MNRAS 440, 134–142 (2014)

doi:10.1093/mnras/stu208

Advance Access publication 2014 March 7

Evolution of extinction curves in galaxies
Ryosuke S. Asano,1‹ † Tsutomu T. Takeuchi,1 Hiroyuki Hirashita2
and Takaya Nozawa3
1 Department of Particle and Astrophysical Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan
2 Institute of Astronomy and Astrophysics, Academia Sinica, PO Box 23-141, Taipei 10617, Taiwan
3 Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, Chiba 277-8583, Japan

Accepted 2014 January 27. Received 2014 January 25; in original form 2013 November 25

We investigate the evolution of extinction curves in galaxies based on our evolution modelof grain-size distribution. In this model, we considered various processes: dust formation byType II supernovae and asymptotic giant branch stars, dust destruction by supernova shocks
in the interstellar medium, metal accretion onto the surface of grains (referred to as grain
growth), shattering and coagulation. We find that the extinction curve is flat in the earliest
stage of galaxy evolution. As the galaxy is enriched with dust, shattering becomes effective,
producing a large abundance of small grains (a  0.01 μm). Then, grain growth becomes
effective at small grain radii, leading to the formation of a bump at a ∼ 10−3 –10−2 μm on
the grain-size distribution. Consequently, the extinction curve at ultraviolet (UV) wavelengths
becomes steep, and a bump at 1/λ ∼ 4.5 μm−1 (where λ is the wavelength) on the extinction
curve becomes prominent. Once coagulation becomes effective, the extinction curves become
flatter, but the UV extinction remains overproduced when compared with the Milky Way
extinction curve. This discrepancy can be resolved by introducing a stronger contributionof coagulation. Consequently, the interplay between shattering and coagulation could be
important in reproducing the Milky Way extinction curve.
Key words: ISM: clouds – dust, extinction – galaxies: evolution – galaxies: general – galaxies:
ISM – stars: formation.
1 I N T RO D U C T I O NDust grains are one of the fundamental ingredients necessary for
an understanding of the formation and evolution of galaxies. The
surfaces of dust grains are the main sites for the formation of hydrogen molecules (e.g. Cazaux & Tielens 2004), which act as aneffective coolant in low-metallicity conditions (e.g. Hirashita &Ferrara 2002; Cazaux & Spaans 2009). Dust is also an important
coolant in star formation, inducing fragmentation into low-massstars (Omukai et al. 2005; Schneider et al. 2006). Thus, dust grains
are strongly related to star formation in galaxies. Furthermore, dust
grains govern the scattering and absorption (i.e. extinction) of stellar
light, in particular at short wavelengths such as the ultraviolet (UV),
and re-emit in the infrared (IR). Consequently, dust grains significantly affect the spectral energy distribution (SED) of galaxies (e.g.
Takagi, Vansevičius & Arimoto 2003).
The extinction curve, which represents the wavelength dependence of dust extinction, is used to relate the intrinsic stellar SED
to the observed SED affected by dust extinction. Thus, the extinction curve is the fundamental tool for interpreting the observational
SEDs of galaxies. Because the extinction curve depends strongly on E-mail: asano.ryosuke@g.mbox.nagoya-u.ac.jp† Fellow of the Japan Society for the Promotion of Science (JSPS).

the physical and optical properties of dust grains [grain size, dust
components, etc.; see, for example, Mathis, Rumpl & Nordsieck
1977 (hereafter MRN); Weingartner & Draine 2001; Nozawa &Fukugita 2013], it is important to understand these properties.
The mean extinction curve of the Milky Way (hereafter the
MW extinction curve) is observationally well investigated (e.g.
Fitzpatrick & Massa 2007), and is widely adopted as a template
extinction curve (e.g. Buat et al. 1999; Matsuoka et al. 2005;
Kobayashi, Inoue & Inoue 2013; Krühler et al. 2013). The MW
extinction curve has a bump at 2175 Å, which is thought to be generated by small carbonaceous grains and/or polycyclic aromatichydrocarbons (PAHs; e.g. Barbaro et al. 2001; Draine 2009a),and shows a steep rise to the far-UV wavelength (called the UV
slope). By fitting the MW extinction curve, MRN derived thegrain-size distribution in the Milky Way, f(a)da ∝ a−3.5 da, witha = 0.005–0.25 μm, where a is the grain radius and f(a)da is
the number density of grains in the size interval [a, a + da].
However, Weingartner & Draine (2001) performed a detailed
fit to the MW extinction curve, finding that the size distributions of carbonaceous and silicate dust grains are quite different
from each other, contra MRN. The extinction curves depend on
the line of sight, and Cardelli, Clayton & Mathis (1989) suggested that the variation of extinction curves can be describedby adopting the parameter RV ≡ AV /E(B − V), where AV is theC 2014 The Authors
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ABSTRACTEvolution of extinction curves

thermore, small grains (a < 0.01 μm) are efficiently destroyed by
sputtering in interstellar shocks driven by SNe (Nozawa, Kozasa &Habe 2006). When grain growth occurs in the ISM, smaller grains
grow more efficiently (e.g. Hirashita & Kuo 2011; A13) because thetime-scale of grain growth is proportional to the volume-to-surface
ratio of a dust grain. In the diffuse ISM, shattering can occur effectively if grains are dynamically coupled with magnetized interstellar turbulence (e.g. Yan, Lazarian & Draine 2004; Hirashita &Yan 2009); in particular, large grains (a  0.1 μm) acquire larger
velocity dispersions than the shattering threshold velocities. Shattering also occurs in SN shocks (e.g. Jones, Tielens & Hollenbach
1996). In dense and cold regions, the coagulation of small grains
can occur (e.g. Hirashita & Yan 2009; Ormel et al. 2009); consequently, the grain-size distribution shifts towards larger sizes (e.g.
Hirashita & Li 2013). In summary, the above processes affecting
grain-size distribution (referred to as ‘dust processes’ in this paper)
are dependent on the metallicity, total dust amount, and grain-size
distribution, and could be interrelated. Thus, it is necessary to construct a model that takes into account all dust processes in a unified
framework.
There are a number of studies on the evolution of the grain-size
distribution in galaxies (e.g. Liffman & Clayton 1989; O’Donnell &Mathis 1997; Hirashita et al. 2010; Yamasawa et al. 2011). However,they do not consider all the dust processes, in order to simplify themodels. Recently, A13 discussed the evolution of the grain-size
distribution, taking into account all the dust processes based on
the chemical evolution of galaxies. A13 showed that the grainsize distribution drastically changes with galactic age because thedominant dust process changes (see Section 2). In view of the
discussion in A13, it is expected that the extinction curve will alsochange with the galactic age owing to the change of the dominant
dust processes. Therefore, in this paper, we examine the evolutionof extinction curves in galaxies using the dust evolution model
developed by A13, and check whether we can reproduce the MW
extinction curve.
This paper is organized as follows. First, we briefly review thedust evolution model constructed by A13 and explain the theoreticaltreatment of the extinction curve in Section 2. In Section 3, we show
the contributions of various dust processes to the extinction curve.
We discuss how we can reproduce the MW extinction curve and thecontribution of different grain species in Section 4. We present ourconclusions in Section 5.
2 MODELIn this section, we first review our dust evolution model for calculating the evolution of the grain-size distribution in a galaxy(A13). Then, we explain the method of calculating the extinction
at wavelength λ, Aλ (in units of magnitude) based on the grain-size
distribution calculated.
2.1 Dust evolution model
We briefly introduce the model constructed by A13 and summarize
their results. A13 investigated the evolution of the grain-size distribution taking into account the dust formation by SNe II and AGB
stars, dust destruction by SN shocks in the ISM, grain growth in
the cold neutral medium (CNM), and grain–grain collisions (shattering and coagulation) in the warm neutral medium (WNM) and
CNM. Grain growth in the WNM was not considered because grain
growth is more efficient in dense, cold regions (e.g. Liffman &Clayton 1989; Draine 2009a). A13 considered the contribution of
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magnitude of the extinction in the V band, and E(B − V) is thereddening (AB − AV , where AB is the extinction in the B band).
Recently, Nozawa & Fukugita (2013) investigated the possible variety of dust properties based on the diversity of extinction curves
observed in the Milky Way. They found that the power-law index
and maximum radius of the grain-size distribution are tightly constrained to be −3.5 ± 0.2 and 0.25 ± 0.05 μm, respectively. Pei(1992) extended the graphite–silicate grain model that can fit the
MW extinction curve (MRN) to the Large and Small Magellanic
Clouds (LMC and SMC), and found that the extinction curves inthese galaxies can be fitted with the MRN grain-size distribution byadjusting only the relative contribution of graphite and silicate.
Many studies have shown that high-z galaxies have different extinction curves from nearby galaxies (e.g. Maiolino et al. 2004;
Liang & Li 2009; Gallerani et al. 2010; Hjorth et al. 2013). The extinction curve of the quasar SDSS104845.05+463718.3 (hereafter
SDSS1048+4637) at redshift z = 6.2 has no 2175-Å bump, and isrelatively flat at λ  1700 Å , rising towards shorter wavelengths at
λ  1700 Å. Maiolino et al. (2004), by using the model in Todini &Ferrara (2001), showed that the extinction curve is consistent withthe dust formation in Type II supernovae (SNe II). Because of theirshort lifetimes (typically 106−7 yr), SNe II are thought to be the origin of dust in the high-z Universe. On the other hand, because of thelong lifetimes of progenitors, asymptotic giant branch (AGB) stars
can be dominant sources of dust in galaxies at age t > 1 Gyr (but see
Valiante et al. 2009). Furthermore, Hirashita et al. (2010) examinedthe extinction curves in starburst galaxies taking into account not
only dust grains produced by SNe II but also the effect of shattering(grain–grain collision) in the warm ionized medium (WIM). They
showed that the shattering can lead to the steepness of the extinction curve at UV wavelengths, and indicated that shattering may
occur effectively in SDSS1048+4637. Liang & Li (2009) showedthat the extinction curves of high-z gamma-ray bursts (GRBs) are
different from those of the Milky Way and LMC. Among them,
one at z = 6.3 appears to have the 2175-Å feature, indicating a
difference from SDSS1048+4637. In order to reveal the origin of
the differences in the extinction curves among galaxies in the highand the low-z Universe, it is necessary to clarify the processes thatgovern the evolution of dust grains in galaxies.
An MRN-like power-law grain-size distribution can be realized
if the grains are processed by grain–grain collisions (shatteringand coagulation; e.g. Tanaka, Inaba & Nakazawa 1996; Kobayashi
et al. 2010). Thus, it is probable that grain–grain collisions are an
important process in the Milky Way and perhaps in nearby galaxies
in general. In addition, if the metallicity in galaxies is larger thana certain value, the accretion of gas-phase metals onto the surface
of pre-existing grains (referred to as ‘grain growth’ in this paper)occurs effectively (Inoue 2011; Asano et al. 2013a). Because grain
growth has the potential to change the grain-size distribution [e.g.
Hirashita & Kuo 2011; Asano et al. 2013b (hereafter A13)], theshape of extinction curves may change as a result of grain growth.
In fact, Hirashita (2012) showed that the UV slope on extinction
curves becomes steeper owing to grain growth if the grain-size
distribution is initially similar to the MRN size distribution.
Various processes have different effects on different grain sizes.
Grains ejected by SNe II into the interstellar medium (ISM) are relatively large (a  0.01 μm) owing to the destruction of small grainsby the sputtering in reverse shocks (e.g. Bianchi & Schneider 2007;
Nozawa et al. 2007). Grains produced in AGB stars may have typicalradii ∼ 0.1 μm (e.g. Winters et al. 1997; Ventura et al. 2012a,b; Di
Criscienzo et al. 2013) and could be described by a log-normal distribution with a peak at a ∼ 0.1 μm (Yasuda & Kozasa 2012). Fur-

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MNRAS 440, 134–142 (2014)

Figure 1. Example of the evolution of the grain-size distribution. Solid,
dotted, dashed and dot–dashed lines represent the cases at t = 0.1, 1, 5,
10 Gyr, respectively, with τ SF = 5 Gyr. The mass fractions, ηWNM andηCNM , are set to be 0.5.

coagulation. For further details and parameter dependences of thegrain-size distribution, see A13.
2.2 Extinction curve
Extinction curves are powerful tools for examining the dust properties in galaxies. In order to analyse extinction curves, the opticalconstants for each dust species are necessary. In this paper, we adopt
the optical constants derived by Draine & Lee (1984) to calculate
the grain extinction cross-section normalized to the geometricalcross-section πa 2 as a function of wavelength and grain radius,
Qext, X (λ, a), where the subscript X represents the grain species(X = carbonaceous dust or silicate dust), and λ is the wavelength.
The optical depth of dust species X at a given wavelength λ, τ X, λ ,is defined as ∞πa 2 CQext,X (λ, a)fX (a)da,(1)
τX,λ =
0where C is a normalization constant and fX (a) is defined so thatfX da is the number density of species X with radii in the range[a, a + da]. The extinction in units of magnitude is proportional tothe optical depth, and is expressed as
AX,λ = 1.086τX,λ ,(2)

where AX, λ is the extinction of dust species X in units of magnitudeat wavelength λ. The total extinction in units of magnitude, Aλ , isexpressed asAX,λ .
(3)Aλ =X

In this work, we consider the extinction curve normalized to the
V-band value, Aλ /AλV , so C and the factor 1.086 cancel out.
3 R E S U LT S
In this section, we show the effects of each dust process on the extinction curve. We add the following processes one by one: the dust
formation by SNe II and AGB stars, dust destruction by SN shocks
in the ISM, grain growth in the ISM, shattering and coagulation in
the ISM. The loss of dust by astration is always included, although
it does not affect the shape of the extinction curve. We adopt a star

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the dust processes in the WNM and CNM by introducing the massfractions of the WNM (∼6000 K, 0.3 cm−3 ) and CNM (∼100 K,
30 cm−3 ), ηWNM and ηCNM , respectively. The sum of ηWNM andηCNM was assumed to be unity in A13, as an equilibrium state of
two thermally stable phases (WNM and CNM) is established in the
ISM if we consider temperatures lower than 104 K (Wolfire et al.
2003). The grain velocities in the two ISM phases derived by Yan
et al. (2004) were adopted to calculate shattering and coagulation.
We assume two dust species, graphite and silicate (Draine &Lee 1984), in the same way as in A13. Although A13 considereda variety of dust species (C, Si, SiO2 , SiC, Fe, FeS, Al2 O3 , MgO,
MgSiO3 , Mg2 SiO4 and Fe2 SiO4 ; Nozawa et al. 2007; Zhukovska,
Gail & Trieloff 2008) for stellar dust production, here carbonaceous
dust and all the other dust species are categorized as graphite andsilicate [we adopt astronomical silicate, Mg1.1 Fe0.9 SiO4 (Draine &Lee 1984)], respectively, in order to avoid chemical complexity
in grain growth, shattering, and coagulation. A13 examined thecontribution of grain growth, shattering and coagulation for two
dust species separately. While the complexity of dust species may
affect the extinction curves (e.g. Nozawa & Fukugita 2013), theaim of this paper is not a detailed fit to a specific extinction curve
but an investigation of the response of extinction curves to theevolution of grain-size distribution. Furthermore, it is thought thatthese two dust species are the main components of dust grains in
the Milky Way (Draine & Lee 1984), and Sofia & Meyer (2001)
suggested that the (Fe+Mg):Si ratio in dust grains is close to 2:1.
Thus, the above grain composition is a reasonable approximation.
Other possible dust species are discussed in Section 4. We assume
that grains are spherical, and that shattering/coagulation occurs if therelative velocity of collisional grains is more/less than the thresholdvelocity (e.g. Yan et al. 2004; Hirashita & Yan 2009). The thresholdvelocity of shattering, v shat , is assumed to be 1.2 and 2.7 km s−1 forcarbonaceous dust and silicate dust, respectively (Jones et al. 1996).
For the threshold velocity of coagulation, we perform calculations
as in Hirashita & Yan (2009), and the threshold velocity is about
10−3 –10−1 km s−1 depending on the grain size (Chokshi, Tielens &Hollenbach 1993). Note that if the radius of the shattered fragments
is less than 3 Å, we remove the fragments, unlike in A13.
For the dust evolution model, A13 assumed that the total baryon
mass (the sum of the stellar mass and the ISM mass in the galaxy)is constant (closed-box model), and formulated the star formation rate (SFR) by introducing the star formation time-scale, τ SF :SFR(t) = MISM (t)/τ SF , where MISM is the ISM mass and t is thegalaxy age. Fig. 1 shows the evolution of the grain-size distribution (the sum of silicate and carbonaceous dust is shown) with all
dust processes considered for τ SF = 5 Gyr. We find that while thegrain-size distribution is dominated by large grains (a  0.1 μm)
produced by stars at <0.1 Gyr, as the galaxy evolution proceeds thegrain-size distribution begins to be regulated by the processes in the
ISM. In particular, once shattering occurs effectively, a large number of small grains (a  0.01 μm) are produced by fragmentation
owing to collisions between large grains. The effect of shattering isseen in the increase of small grains at 0.1–1 Gyr. Owing to the large
number of small grains, grain growth occurs effectively because
the surface-to-volume ratio of smaller grains is larger than that oflarger grains. Consequently, the bump at a ∼ 0.01 μm emerges at
1–10 Gyr. Smaller grains can acquire lower velocity dispersions because they are coupled with smaller-scale turbulence (e.g. Yan et al.
2004). Therefore, after the amount of small grains are increased,
the coagulation occurs mainly by collisions between small grains
whose velocity dispersions are smaller than the coagulation threshold. The shift of the bump position from 1 to 10 Gyr is a result ofEvolution of extinction curves

Figure 4. As Fig. 3, but including grain growth. We adopt ηCNM = 0.5.

(A13). Thus, we cannot observe the difference between the extinction curves with and without dust destruction at 0.1 Gyr (solidlines in Figs 2 and 3). The extinction curves with dust destruction
are slightly flatter than those produced by stardust (see Fig. 2) att  1Gyr. Because smaller grains are more easily destroyed by SN
shocks (Nozawa et al. 2006; Yamasawa et al. 2011; A13), the extinction curves become flatter than in the case without dust destruction
by SN shocks in the ISM (Fig. 2).
A13 examined the effect of dust destruction by SN shocks in the
ISM for various nSN , and showed that the effect is larger for larger
nSN . Thus, the extinction curve becomes flatter for larger nSN .
3.3 Grain growth

Figure 3. As Fig. 2, but including dust destruction by SN shocks in the
ISM.

formation time-scale of τ SF = 5 Gyr, and mass fractions of the
WNM and CNM of ηWNM = ηCNM = 0.5 unless otherwise stated.

Fig. 4 shows the evolution of extinction curves with dust formation
by SNe II and AGB stars, dust destruction by SN shocks in the ISM,
and grain growth in the CNM. We find that the extinction curves arealmost the same as in Fig. 3. In this case, because the total surface
area of grains is dominated by large grains with a > 0.3 μm (A13),
the effect of grain growth is prominent at large sizes (a  0.1 μm).
Because the grains are already large, grain growth just keeps theextinction curve flat.
3.4 Shattering

3.1 Dust formation by SNe II and AGB stars
In Fig. 2, we show the evolution of extinction curves with dust
formation by SNe II and AGB stars. From Fig. 2, it can be seen thatthe extinction curves are flat throughout galactic ages, and do notchange significantly with time. This is because the size distributionof grains produced by SNe II and AGB stars is dominated by large
grains (a  0.1 μm) (Nozawa et al. 2007; Yasuda & Kozasa 2012)and does not change appreciably with galactic age (fig. 1 of A13).
A13 showed that, although the size distribution of grains produced by SNe II depends on the hydrogen number density of the
ISM surrounding the SNe II, nSN (we adopt nSN = 1.0 cm−3 in thispaper), the tendency that the grain-size distribution is dominated bylarge grains is unchanged.
3.2 Dust destruction
In Fig. 3, we show the evolution of an extinction curve with dust
destruction by SN shocks in the ISM in addition to the dust formation by stars. The time-scale on which the dust destruction affects
the grain-size distribution in galaxies, τ SN , is about τ SN ∼ 0.1τ SF

We now consider the evolution of an extinction curve with shattering in addition to all the dust processes considered in Section 3.3.
From Fig. 5, we observe that the extinction curve is almost flat
until galactic age t ∼ 1 Gyr; at t  1 Gyr, shattering is ineffective
owing to the small dust abundance, and the grain-size distribution
is similar to in the case without shattering (A13). A13 discussed thetime-scale on which the grain-size distribution changes owing to theprocesses in the ISM (especially shattering), and obtained the timescale τ shat ∼ 1(τ SF /Gyr)1/2 Gyr from a rough order-of-magnitude
estimate (see their appendix B). Thus, there is only a small difference between the cases with and without shattering at t < 1 Gyr forτ SF = 5 Gyr. At t = 5 Gyr, the extinction curve drastically changes
and starts to have a prominent bump at 1/λ ∼ 4.5 μm−1 (the socalled 2175-Å bump) and a steep slope towards shorter wavelengths.
This is because the grain-size distribution changes considerably owing to the interplay between grain growth and shattering (A13). A13
showed that once shattering occurs effectively, the number of small
grains increases with the decrease in the number of large grains.
Thus, the total surface area of grains per grain mass becomes large,
and grain growth occurs effectively, especially at small grain radii
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Figure 2. Evolution of extinction curves with dust formation by SNe II and
AGB stars. Solid, dotted, dashed, and dot–dashed lines represent the cases
at t = 0.1, 1, 5, 10 Gyr, respectively, with τ SF = 5 Gyr.

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distribution shifts to larger sizes by coagulation, and finally moves
to a ∼ 0.03–0.05 μm at 10 Gyr. As the abundance of large grains
is increased by coagulation, the bump at 1/λ ∼ 4.5 μm−1 and theUV slope become smaller and flatter, respectively.
For reference, we show the MW extinction curve taken from
Whittet (2003) in Fig. 6. We find that our calculated extinction
curve at the age of the Milky Way (∼10 Gyr) is steeper and has alarger 2175-Å bump than the observed extinction curve of the MilkyWay. The main aim of this paper is not to reproduce the MW extinction curve precisely but to examine the trend of evolution of theextinction curve. Nevertheless, we will propose and examine somepossibilities of reproducing the MW extinction curve by slightly
modifying our model in Section 4.1.

Figure 5. As Fig. 4, but including shattering. We adopt ηWNM =ηCNM = 0.5.

(0.01 μm), forming a bump at a ∼ 0.01 μm in the grain-size distribution. Consequently, the 2175-Å bump and UV slope on theextinction curve become larger and steeper, respectively. In fact,
Hirashita (2012) also showed that the 2175-Å bump and the UV
slope are enhanced by grain growth. At t = 10 Gyr, although grain
growth becomes ineffective owing to the large depletion of heavy
elements (e.g. Asano et al. 2013a), the abundance of small grains
still increases by shattering. Consequently, the 2175-Å bump andthe UV slope become larger and steeper at 10 Gyr than at 5 Gyr.
3.5 Coagulation
In Fig. 6, we show the evolution of the extinction curve with all
the dust processes considered in our model. We find that the extinction curves at t = 0.1 and 1 Gyr are flat, and that there is little
difference between the extinction curves with and without coagulation (see Fig. 5). As mentioned in Section 2.1, because coagulation
occurs when the relative velocity of colliding grains is less thanv coag [fiducial v coag ∼ 10−3 –10−1 km s−1 , depending on the grain
size (Chokshi et al. 1993)], coagulation does not occur for large
grains (a  0.1 μm) whose velocities are above the coagulationthreshold. At t = 5 and 10 Gyr, the 2175-Å bump is smaller, andthe UV slope is flatter than those in the case without coagulation(Fig. 5). A13 showed that the bump at a ∼ 0.01 μm in the grain-size

So far, we have adopted the star formation time-scale τ SF = 5 Gyr,and mass fractions of the WNM and CNM of ηWNM = ηCNM = 0.5
as a fiducial case. Here, we demonstrate how the extinction curve
depends on these parameters.
In panels (a) and (b) of Fig. 7, we show the time evolution of
the extinction curve in the cases of τ SF = 0.5 and 50 Gyr, respectively, for ηWNM = ηCNM = 0.5. Focusing on (a), we find that theextinction curve at 1 Gyr is different from that in Fig. 6, whereτ SF = 5 Gyr is adopted. If τ SF is short, the total dust amount ejectedby stars is large at younger galactic ages. This early increase in dust
abundance shortens the time-scale on which shattering becomes effective according to τ shat ∼ 1(τ SF /Gyr)1/2 Gyr (A13; Section 3.4).
Thus, shattering is already able to produce a large amount of small
grains at <1 Gyr for τ SF = 0.5 Gyr. Consequently, the extinction
curve has the 2175-Å bump and steep UV slope at earlier phasesof galaxy evolution. We also observe that the extinction curves at
t = 5 and 10 Gyr are almost identical to each other. This meansthat the grain-size distribution does not change at these ages forτ SF = 0.5 Gyr. Because the total dust amount decreases rapidly by
astration in the case with the short star formation time-scale (e.g.
Asano et al. 2013a), shattering and coagulation become inefficientat those ages. As a result, the shape of the grain-size distribution
does not change significantly. On the other hand, Fig. 7(b) shows
the result for τ SF = 50 Gyr. We find that the 2175-Å bump continues
to grow even after 5 Gyr. This is because the increase of the total
dust amount is slower owing to the longer star formation time-scalethan in (a). Indeed, τ shat ∼ 7 Gyr is consistent with the rapid increase of small grains at 5–10 Gyr (see also fig. 9 in A13). Thus, thetime-scales of all dust processes we considered become long (e.g.
A13), and the evolution of the extinction curve slows down.
Panels (c) and (d) of Fig. 7 show the cases with (ηWNM ,ηCNM ) = (0.1, 0.9) and (0.9, 0.1), respectively, for τ SF = 5 Gyr.
Because a large ηCNM is adopted in (c), the time-scale of coagulation is short. Consequently, the 2175-Å bump becomes small atearlier phases than in the case with a small ηCNM . In (d), where asmall ηCNM is adopted, the time-scale of grain growth is longer thanthat for (c). As mentioned in Section 3.4, the 2175-Å bump becomesprominent owing to grain growth. Thus, the 2175-Å bump becomes
more prominent at later phases than in the case with a larger ηCNM .
4 DISCUSSION
4.1 Reproducing the MW extinction curve

Figure 6. As Fig. 5, but including coagulation. Filled diamonds represent
the extinction curve of the Milky Way (Whittet 2003).

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In the previous section, we examined the evolution of theextinction curve in galaxies taking into account various dust

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3.6 Parameter dependenceEvolution of extinction curves

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processes. We found that because stellar dust is biased to large
grains (a  0.1 μm), the extinction curve at the earliest stage ofgalaxy evolution is flat. After t ∼ τ shat ∼ 1(τ SF /Gyr)1/2 Gyr, shattering and grain growth occur effectively, and the extinction curve
becomes steeper and has a larger bump at 1/λ ∼ 4.5 μm−1 than in
the case of the MW extinction curve. After coagulation becomes
effective, the bump becomes small. However, compared with the
MW extinction curve, the calculated extinction curves are too steep
and have too large a bump. The reason for this is too abundant small
grains, forming the bump at a ∼ 0.01 μm in the grain-size distribution (Fig. 1). Thus, it is crucial to weaken this bump in order toreproduce the MW extinction curve. As explained in Section 3.4, thebump in the grain-size distribution is formed by grain growth. Thismeans that our models may have overestimated grain growth. Thus,
as the first possibility of reproducing the MW extinction curve, we
examine a model in which only grain growth is turned off. This isreferred to as (i) the no grain growth model.
There is another way of weakening the bump in the grain-size
distribution. In the current model, coagulation takes place only forgrains with radii less than ∼0.05 μm, because large grains havelarger velocities than the coagulation threshold (Section 3.5). However, if the grains can coagulate beyond 0.05 μm, the bump in thegrain-size distribution could be smoothed out. There are some indications that the grains could coagulate beyond 0.05 μm: Ossenkopf(1993) argued that fluffy grains can be formed in dense regions bycoagulation. Fluffiness enhances the cross-section in grain–grain
collisions, raising the coagulation rate. Fluffy grains can also absorb
collision energy, raising the coagulation threshold (e.g. Ormel et al.
2009). Enhancement of coagulation efficiency is also suggested byHirashita & Li (2013) to explain μm-sized grains in dense molecular
cloud cores. Thus, to investigate the possibility of strong coagulation, we examine a model with no coagulation threshold velocity.
This model is referred to as (ii) the strong coagulation model.
Fig. 8 shows the extinction curves and the grain-size distributions
for carbon and silicate dust at the galactic age of 10 Gyr for (i) theno grain growth model and (ii) the strong coagulation model. We
adopt τ SF = 5 Gyr, ηWNM = ηCNM = 0.5. For comparison, weshow the grain-size distribution using the same parameters as in
Section 3.5 for a fiducial case. In both panels, we find that the MW
extinction curve is better reproduced than for the cases consideredin Section 3. For the no grain growth model (top panels), we observethat the calculated extinction curve is similar to that of the MilkyWay, as the shape of the grain-size distribution is similar to the MRN
distribution. This is because when grain growth is not considered,
the bump produced at a < 0.01 μm in the size distribution (seeFig. 1) cannot be formed. Consequently, the grain-size distribution
approaches a power-law size distribution with a power index ∼−3.5
because shattering and coagulation are dominant processes in theformation of the grain-size distribution (e.g. Kobayashi et al. 2010).
However, if we do not consider the contribution of grain growth to

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Figure 7. As Fig. 6, but with (a) τ SF = 0.5 Gyr and ηWNM = ηCNM = 0.5; (b) τ SF = 50 Gyr and ηWNM = ηCNM = 0.5; (c) τ SF = 5 Gyr, ηWNM = 0.1 andηCNM = 0.9; and (d) τ SF = 5 Gyr, ηWNM = 0.9 and ηCNM = 0.1.

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the evolution of the total dust mass in galaxies, it is hard to reproduce
the total dust mass of the Milky Way, as has been pointed out by,for example, Zhukovska et al. (2008), Draine (2009a), Inoue (2011)
and Asano et al. (2013a). For example, Inoue (2011) showed that in
the case without grain growth, the total dust amount in a galaxy is
about 10 times lower than that in the Milky Way, even if dust grains
are not destroyed by SN shocks. Thus, with this scenario we canobtain an extinction curve similar to that of the Milky Way, but wehave difficulty in reproducing the total dust mass.
Next, we discuss the extinction curve for the case of strong coagulation (bottom panels). We observe that the extinction curve has aslightly larger bump and a steeper UV slope than the MW extinction
curve, as the abundance of small grains (a < 0.01 μm) is slightly
enhanced compared with the MRN size distribution. However, wefind that the bump in the grain-size distribution can disappear. In
this model, we remove the coagulation threshold, so the bump in thegrain-size distribution can shift towards larger sizes than in the fiducial case. When the radius of grains reaches ∼0.2 μm, the grains areshattered effectively, and consequently the bump vanishes. Furthermore, owing to the supply of large grains by strong coagulation,
shattering is induced, and, as a result, small grains are supplied.
Hence, the number of small grains is larger than in the fiducial case.
With such an interplay, the size distribution is expected to approach
MNRAS 440, 134–142 (2014)

the power-law size distribution with a power index −3.5 (Tanaka
et al. 1996; Kobayashi et al. 2010). Consequently, the extinction
curve is nearer to the MW extinction curve than the cases in Section 3. In addition, unlike case (i), case (ii) may naturally accountfor the evolutions of the total dust mass, grain-size distribution andextinction curve in a galaxy at the same time. Note that power-lawlike grain-size distributions can also reproduce the LMC and SMC
extinction curves (Pei 1992). Thus, we conclude that the strong interplay of shattering and coagulation is important in reproducing
the extinction curves observed in nearby galaxies.

4.2 Other possible grain species
As mentioned in Section 1, we do not consider PAHs in this work.
PAHs are important in reproducing the SEDs at near- and midinfrared wavelengths in galaxies (e.g. Li & Draine 2001), and they
may contribute to the 2175-Å bump in the MW extinction curve(e.g. Weingartner & Draine 2001). However, what kinds of grains
dominate the bump at 2175 Å is still a matter of debate (e.g. Draine &Malhotra 1993; Li & Draine 2001; Draine 2003). Furthermore,
although carbon-rich AGB stars or shattering processes in grain–
grain collisions are considered as possible sources of PAHs (e.g.

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Figure 8. Extinction curves (left panels) and grain-size distributions (right panels) at galactic age 10 Gyr for total grains (solid line), carbon dust (dotted line)
and silicate dust (dashed line). From top to bottom, the two panels represent the cases (i) without grain growth, and (ii) without a coagulation threshold velocity.
For reference, we have plotted the results of the grain-size distribution using the same parameters as in Section 3.5 (thin solid line), and the MW extinction
curve (filled diamonds; Whittet 2003). We adopt τ SF = 5 Gyr and ηWNM = ηCNM = 0.5.

Evolution of extinction curves

4.3 Remarks on extinction in high-z galaxies
In Section 4.1, we compared our models with the MW extinction
curve, and investigated the important physics that reproduces the
MW extinction curve. Here, we consider the implications of ourresults for observed extinction curves in high-z galaxies.
The extinction curves in high-z galaxies have recently been explored by many authors (e.g. Maiolino et al. 2004; Gallerani et al.
2010; Hjorth et al. 2013). Indeed, from observations, Gallerani
et al. (2010) showed that the extinction curves in high-z (z > 4)
quasi-stellar objects (QSOs) tend to be flatter than those in nearby
galaxies. In addition, these flat extinction curves in high-z galaxies are thought to be dominated by dust grains ejected by SNe II
whose lifetimes are short (typically 106−7 yr) (e.g. Maiolino et al.
2004; Hirashita et al. 2008). Our result, that the extinction curvesin the early stage of galaxy evolution are relatively flat, is probably
consistent with their results, although our extinction curves may be
too flat. However, there is a suggestion that the large quantities ofdust in dusty QSOs in the high-z Universe are regulated by grain
growth (e.g. Michałowski et al. 2010; Kuo & Hirashita 2012). This
claim is strengthened by the fact that these dusty QSOs in the highz Universe have metallicity Z > Z (e.g. Juarez et al. 2009). As
shown by our calculations, once grain growth occurs, the extinction
curve changes significantly. Thus, as investigated in Section 4.1,
strong coagulation may be worth considering also for high-z QSOs.
However, we should note that QSOs with steep extinction curves
may be missing from the samples that have been observed by otherauthors because of the strong UV (optical in the observer’s frame)extinction.
The rest-UV SEDs in high-z galaxies (z  4) have been investigated by a number of authors (e.g. Bouwens et al. 2009; González
et al. 2012). In particular, it is reported that UV colours in thesegalaxies become bluer with increasing redshift (e.g. Bouwens et al.
2009). This fact is generally interpreted as reflecting low dust
amounts in galaxies. However, the relation between extinction and
UV colour depends on the extinction curve (Wilkins et al. 2013).
In particular, the blue colour may be explained with a flat extinction curve as shown by our results in the early stage of galaxy
evolution. Thus, there may be a significant amount of dust even in
high-z galaxies with blue colours, which implies that the intrinsic
luminosity may be underestimated.
The strong 2175-Å feature in star-forming galaxies at 1 < z < 2.5(the age of the Universe is ∼3–6 Gyr in a CDM cosmologicalmodel with  = 0.7, M = 0.3 and H0 = 70 kms−1 Mpc−1 ) is
reported by Noll et al. (2007). They found robust evidence of the2175-Å bump in one-third of their sample. The existence of the2175-Å bump in galaxies at these redshifts supports our results thatthe bump becomes strong in galaxies at t ∼ τ shat ∼ 1 (τ SF /Gyr)1/2
Gyr with τSF  10 Gyr. Thus, shattering and grain growth may beeffective in these galaxies, and the variation of the extinction curve
can be explained by the difference of the galactic age and the starformation time-scale.
5 CONCLUSION
We investigated the evolution of extinction curves in galaxies basedon the evolution model of grain-size distribution (A13). We took
into account various dust processes: dust formation by SNe II and
AGB stars, dust reduction through astration, dust destruction by
SN shocks in the ISM, grain growth in the CNM, and grain–grain
collisions (shattering and coagulation) in the WNM and CNM.
We found that the extinction curve is flat in the early stage ofgalaxy evolution because of the large grains (a  0.1 μm) produced
by stars. As galactic evolution proceeds, shattering becomes effective, and the number of small grains increases. Later, these small
grains grow owing to grain growth, forming a bump at a ∼ 10−3 –10−2 μm in the grain-size distribution. Because the relatively small
grains are dominant in grain-size distribution, the extinction curve
has a large bump at 1/λ ∼ 4.5 μm−1 and a steep UV slope. The
time-scale on which dust processes in the ISM (grain growth, shattering and coagulation) begin to control the grain-size distribution isestimated as t ∼ 1(τ SF /Gyr)1/2 Gyr. After coagulation occurs effectively, the extinction curves become flatter but still tend to be steeper
than the MW extinction curve. We also found that for reproductionof the MW extinction curve, it may be important to consider a larger
contribution of coagulation than that which we have assumed. Thismeans that the strong interplay between shattering and coagulationinduced by strong coagulation could be essential to reproduce the
MW extinction curve. We conclude that the extinction curves ingalaxies drastically change with galactic age because of the evolution of the grain-size distribution.
AC K N OW L E D G E M E N T S
We thank the anonymous referee for her/his helpful comments,
which improved the quality and clarity of this paper. We are gratefulto Takashi Kozasa, Anthony P. Jones, Daisuke Yamasawa and Asao
Habe for fruitful discussions. We thank H. Kobayashi for helpful
discussions on the process of the grain–grain collisions and the
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Latter 1991; Jones et al. 1996; Seok, Hirashita & Asano 2014), themain formation mechanism of PAHs is still controversial.
We do not consider other carbonaceous dust, such as glassy carbon or amorphous carbon. Nozawa & Fukugita (2013) discussed
the effect of these species on extinction curves, and showed thatthey make the bump at 2175 Å small. This means that the 2175Å bump that we calculated may be overestimated. Recently, Joneset al. (2013) constructed a dust model considering hydrogenatedamorphous carbon without graphite, and the model reproduced theextinction curve, IR extinction, IR–mm dust emission, and albedo
of the Milky Way. In fact, their best-fitting model also depends onthe dust properties adopted. However, the trend that the small carbonaceous grains produce the 2175-Å bump and that large grains
are responsible for flat extinction does not change. Thus, the evolutionary trend of the shape of extinction curves would not change
significantly even if species other than graphite were adopted as
representative carbonaceous species.
In this paper, we consider astronomical silicate as one of themain dust species. However, many observations have suggested
that other species of cosmic silicate may exist as iron-poor ones
such as enstatite (MgSiO3 ) and forsterite (Mg2 SiO4 ) (e.g. Draine
2003). In addition, many of the iron atoms are likely to be locked
up in pure Fe grains (Draine 2009b) and other Fe-bearing grains
such as magnetite (Fe3 O4 ) and iron sulfide (FeS) (e.g. Cowie &Songaila 1985). Recently, Nozawa & Fukugita (2013) showed thatthe extinction curve produced by Fe or Fe3 O4 or FeS grains inaddition to the combination of graphite and Mg2 SiO4 is similar tothat led by the combination of graphite and astronomical silicate,
as long as most of the Mg, Si, and Fe atoms are locked up in thesegrains. This indicates that astronomical silicate and the combinationof other dust species have similar optical properties to each other.
Thus, even if we were to consider other species of silicate grains,
it is unlikely that this would significantly modify the discrepancy
between the modelled and MW extinction curves.

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evolution of the grain-size distribution. RSA was supported by aGrant-in-Aid for JSPS Research under Grant No. 23-5514. TTT
was supported by the Strategic Young Researcher Overseas Visits
Program for Accelerating Brain Circulation No. R2405 and theGrant-in-Aid for Scientific Research Fund (20740105, 23340046)
commissioned by the MEXT. HH is supported by NSC grant 1022119-M-001-006-MY3. TN was supported by the World PremierInternational Research Center Initiative (WPI Initiative), MEXT,
Japan, and by the Grant-in-Aid for Scientific Research of the JSPS
(22684004, 23224004).
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