factors: peculiar velocities (PVs) of the source and observer, gravitationallensing and integrated Sache-Wolf’s effect.

The velocity perturbations change the luminosity distance fluctuation (the relative

8

difference between measured dmeL and modeled theoretical dthL distance) as [18, 19, 20]:

δdLdL

=dmeL − dthL

dthL= r ·

(v − (1 + z)2

H(z)dL(v − vo)

), (14)

where dL is given by Eq. (6) and z is the redshift we measure. Velocities v and vo arepeculiar velocities of the source and the observer respectively. The projection of thosevelocties along the line of sight is the only one we can observe. We can also account forthe observer’s velocity vo from the CMB dipole. Using Eq. (14) one can derive the localPV field (r · v(r)) of the SNe Ia.

The PVs are drawn from a distribution beacuse they arise from some initial guassianperturbations. Thus the mean of peculiar velocity is zero 〈(v(r) · r)〉 = 0. But thereexitsts nonzero variance. Written as a cross-correlation funciton it is given by ξ(ri, rj) =〈(v(ri)·ri)(v(rj)·rj)〉. Since it is rotationally invariant it can be decomposed into a paralleland perpendicular part [17, 12]:

ξ(ri, rj) = sin θi sin θjξ⊥(r, zi, zj) + cos θi cos θjξ‖(r, zi, zj), (15)

where rij ≡ ri − rj, r = |rij|, cos θi ≡ ri · rij and cos θj ≡ rj · rij. In linear theory thevelocity field is given by the density fluctuations field δ(r).

O

ri

rj

rijθi

θj

Figure 7 — Relative position of the two SNe to the observer (O) is given by ri and rj. The angleseperation between them is equal to θj − θi.

In Fourier space we can write [12, 20] δ(k, z) = D(z)D(z=0)δ(k, z = 0) = D(z)δ(k, z = 0),

where D(z) is the linear growth factor. From the linearized equation of motion (andassuming that the universe has no vorticity (∇ × v = 0)) one can derive the relationbetween the density and velocity field that yields

vk = −iD′(z)δ(k, z = 0)k2

k, (16)

where the derivate is with respect to the conformal time (dη = dt/a). For the matterdensity fluctuations we know that 〈δ(k, z = 0)δ(k′, z = 0)〉 = 2πδ3(k − k′)P (k, z = 0),where P (k, z) is the matter power spectrum which can be evaluated either numerically(CAMB) or using analyitical approximations [12, 20]. The Dirac delta function δ3(k−k′)tells us the independance of different modes. We also note that P (k) depends only thesize not the direction of the wave vector. This is due to the fact that universe is nearlyisotropic. The PV cross correlation function can thus be written as

ξ‖,⊥(r, zi, zj) = D′(zi)D′(zj)∫ ∞

0

dk

2π2P (k)K‖,⊥(kr), (17)

9

(a) (b)

Figure 8 — The ratio of the covariance from peculiar velocities Cv, compared to the random errorsσ, for a pair of supernovae, over a range of angular separations. In the left panel (a) we vary theseparation on the sky, θ. In the right panel (b) we vary the redshift, with them both at the same z orwith one supernovae fixed at z = 0.03 (dash-dot) [20]

where, for an arbitrary variable x, the kernels K‖,⊥ have the form, K‖(x) ≡ j0(x)− 2j1(x)x ,

K⊥(x) ≡ j1(x)x .

From Eqs. (15) and (17) we can see that measuring the variance of the PVs measuresthe strength of the underlying density field. Beacuse in the autocorrelation function ofthe PVs we have rij = 0 and cos θi,j = 0 so only the perpendicular component survives.Moreover, since r = 0 the variance is given by

〈(v(r) · r)2〉 =(D′(z))2

6π2

∫ ∞0

P (k)dk. (18)

The above estimate of ξ(ri, rj) is based on linear theory. On scales smaller than about10h−1 Mpc, nonlinear contributions dominate. For typical values of h ∈ [0.5, 0.7] weget scales of 15 − 20 Mpc. These are usually modeled as an uncorrelated term which isindependent of redshift, often set to σv ∼ 300 km/s. Comparison with N-body simulationsindicate that this is an effective way of accounting for the non-linearities [5, 6, 17, 20].The physic behind it is quite straightforward. From Eq. (16) we see that in the lineartheory the velocity grows as δ/k which is a contribution from the large scale perturbations(smaller k gives larger scales and larger velocity). In the linear theory δ scales as a atlater times. But in reality there comes a point, at sufficiently high δ, when the growth ofperturbations decouples from the growth of the universe and after an exponential decayin growth the overdensity forms virialised dark matter objects called dark matter halos.Inside these halos reside galaxies and in them the observed SNe Ia. Since in the lineartheory the halos don’t exist we add their contribution by hand as σv which is the averagevelocity of the halos (around 300 km/s).

The cross-correlation function ξ(ri, rj) measures correlations between two SNe that areat an angle seperation of θj − θi and the redshift (radial) seperations of |zj − zi|. Thecorrelations arise due to the local underlying mass density field near the SNe and areresponsible for the local kinematics of the observed SNe.

The PVs correlations lead to luminosity distance correlations and the covariance forthe δdL/dL for a pair of SN can be written as [20]:

Cv(i, j) =(

1− (1 + z)2

H(z)dL

)i

(1− (1 + z)2

H(z)dL

)j

ξ(ri, rj). (19)

The Eq. (19) has impact on our analysis of the observed data (x = δdL/dL). Whichmeans that the covariance matrix of the luminosity distance fluctuations is not diagonalanymore, i.e. correlations between different magnitudes is no longer zero. So the χ2

statistics function that we want to minimize over the cosmological parameters of interest

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can no longer be written as in Eq. (12) but rather as χ2 = xTC−1x and logL = −χ2/2−log(det C)/2 [17], where x is the vector of the observed data and C is the covariancematrix, representing correlations between the luminosity distances, given by

Cij = Cv(i, j) + δijσ2. (20)

The covariance of correlated PVs (Cv(i, j)) is given by Eq. (19), while the standarduncorrelated errors (σ) are those from the lightcurve fitting and intrinsic magnitude scatteralong with the uncorrelated σv term accouting for the nonlinearities of the PV field. We seethat the PV covariance is comparable with the uncorrelated errors for low-z (Figure (8)).The correlation also drops as we vary angular sepeartion between the two SNe. This wouldbe expected since the gravitational interaction, that governs kinematics of the SNe, lossesstrength with distance as ∝ r−2 and can be neglected at sufficient seperation.

An analysis of the SNe Ia with only uncorrelated errors allows us to constrain thecosmological parameters throught the Hubble rate in Eq. (9). Namely Ωm and w for aflat universe (in a flat universe holds the relation ΩΛ = 1 − Ωm). However by fitting forthe PV covariance we probe the matter power spectrum P (k) and can therefore constrainfurther parameters such as the baryon density (Ωb), the scalar spectral index (ns) andthe fluctuation amplitude of the inital density fluctuations at 8h−1 Mpc (σ8). The scalarspectral index is equal to unity in scale invaraint models (ns = 1) and slightly shallower(ns < 1) in inflatory models. In inflation, the ns deviation from unity closely is closelyrelated to the inflatory potential. As we see the matter power spectrum P (k) is a veryimportant quantity combining baryonic to inflatory physics and presents an importanttool to constrain additional cosmological parameters.

It has been shown ([20, 17]) that when no redshift cutoff is imposed the current SNedata detects correlations in PVs at the 3.6σ level. Moreover, the future large data setswill be sensitive to the PVs even if a lower bound of z ≥ 0.03 is used. Thus enabling usto constrain the matter power spectrum from the actual SNe Ia data.

An alternative approach of accouting for the correlated peculiar velocities is to estimatethe underlying density field from galaxy redshift surveys and the use this to try and removepeculiar velocities at each SNe [21].

5 Conclusion

Using SNe Ia data it has been proven that the universe is undergoing an accelerated ex-pansion [5, 6] and indicating unexpected gravitational physics, frequently attributed tothe ’dark energy’ with negative pressure. SNe Ia remain one of our best tools for un-ravelling the properties of dark energy because their individual measurement precision isunparalleled and they are readily attainable in statistically suffiecient samples (∼ 100) tomeasure dark-energy-induced changes to the expansion rate of ∼ 1%. Density inhomo-geneities cause the SNe to deviate from the Hubble flow, as gravitational instability leadsto matter flowing out of underdense regions into overdense ones. These peculiar velocitieslead to an increased scatter in the Hubble diagram. However, in the limit of low redshifts(z ≤ 0.1) and large sample size [20], the correlations between SNe PVs contribute signifa-cantly to the overall error. Furthermore, the systematic uncertainties in relative distancesdue to unkown evolution in redshift of SN Ia, intervining dust that obsurecs SN light andinstrumental calibration are going to be a limiting factor in future surveys.

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References

[1] D. Branch, Memo 2.2.1: Type ia supernovae as standard candles,http://supernova.lbl.gov/evlinder/Ia candle.pdf.

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