STAR FORMATION HISTORY, DUST ATTENUATION, AND EXTRAGALACTIC BACKGROUND LIGHT

The proper specific volume emissivity of the radiating sources can be written as

$$\begin{eqnarray}&&{{\epsilon }_{\nu }}(z)={{\epsilon }_{\nu ,Q}}(z)+{{\epsilon }_{\nu ,G}}(z),\end{eqnarray} \tag{ 3 }$$

where ${{\epsilon }_{\nu ,Q}}(z)$ and ${{\epsilon }_{\nu ,G}}(z)$ are the proper specific volume emissivity of QSOs and galaxies, respectively. For QSOs, we use the parametric form for ${{\epsilon }_{\nu ,Q}}(z)$ as given in HM12 at $912\;\overset{\circ}{\rm A}$, which is consistent with the QSO luminosity function of Hopkins et al. (2007),

$$\begin{eqnarray}&&\frac{{{\epsilon }_{912,Q}}(z)}{{{(1+z)}^{3}}}={{10}^{24.6}}{{(1+z)}^{4.68}}\frac{{\rm exp} (-0.28z)}{{\rm exp} (1.77z)+26.3}\;,\end{eqnarray} \tag{ 4 }$$

in units of ergs s$^{-1}$ Mpc$^{-3}$ Hz$^{-1}$. To obtain ${{\epsilon }_{\nu ,Q}}$ at different wavelengths, we use an SED given by the broken power law, L_ν ∝ ν^-0.44 for $\lambda \;\gt$ 1300 Å and ${{L}_{\nu }}\propto {{\nu }^{-1.57}}$ for $\lambda \;\lt$ 1300 Å (Vanden Berk et al. 2001; Telfer et al. 2002). It is well known that stellar emission from galaxies dominates the EBL in the optical regime in all redshifts. Therefore, we place more emphasis on estimating ${{\epsilon }_{\nu ,G}}(z)$ accurately. We discuss this in detail in the following section.

We need to compute the galaxy emissivity, ${{\epsilon }_{\nu ,G}}(z)$, which is consistent with the observed luminosity functions of galaxies at different wavelengths and redshifts. The luminosity function, ${{\phi }_{\nu }}(L,z)$, observed at different z and frequency ν is usually specified in the form of a Schechter function. The comoving luminosity density, ${{\rho }_{\nu }}(z)$, for galaxies, which is nothing but the space-averaged comoving specific emissivity,

$$\begin{eqnarray*}&&{{\rho }_{\nu }}(z)=\frac{{{\epsilon }_{\nu ,G}}(z)}{{{(1+z)}^{3}}},\end{eqnarray*}$$

is given by an integral,

$$\begin{eqnarray}&&{{\rho }_{\nu }}=\int _{{{L}_{{\rm min} }}}^{\infty }L{{\phi }_{\nu }}(L)dL=\phi _{\nu }^{*}\;{{L}^{*}}\;{\Gamma }\left( \alpha +2,\;{{L}_{{\rm min} }}/{{L}^{*}} \right).\end{eqnarray} \tag{ 5 }$$

Here, $\phi _{\nu }^{*}$, L^*, and α are the Schechter parameters, L_min is the luminosity corresponding to the faintest galaxy at a redshift z, and Γ is the incomplete gamma function. We dropped the subscript z in the above equation for clarity. The ${{\rho }_{\nu }}$ depends on the choice of L_min. In principle, one can always take ${{L}_{{\rm min} }}=0$ for $\alpha \gt -2.0$ where the integral in Equation (5) converges. Generally, for galaxies at $z\lt 2.5$, one finds $\alpha \gt -1.3$ (Cucciati et al. 2012). In this case, the change in ${{\rho }_{\nu }}$, when one changes the L_min from 0 to 0.01L^*, is less than 10%. We discuss the effect of adopting different L_min values in Section 8.

The ${{\rho }_{{{\nu }_{0}}}}(z)$ measurements are used to determine the global star formation history (see Lilly et al. 1996; Madau et al. 1996) of the universe, provided that the magnitude of the dust attenuation, ${{A}_{{{\nu }_{0}}}}(z)$, at any frequency ${{\nu }_{0}}$ and redshift z is known. The average SFRD, in units of ${{M}_{\odot }}$ yr⁻¹ Mpc⁻³, is connected to ${{\rho }_{{{\nu }_{0}}}}$ through the relationship (Kennicutt 1998),

$$\begin{eqnarray}&&{\rm SFRD}(z)={{\zeta }_{{{\nu }_{0}}}}\times {{\rho }_{{{\nu }_{0}}}}(z){{10}^{0.4{{A}_{{{\nu }_{0}}}}(z)}}.\end{eqnarray} \tag{ 6 }$$

Here, ${{\zeta }_{{{\nu }_{0}}}}$ is a constant conversion factor that depends on ${{\nu }_{0}}$ and the initial mass function (IMF) assumed for galaxies. However, note that the relation given in Equation (6) is an approximation, as ${{\rho }_{{{\nu }_{0}}}}(z)$ can also have contributions from the old stellar population where the stars that are formed earlier are still shining at z.

The derived SFRD(z) and the SED produced from the instantaneous burst of star formation can be used to obtain the luminosity density at different ν and z. For an assumed IMF and metallicity Z, the population synthesis models provide an SED in terms of the specific luminosity, l_ν (τ ,Z) (in units of ergs s⁻¹ Hz⁻¹ per unit mass of stars formed) at different ages, τ, of the stellar population. Because the timescales involved in the process of star formation (10⁵–10⁷ yr) are relatively small, the SED from the instantaneous starburst can be directly convolved with the global SFRD(z) to obtain the ${{\rho }_{\nu }}\left( {{z}_{0}} \right)$ by solving the following convolution integral (see, e.g., Kneiske et al. 2002, HM12),

$$\begin{eqnarray}&&{{\rho }_{\nu }}\left( {{z}_{0}} \right)={{C}_{\nu }}\left( {{z}_{0}} \right)\int _{{{z}_{0}}}^{{{z}_{{\rm max} }}}{\rm SFRD}(z){{l}_{\nu }}[t\left( {{z}_{0}} \right)-t(z),Z]\;\frac{dt}{dz}dz,\end{eqnarray} \tag{ 7 }$$

where $\tau =t\left( {{z}_{0}} \right)-t(z)$ is the age of the stellar population at the redshift z₀ that went through an instantaneous burst of star formation at a redshift z, ${{C}_{\nu }}\left( {{z}_{0}} \right)$ is the dust correction factor at z_0, and $dt/dz={{[(1+z)H(z)]}^{-1}}.$ The fact that the burst of star formations occurs at all epochs, t, but with the average star formation rates equal to the global SFRD$(t)$, is captured by the product of ${\rm SFRD}(z)$ and ${{l}_{\nu }}\left[ t\left( {{z}_{0}} \right)-t(z),Z \right]$ in the convolution integral. We use ${{z}_{{\rm max} }}=\infty$, as often used in the literature (e.g., Gilmore et al. 2009; Inoue et al. 2013, HM12). Later in Section 8, we discuss the validity of the ${{z}_{{\rm max} }}=\infty$ assumption and the effect of using different z_max. The dust correction factor, ${{C}_{\nu }}\left( {{z}_{0}} \right)$, for $\lambda \lt 912$ Å is assumed to be equal to the escape fraction of hydrogen-ionizing photons from galaxies, as given in HM12. For $\lambda \gt 912$ Å, we use ${{C}_{\nu }}\left( {{z}_{0}} \right)={{10}^{-0.4\,{{A}_{\nu }}\left( {{z}_{0}} \right)}}$, where ${{A}_{\nu }}\left( {{z}_{0}} \right)$, which is normalized at ${{\nu }_{0}}$, is given by

$$\begin{eqnarray}&&{{A}_{\nu }}\left( {{z}_{0}} \right)={{A}_{{{\nu }_{0}}}}\left( {{z}_{0}} \right)\frac{{{k}_{\nu }}}{{{k}_{{{\nu }_{0}}}}}.\end{eqnarray} \tag{ 8 }$$

Here, ${{k}_{\nu }}$ is a frequency-dependent dust extinction curve.

Motivated by the previous works of Stecker et al. (2012) and Helgason et al. (2012), we have compiled available observations of the galaxy luminosity functions and the corresponding ${{\rho }_{\nu }}$ at different rest wavelengths and z. In Table 1, we have given references along with the rest waveband and a redshift range over which the luminosity functions have been determined. In Table 5 in the Appendix, we list the faint end slopes of the luminosity functions with the rest waveband and redshift, along with the L_min values we used to determine ${{\rho }_{\nu }}.$ In general, we preferred the references where the luminosity functions are determined in different wavebands from the FUV (centered at $\lambda =0.15\;\mu {\rm m}$) to the K (2.2 μm) band and with the largest possible coverage in redshift. This compilation has luminosity functions determined in the FUV band up to z = 8, in the NUV and H band up to z = 3.5, and for all other bands the measurements are available up to $z\sim 2.5.$ We take the ${{\rho }_{\nu }}(z)$ with the errors from the references where it is explicitly calculated. We use luminosity functions given in other references and compute ${{\rho }_{\nu }}$(z) (using Equation (5)) with ${{L}_{{\rm min} }}=0.01{{L}^{*}}.$ Because there are more measurements of the ${{\rho }_{\nu }}$ in the FUV band and covering a large z range, we choose ${{\nu }_{0}}={{\nu }_{{\rm FUV}}}$, the frequency we use to determine SFRD(z) as a frequency corresponding to the FUV band.

Table 1.  Details of the Observed Galaxy Luminosity Functions Used to Obtain the ${{\rho }_{\nu }}$ in Our Study

Reference	Wavebanda	Redshift Range	Plotting Symbolb
Schiminovich et al. (2005)	FUV	0.2–2.95	cyan triangle
Reddy & Steidel (2009)	FUV	1.9–3.4	orange triangle
Bouwens et al. (2007)	FUV	3.8–5.9	red diamond
Bouwens et al. (2011)	FUV	6.8–8	red diamond
Dahlen et al. (2007)	FUV	0.92–2.37	blue square
	NUV	0.29–2.37	blue square
Cucciati et al. (2012)	FUV	0.05–4.5	green circle
	NUV	0.05–3.5	green circle
Tresse et al. (2007)	FUV, NUV, U, V, B, R, I	0.05–2	red circle
Wyder & Treyer (2005)	NUV	0.055	black triangle
Faber et al. (2007)	B	0.2–1.2	orange star
Dahlen et al. (2005)	U, B, R	0.1–2	blue square
	J	0.1–1	blue square
Stefanon & Marchesini (2013)	J, H	1.5–3.5	green square
Pozzetti et al. (2003)	J, K	0.2–1.3	orange triangle
Arnouts et al. (2007)	K	0.2–2	green diamond
Cirasuolo et al. (2007)	K	0.25–2.25	blue triangle

Notes.

^aCentral wavelengths corresponding to different wavebands are as follows: FUV = 0.15 μm, NUV = 0.28 μm, U = 0.365 μm, B = 0.445 μm, V = 0.551 μm, R = 0.658 μm, I = 0.806 μm, J = 1.27 μm, H = 1.63 μm, and K = 2.2 μm. ^bThese plotting symbols are used in Figures 5 and 17 for the ${{\rho }_{\nu }}$ obtained using different luminosity functions.

Download table as:  ASCIITypeset image

We use a population synthesis model, "Starburst99"¹ (Leitherer et al. 1999), to obtain the specific luminosity from the stellar population of a typical galaxy, ${{l}_{\nu }}(t,Z)$, at an age t and a metallicity Z with an instantaneous burst of star formation. In these simulations, we consider a constant metallicity of Z = 0.008 over all z. Later in Section 8, we also discuss the effect of using different values of metallicity. We use the Salpeter IMF with an exponent of 2.35 and the stellar mass range from 0.1 to 100 ${{M}_{\odot }}$. For this particular galaxy model, we find the conversion factor for connecting ${{\rho }_{{\rm FUV}}}(z)$ and SFRD(z) (see Equation (6)) to be ${{\zeta }_{{{\nu }_{0}}}}=1.25\times {{10}^{-28}}$. As described before, the reference frequency ν₀ that we use corresponds to the frequency of the FUV band. Note that this conversion factor 1.25 × 10⁻²⁸ is 11% smaller than widely used, 1.4 × 10⁻²⁸, quoted by Kennicutt (1998). This difference is mainly because of the updated population synthesis model and the assumed metallicity.

We fit a functional form to the compiled ${{\rho }_{{\rm FUV}}}$ data and obtained its parameter using the mpfit IDL routine² that uses ${{\chi }^{2}}$ minimization. At high redshifts, we take 20% errors on the ${{\rho }_{{\rm FUV}}}$ calculated from the luminosity function given by Bouwens et al. (2011). We convert the asymmetric errors into symmetric errors by taking their average. To fit ${{\rho }_{{\rm FUV}}}(z)$, we use the following functional form, which was originally used by Cole et al. (2001) to fit the SFRD(z),

$$\begin{eqnarray}&&{{\rho }_{{\rm FUV}}}(z)=\frac{a+bz}{1+{{(z/c)}^{d}}}.\end{eqnarray} \tag{ 9 }$$

There is a large scatter in the ${{\rho }_{{\rm FUV}}}(z)$ data. Therefore, along with this fit (hereafter, the median fit), we construct 1σ upper- and lower-limit fits (hereafter we refer to them as the high and low ${{\rho }_{{\rm FUV}}}$ fits, respectively). These ${{\rho }_{{\rm FUV}}}$ fits multiplied by $1.25\times {{10}^{-28}}$ are nothing but the different SFRD(z) with the ${{A}_{{\rm FUV}}}(z)=0$ (see Equation (6)), which are plotted in Figure 1. The values of fitting parameters for the median ${{\rho }_{{\rm FUV}}}\times 1.25\times {{10}^{-28}}$ fit are $a=(6\pm 1)\times {{10}^{-2}}$, $b=(11\pm 2)\times {{10}^{-2}}$, $c=4.41\pm 0.58$, and $d=3.15\pm 0.62.$ We construct 1σ high and low ${{\rho }_{{\rm FUV}}}$ fits by adding and subtracting the error in each parameter from its respective best-fit value (see, Figure 1). We determine the combinations of SFRD(z) and ${{A}_{{\rm FUV}}}(z)$ for all three (low, median, and high) ${{\rho }_{{\rm FUV}}}$ fits with different extinction curves, as discussed below.

Zoom In Zoom Out Reset image size

The average extinction curve for high-redshift galaxies is one of the key unknowns in astronomy. However, the mean extinction curves for our galaxy, SMC, and LMC and for some low-redshift starburst galaxies are well known (Lequeux et al. 1982; Clayton & Martin 1985; Calzetti et al. 1994). It is a general practice to use the average extinction curve determined for the nearby starburst galaxies by Calzetti et al. (2000) ³ for the high-redshift galaxies. Here, along with the Calzetti et al. (2000) extinction curve, we use extinction curves determined for the SMC, LMC, and LMC supershell (LMC2) from Gordon et al. (2003) and for the Milky Way (MW) from Misselt et al. (1999). In particular, this set of extinction curves encompasses a wide range of dust properties typically present in the astronomical domain. Because we are using ${{\rho }_{{\rm FUV}}}$ measurements for determining the SFRD(z), we normalize all the extinction curves, ${{k}_{\nu }}$, at ν corresponding to the FUV band ($0.15\;\mu {\rm m}$). In Figure 2, we have plotted the ${{k}_{\nu }}/{{k}_{{\rm FUV}}}$ for different extinction curves as a function of ${{\lambda }^{-1}}$, along with the respective measured data points from Gordon et al. (2003) for SMC, LMC, and LMC2. In Figure 2, we also mark the different ${{\lambda }^{-1}}$ for the wavebands at which we have compiled the ${{\rho }_{\nu }}$ measurements to determine the ${{A}_{{\rm FUV}}}$ and the SFRD.

Zoom In Zoom Out Reset image size

From Equation (6) it is clear that SFRD(z) and ${{A}_{{\rm FUV}}}(z)$ are degenerate quantities, and different combinations of them can give the same ${{\rho }_{{\rm FUV}}}$. However, the measured ${{\rho }_{\nu }}$ values at different frequencies other than the FUV band, along with the assumed extinction curve, break this degeneracy. Here we introduce a novel method that, by using the multiwavelength and multiepoch luminosity functions, determines the ${{A}_{{\rm FUV}}}(z)$ and SFRD(z) uniquely for an assumed extinction curve. In this method we initially fix the ${{A}_{{\rm FUV}}}$ and SFRD at some higher redshifts, and then, using this we progressively determine ${{A}_{{\rm FUV}}}$ and SFRD at lower redshifts. This "progressive fitting method" is described below in detail.

Combining Equations (6)–(8), the ${{\rho }_{\nu }}$(z₀) can be written as

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} {{\rho }_{\nu }}\left( {{z}_{0}} \right) & = & 1.25\times {{10}^{-28}}\times {{10}^{\left[ -{{A}_{{\rm FUV}}}\left( {{z}_{0}} \right)\frac{{{k}_{\nu }}}{{{k}_{{\rm FUV}}}} \right]}} \\ {} & {} & \times \int _{{{z}_{0}}}^{\infty }{{\rho }_{{\rm FUV}}}(z)\ {{10}^{0.4{{A}_{{\rm FUV}}}(z)}}\; \\ {} & {} & \times {{l}_{\nu }}\left[ t\left( {{z}_{0}} \right)-t(z),Z \right]\;\frac{dt}{dz}dz. \\ \end{array}\end{eqnarray} \tag{ 10 }$$

For a given extinction curve, ${{k}_{\nu }}$, and our ${{\rho }_{{\rm FUV}}}(z)$ fits, the only unknown in the above equation is ${{A}_{{\rm FUV}}}(z)$ for z ≥ z₀. Therefore, to obtain the ${{\rho }_{\nu }}$(z₀) one needs to know the ${{A}_{{\rm FUV}}}(z)$ for all z ≥ z₀. The procedure we followed to obtain the ${{A}_{{\rm FUV}}}(z)$ for each extinction curve, ${{k}_{\nu }}$, using the ${{\rho }_{\nu }}(z)$ measurements is given below:

• 1.  
  We choose the highest possible redshift z_th where we have ${{\rho }_{\nu }}\left( {{z}_{{\rm th}}} \right)$ measurements in most of the wavebands.
• 2.  
  For all $z\geqslant {{z}_{{\rm th}}}$, we assume a functional form for ${{A}_{{\rm FUV}}}(z).$
• 3.  
  We fix the normalization of this function and hence the value of ${{A}_{{\rm FUV}}}\left( {{z}_{{\rm th}}} \right)$ by matching the predicted ${{\rho }_{\nu }}\left( {{z}_{{\rm th}}} \right)$ with the measured ones at different wavebands (other than the FUV band), using least-squares minimization. This fixes the ${{A}_{{\rm FUV}}}(z)$ for $z\geqslant {{z}_{{\rm th}}}.$ Then we call z_th as z₁.
• 4.  
  We choose the next redshift ${{z}_{0}}\lt {{z}_{1}}$, which is the next nearby lower redshift where we have multiwavelength ${{\rho }_{\nu }}\left( {{z}_{0}} \right)$ measurements.
• 5.  
  We assume ${{A}_{{\rm FUV}}}(z)$ is constant and equal to ${{A}_{{\rm FUV}}}\left( {{z}_{0}} \right)$ in between the redshifts z₀ and z₁. For z ≥ z₁ we use ${{A}_{{\rm FUV}}}(z)$, as determined earlier. Then we calculate the ${{\rho }_{\nu }}\left( {{z}_{0}} \right)$ for different values of ${{A}_{{\rm FUV}}}\left( {{z}_{0}} \right)$.
• 6.  
  We compare the resultant ${{\rho }_{\nu }}\left( {{z}_{0}} \right)$ with the measured one at different wavebands and determine the best-fit ${{A}_{{\rm FUV}}}\left( {{z}_{0}} \right)$ by least-squares minimization. This fixes the ${{A}_{{\rm FUV}}}(z)$ for z ≥ z₀. Then we call this z₀ as z₁.
• 7.  
  We repeat steps 4 to 6 until we reach the lowest z where we have multiwavelength ${{\rho }_{\nu }}(z)$ measurements. This provides us the best-fit values of ${{A}_{{\rm FUV}}}(z)$ and SFRD(z) over the whole redshift range for a given extinction curve.

In Figure 3 we show the ${{A}_{{\rm FUV}}}(z)$ obtained (histograms) using the progressive fitting method described above for different extinction curves. We fit a continuous function through the resultant ${{A}_{{\rm FUV}}}(z)$ using a functional form that is the same as the one we used to fit the ${{\rho }_{{\rm FUV}}}(z)$ measurements (given in Equation (9)). For fitting this functional form we use the mpfit idl routine by taking 10% errors for all. To demonstrate the procedure described here, in Figure 3, we also show the resultant ${{A}_{{\rm FUV}}}$(z) obtained using the high, low, and median ${{\rho }_{{\rm FUV}}}$ fits (histograms), along with its fitted functional form for different extinction curves. Because we show Figure 3 for the purpose of demonstrating our "progressive fitting method," for clarity, we do not show ${{A}_{{\rm FUV}}}(z)$ obtained for the MW extinction curve. Note that this resultant ${{A}_{{\rm FUV}}}(z)$ will directly give the corresponding SFRD$(z)$ (see Equation (6)). We also fit SFRD$(z)$ using the same functional form (see Equation (9)).

Zoom In Zoom Out Reset image size

Our aim is to obtain the combinations of ${{A}_{{\rm FUV}}}$(z) and SFRD(z) that will reproduce the measured ${{\rho }_{\nu }}(z)$ obtained using the observed luminosity functions at different wavebands and different z. The ${{\rho }_{\nu }}(z)$ measurements are taken from different references, and they have different biases and error estimates. Therefore, to minimize the uncertainty and determine the ${{A}_{{\rm FUV}}}$ over a large z range uniquely, we have to choose ${{\rho }_{\nu }}(z)$ measurements that span many wavebands and a large z range and possibly reported by the same group, so that the effect of various biases will be minimum. Fortunately, this requirement is satisfied by the ${{\rho }_{\nu }}$ measurements reported in Tresse et al. (2007), where the ${{\rho }_{\nu }}$ is measured over seven different wavebands (from FUV to I band) and at the same redshift bins spanning up to z = 2. Therefore, to obtain a robust ${{A}_{{\rm FUV}}}(z)$ and SFRD(z) combination we choose the observed ${{\rho }_{\nu }}(z)$ given by Tresse et al. (2007) and take ${{z}_{{\rm th}}}=2$. We assume that the form of the ${{A}_{{\rm FUV}}}$(z) for $z\geqslant 2$ becomes $1/(1+z)$ and is independent of the extinction curve used. We show later that this assumed form gives ${{A}_{{\rm FUV}}}$(z) consistent with other independent measurements. This trend of decreasing ${{A}_{{\rm FUV}}}$ at higher z has been previously observed (see. e.g., Takeuchi et al. 2005; Bouwens et al. 2009; Cucciati et al. 2012; Burgarella et al. 2013). This is consistent with the picture of a gradual buildup of dust in galaxies with cosmic time, as evident from the fact that galaxies at very high redshifts ($z\gt 5$) are bluer than the $z\sim 2$–4 galaxies (Bouwens et al. 2009).

We calculate the ${{A}_{{\rm FUV}}}(z)$ and corresponding SFRD(z) for all five extinction curves used in this paper using the low, high, and median ${{\rho }_{{\rm FUV}}}(z)$ fits. As we show later, we use the ${{\rho }_{\nu }}(z)$ obtained using the combinations of ${{A}_{{\rm FUV}}}(z)$ and SFRD(z) to estimate the ${{\rho }_{\nu }}(z)$ at FIR wavelengths and the EBL at different redshifts. We denote the obtained combinations of ${{A}_{{\rm FUV}}}(z)$ and SFRD(z), the ${{\rho }_{\nu }}(z)$, and the EBL using different extinction curves as the "smc," "lmc," "lmc2," "mw," and "cal" models, based on the SMC, LMC, LMC2, MW, and Calzetti et al. (2000) extinction curves used, respectively. For most comparisons we use our default models, which are obtained using median fits through the ${{\rho }_{{\rm FUV}}}$ points. We use the predictions of the high and low fits only when we discuss the spread. For clarity in the subsequent discussions, whenever we use the "high (low) model," we mean the relevant quantity (e.g., ${{\rho }_{\nu }}$, ${{A}_{{\rm FUV}}}$, SFRD, and EBL) obtained with the high (low) ${{\rho }_{{\rm FUV}}}$ fit and the "model" extinction curve. When we denote only "model," we mean the relevant quantity obtained using the median ${{\rho }_{{\rm FUV}}}$ fit and that "model" extinction curve.

In the following section we discuss the resultant ${{\rho }_{\nu }}(z)$, ${{A}_{{\rm FUV}}}(z)$, and SFRD(z) determined using the method described in this section.

In Figure 4, we plot the ${{\rho }_{\nu }}$ obtained using the convolution integral (Equation (7)) for the best-fit combinations of SFRD and ${{A}_{{\rm FUV}}}$ at different z, along with the measurements of Tresse et al. (2007). Note that to obtain the ${{A}_{{\rm FUV}}}$ by least-squares minimization we use the ${{\rho }_{\nu }}$ measurements of Tresse et al. (2007) in all wavebands except at the FUV band. However, these ${{\rho }_{{\rm FUV}}}$ measurements, along with many others reported in the literature up to z = 8 (see Tables 1 and 5), go into fitting the ${{\rho }_{{\rm FUV}}}$ (as shown in Figure 1). All our five models show very good agreement with the measurements of Tresse et al. (2007) in all wavebands (including the FUV band). The difference in the strength of the 2175 Å absorption feature arises because of using different extinction curves.

Zoom In Zoom Out Reset image size

In Figure 5, along with the compiled measurements, we plot the ${{\rho }_{\nu }}$ at the FUV band obtained by using the combination of the SFRD$(z)$ and the ${{A}_{{\rm FUV}}}(z)$ for the low, median, and high "lmc2" model. There are negligible differences in the ${{\rho }_{{\rm FUV}}}$ obtained for different models. Therefore, for clarity, we do not show the similar ${{\rho }_{{\rm FUV}}}$ plots for other models.

Zoom In Zoom Out Reset image size

For the high z and all other wavebands, we show our estimated ${{\rho }_{\nu }}$, along with the compiled measurements in Figure 17 in the Appendix (see Table 1 for the references and plotting symbols). Even though we use measurements of ${{\rho }_{\nu }}$ up to $z\sim 2$ and up to the wavelength corresponding to the I band to obtain the ${{A}_{{\rm FUV}}}(z)$, our estimated ${{\rho }_{\nu }}$ matches well with the various measurements up to $z\sim 4$ from the NUV to the K band. This implies that our determined combinations of the ${{A}_{{\rm FUV}}}(z)$ and the SFRD(z) are valid over a large z range and suggests that our assumption of decreasing dust attenuation at high z is also valid. However, note that at the high redshifts (i.e., $z\gt 4$) there are no measurements of ${{\rho }_{\nu }}$ except at the FUV band. In the H band, our calculated ${{\rho }_{\nu }}$(z) is slightly overestimated than the measured ones. However, as there are very few measurements we do not attempt to address this disagreement.

The good matching between the observations and the model predictions suggests that we have a consistent combination of the ${{A}_{{\rm FUV}}}(z)$ and SFRD(z) for each extinction curve under consideration. The evolution of our best-fit ${{A}_{{\rm FUV}}}$ and the corresponding SFRD with z is discussed in the next section.

Understanding the dust attenuation and its wavelength and redshift dependences is very important to derive the intrinsic SFRD(z) accurately. Dust attenuation is measured by using either of the SED fitting techniques, the Balmer decrement method, or by comparing the FUV and the IR luminosity function measurements. It has also been noticed that at any given z, the derived ${{A}_{{\rm FUV}}}$ may also depend on the galaxy luminosity and the stellar mass of the galaxy (see, e.g., Bouwens et al. 2012). Recently, it has been shown that the shape of the extinction curve strongly depends on the distribution of dust in the galaxies and the viewing geometries where scattering plays an important role (Chevallard et al. 2013). As our main purpose is to calculate the EBL, we are mainly interested in the volume-averaged star formation rates and emissivity. Therefore, to calculate the average dust correction as a function of z, for simplicity we do not consider the dependence of ${{A}_{{\rm FUV}}}$ on galaxy luminosity or stellar mass and the dependence of ${{k}_{\nu }}$ on scattering and viewing geometries. In this section, we compare ${{A}_{{\rm FUV}}}(z)$ obtained for different extinction curves with the ${{A}_{{\rm FUV}}}$ measurements in the literature based on other independent approaches.

The fitting parameters for ${{A}_{{\rm FUV}}}(z)$ for different extinction curves are given in Table 2. In Figure 6, we plot the range of ${{A}_{{\rm FUV}}}(z)$ for different extinction curves, along with the measurements of Takeuchi et al. (2005), Cucciati et al. (2012), Burgarella et al. (2013), and Bouwens et al. (2012). Takeuchi et al. (2005) and Burgarella et al. (2013) determined the ${{A}_{{\rm FUV}}}$ using the ratio of the FUV to FIR band luminosity density. Cucciati et al. (2012) have calculated the ${{A}_{{\rm FUV}}}$ using the Calzetti et al. (2000) extinction curve and used the SED fitting technique. At very high redshifts, Bouwens et al. (2012) determined the effective dust extinction using the UV-continuum slope $\beta$ distribution and the IRX-$\beta$ relationship (see Meurer et al. 1999). We take the effective extinction calculated for the luminosity function integrated up to −17.7 magnitude from Bouwens et al. (2012, from Table 6 of their paper).

Zoom In Zoom Out Reset image size

Table 2.  Fitting Parameters for the ${{A}_{{\rm FUV}}}$ ^a

Extinction Curve	${{\rho }_{{\rm FUV}}}$ Fitb	a	b	c	d
SMC	Low	1.41	0.79	2.51	2.32
	Median	1.03	1.01	1.87	2.16
	High	0.85	0.86	1.77	2.16
LMC2	Low	1.91	0.85	2.40	2.23
	Median	1.42	0.93	2.08	2.20
	High	1.00	1.16	1.66	2.14
LMC	Low	2.24	0.79	2.51	2.21
	Median	1.81	0.82	2.12	2.06
	High	1.53	0.73	1.93	2.05
Milky Way	Low	2.76	0.44	2.98	2.14
	Median	2.39	0.48	2.44	1.97
	High	2.02	0.49	2.15	1.97
Calzetti	Low	2.45	0.79	2.48	2.15
	Median	1.96	1.00	1.94	2.02
	High	1.61	0.91	1.81	2.03

Notes.

^aThe fitting form is ${{A}_{{\rm FUV}}}(z)=\frac{a+bz}{1+{{(z/c)}^{d}}}.$ ^bNote that the models with low (high) ${{\rho }_{{\rm FUV}}}$ fit give higher (lower) ${{A}_{{\rm FUV}}}$ than the median model, as explained in the text.

Download table as:  ASCIITypeset image

The shaded region in Figure 6 is obtained by using the low, high, and median ${{\rho }_{{\rm FUV}}}$ fits to determine the ${{A}_{{\rm FUV}}}$. Since the SFRD is directly related to the ${{\rho }_{{\rm FUV}}}$ and ${{A}_{{\rm FUV}}}$, when we use the low (high) ${{\rho }_{{\rm FUV}}}$ fits, to obtain the same ${{\rho }_{\nu }}$ at different wavebands we need higher (lower) SFRD and hence higher (lower) ${{A}_{{\rm FUV}}}$. In other words, the low (high) ${{\rho }_{{\rm FUV}}}$ implies that galaxies are more red (blue), which suggests that these galaxies should have more (less) dust extinction. This trend is evident from Figure 6, where the dotted and dashed curves show the ${{A}_{{\rm FUV}}}$ obtained using the low and high ${{\rho }_{{\rm FUV}}}$ fits, respectively.

The shaded region in Figure 6 represents the allowed range of ${{A}_{{\rm FUV}}}$. For each assumed extinction curve, we obtain a different allowed range for the ${{A}_{{\rm FUV}}}$, and the difference is prominent at redshifts $z\lt 1$. For redshifts $1\lt z\lt 2$, we find that the ${{A}_{{\rm FUV}}}$ values remain constant or show a mild decrease with increase in $z$. At high redshifts, i.e., $z\gt 2$, our assumption of decreasing dust attenuation plays a role in obtaining a similar allowed range of the ${{A}_{{\rm FUV}}}$ for all assumed extinction curves. As can be seen from Figure 6, the allowed ${{A}_{{\rm FUV}}}$ range for $z\gt 2$ nicely follows that of other independent measurements, rendering support to our assumption. Apart from the ${{A}_{{\rm FUV}}}$ determined for the "mw" model, for all the other models we find a moderate increase in the ${{A}_{{\rm FUV}}}$, with redshift up to $z=1$ from $z=0$. This trend of increasing FUV band dust attenuation magnitude has been detected previously (see Takeuchi et al. 2005; Cucciati et al. 2012; Burgarella et al. 2013), as shown in Figure 6. However, at $z\leqslant 0.8$, our estimated ${{A}_{{\rm FUV}}}(z)$ for the "cal," "mw," and "lmc" models are higher than these measurements. The ${{A}_{{\rm FUV}}}$ determined for the "smc" model matches well in all redshifts, except that it underpredicts ${{A}_{{\rm FUV}}}$ at $1\lt z\lt 2$. Overall, a good match with these measurements of the ${{A}_{{\rm FUV}}}$ is obtained over the large $z$ range for the "lmc2" model.

From the very good agreement between the ${{A}_{{\rm FUV}}}(z)$ determined for the "lmc2" model and the measurements of Burgarella et al. (2013) and Takeuchi et al. (2005), we conclude that the average extinction curve, which is applicable for galaxies over a wide range of redshifts, is most likely to be similar to the LMC2 extinction curve.

Recently, Kriek & Conroy (2013), using the SED of the galaxies, investigated the dust extinction curves for 32 different spectral classes of galaxies over $0.5\leqslant z\leqslant 2$. They found that the MW and Calzetti extinction curves provide poor fits to the UV wavelengths for all SEDs. They concluded that the SED with the $2175\;\overset{\circ}{\rm A}$ UV bump, albeit weaker in strength compared to the MW, is preferred. Buat et al. (2012) studied a sample of 751 galaxies with redshift $0.95\lt z\lt 2.2$ and found that the mean parameters describing the dust attenuation curves are similar to those found for the LMC2 extinction curve. This is consistent with what we find here. It is interesting to note that in the case of intervening absorption systems seen in the QSO spectra, the high percentages of systems with the $2175\;\overset{\circ}{\rm A}$ absorption feature detection favors the LMC2 extinction curve (see, e.g., Srianand et al. 2008; Noterdaeme et al. 2009; Jiang et al. 2013). It has also been observed that the extinction curves for individual galaxies depend on the type and other galaxy properties (Wild et al. 2011; Chevallard et al. 2013; Kriek & Conroy 2013). Therefore, a single universal extinction curve for all galaxies may not be realistic. However, our study suggests that for estimating the volume-averaged properties such as the SFRD, ${{A}_{{\rm FUV}}}$, and EBL, the LMC2 extinction curve should be preferred.

We assume that the SFRD is a smooth and continuous function of $z$ and fit it with the same functional form (using Equation (9)) we are using to fit the ${{\rho }_{{\rm FUV}}}$ and ${{A}_{{\rm FUV}}}$. The fitting parameters for the SFRD$(z)$ are given in Table 4. In Figure 7, we plot the SFRD$(z)$ for all five extinction curves with their high and low models. As explained earlier, we obtain the high (low) SFRD for low (high) ${{\rho }_{{\rm FUV}}}$ fits. Because the SFRD is directly proportional to the ${{\rho }_{{\rm FUV}}}$ and ${{A}_{{\rm FUV}}}$, at $z\gt 3$, where differences in the ${{A}_{{\rm FUV}}}$ for all high and low models are small, the ${{\rho }_{{\rm FUV}}}$ term dominates, and the SFRD$(z)$ curves cross each other, as demonstrated in Figure 7.

Zoom In Zoom Out Reset image size

Table 3.  SFRD from Different Observations

Reference	Technique	Redshift Range	Plotting Symbols
Shim et al. (2009)	H-$\alpha$	0.7–1.9	cyan diamond
Tadaki et al. (2011)	H-$\alpha$	2.2	red diamond
Sobral et al. (2013)	H-$\alpha$	0.4–2.3	green diamond
Ly et al. (2011)	H-$\alpha$	0.8	orange diamond
Condon et al. (2002)	1.4 GHz	0.02	cyan squares
Smolčić et al. (2009)	1.4 GHz	0.1–1.3	red and orange squares
Rujopakarn et al. (2010)	FIR	0–1.3	blue circles
Burgarella et al. (2013)	FIR	0–4	red circles

Download table as:  ASCIITypeset image

Table 4.  Fitting Parameters for the SFRD$(z)$ ^a

Extinction	${{\rho }_{{\rm FUV}}}$	a	b	c	d
Curve	Fitb	(${{10}^{-2}}$)	(${{10}^{-2}}$)
SMC	Low	1.55	7.14	2.53	3.10
	Median	1.38	6.24	2.65	3.01
	High	1.50	5.12	3.08	3.09
LMC2	Low	2.54	10.9	2.22	3.07
	Median	2.01	8.48	2.50	3.09
	High	1.67	7.09	2.74	3.02
LMC	Low	3.57	13.6	2.15	3.13
	Median	3.13	9.88	2.37	3.03
	High	3.03	7.37	2.70	3.01
Milky Way	Low	6.27	15.2	2.14	3.16
	Median	5.78	11.2	2.28	3.02
	High	5.03	8.33	2.59	2.99
Calzetti	Low	4.44	15.8	2.06	3.11
	Median	3.62	12.0	2.20	2.97
	High	3.23	8.78	2.54	2.97

Notes.

^aThe fitting form is ${\rm SFRD}(z)=\frac{a+bz}{1+{{(z/c)}^{d}}}$ ${{M}_{\odot }}\;{\rm y}{{{\rm r}}^{-1}}\;{\rm Mp}{{{\rm c}}^{-3}}$. ^bNote that, as explained in the text, the models with low and high ${{\rho }_{{\rm FUV}}}$ fit need not give higher and lower SFRD$(z)$ than the median model for all $z$, respectively.

Download table as:  ASCIITypeset image

In Figure 7, we also plot the SFRD determined through different observations. There are different indicators of star formation, but not all are independent of the assumed dust correction. Therefore for comparison we use the SFRDs determined through the radio, H-$\alpha$, and FIR emission from galaxies. We do not consider the SFRD determined with observations such as the UV luminosity, where it mainly depends on the assumed values of ${{A}_{{\rm FUV}}}$ and the extinction curve. We select the measurements where the luminosity densities are converted to the SFRD using conversion laws (see Equation (6)) with assumed Salpeter IMF with stellar mass with range 0.1 to 100 ${{M}_{\odot }}$, similar to the one we use. Table 3 summarizes the references to such data and indicates the plotting symbols used for it in Figure 7. Our inferred SFRD$(z)$ using the "cal," "lmc," and "mw" models are higher than these SFRD measurements with different techniques at $z\lt 1$ (see, Figure 7). The SFRD$(z)$ calculated for the "smc" model is consistent with SFRD measurements using radio observations at low $z$ but underpredicts the SFRD at $2\gt z\gt 1$. We find that the SFRD$(z)$ determined using the "lmc2" model fits the SFRD data well at all $z$. Like the case of ${{A}_{{\rm FUV}}}$, determination of SFRD$(z)$ based on the LMC2 extinction curve provides the best fit to the different independent measurements, compared to those of other models.

From Figures 6 and 7, it is clear that, irrespective of the extinction law used, we find the $z$ at which the SFRD$(z)$ peaks is higher than the $z$ beyond which ${{A}_{{\rm FUV}}}(z)$ begins to decline. A similar trend in peaks of SFRD and ${{A}_{{\rm FUV}}}$ has been reported in Cucciati et al. (2012) and Burgarella et al. (2013).

In Figure 8 we plot our SFRD obtained using the LMC2 extinction curve, along with the SFRD determined by Madau & Dickinson (2014). The shaded region in Figure 8 represents the SFRD range covered when we use low and high "lmc2" models to determine the SFRD and ${{A}_{{\rm FUV}}}$. Madau & Dickinson (2014) used the ${{\rho }_{{\rm FUV}}}$ measurements from the literature and converted them into the SFRD, using the conversion constant, $\zeta =1.15\times {{10}^{-28}}$, which is 10% smaller than what we use. For the dust correction they use the ${{A}_{\nu }}$ provided by the different surveys from where the luminosity functions are used to obtain the ${{\rho }_{{\rm FUV}}}$. They calculate the ${{\rho }_{{\rm FUV}}}$ by integrating the luminosity function from ${{L}_{{\rm min} }}=0.03{{L}^{*}}$, whereas in our case we directly take the ${{\rho }_{{\rm FUV}}}$ given in different references, where it is often calculated with ${{L}_{{\rm min} }}=0$ (for the ${{L}_{{\rm min} }}$ values used here, see Table 5 in the Appendix). Madau & Dickinson (2014) use the same IMF we used, but take different metallicities and consider metallicity evolution with $z$. As compared to our preferred SFRD$(z)$ for the LMC2 model, the SFRD of Madau & Dickinson (2014) shows rapid increase and decrease in low and high $z$, respectively. However, the difference between both is within 0.1–0.2 dex for $z\lt 5$. The peak of SFRD$(z)$ of our preferred "lmc2" model matches exactly with that of Madau & Dickinson (2014). The peak of our SFRD$(z)$ is at $z=1.9_{-0.3}^{+0.2}$, which is also consistent with the peak of SFRD reported by Cucciati et al. (2012).

Zoom In Zoom Out Reset image size

For comparing the SFRD$(z)$ shapes, we also plot the fit to the SFRD measurements compiled from the different observational data reported in the literature given by Behroozi et al. (2013) (see Figure 2 and Table 4 of their paper). Behroozi et al. (2013) provide the fit for the recent data and the old data used by Hopkins & Beacom (2006). Both of these fits are obtained for the Chabrier (2003) IMF and show a rapid increase at low $z$ as compared to our SFRD$(z)$. At high $z$, the fit through the compiled SFRD data used by Hopkins & Beacom (2006) shows a slow decrease, whereas the Behroozi et al. (2013) fit shows a rapid decrease as compared to our SFRD. The peak of the compiled SFRD measurements of Behroozi et al. (2013) for the old and new measurements is at $z=1.7$, which is consistent with our SFRD$(z)$ peak within the allowed uncertainties. The main differences between our SFRD$(z)$ estimated here and the compiled SFRD measurements of Behroozi et al. (2013) and SFRD$(z)$ estimated by Madau & Dickinson (2014) is that we self-consistently calculate the ${{A}_{{\rm FUV}}}(z)$, which gives SFRD$(z)$ consistent with ${{\rho }_{\nu }}$ measurements at different wavebands and redshifts.

Having obtained the best-fit ${{A}_{{\rm FUV}}}(z)$ and SFRD$(z)$, we use the stellar population synthesis models to calculate the emissivity from the UV to NIR regime. However, to generate the complete EBL, in addition to this we need the IR emissivity. We predict the IR emissivity using our best-fit ${{A}_{{\rm FUV}}}(z)$ and SFRD$(z)$, which is explained in the following section.

First, we compute the EBL at $z=0$ for our five different models, which are plotted in Figure 10. Apart from the wavelengths near the 2175 Å absorption, different EBLs are quite indistinguishable from each other for $\lambda \lt 0.8\;\mu {\rm m}$. As expected, we see a clear difference appearing at higher wavelengths between different models. Apart from the extinction curves, the difference is also because of the differences in the ${{A}_{{\rm FUV}}}$ values for the different models (see Figure 6). Therefore, even though the estimated EBL in the UV and optical parts are similar (because of the way we determine SFRD and ${{A}_{{\rm FUV}}}$), there are clear differences in the IR wavelengths where the dust re-emission is important.

Zoom In Zoom Out Reset image size

The shaded area in Figure 10 represents the range of the allowed EBL intensity determined by the local EBL observations. The data used in Figure 10 are taken from the compilation of Dwek & Krennrich (2013, see their Figure 7). The lower limits are determined by the intensity of the integrated galaxy light (IGL). The IGL is obtained by adding the light emitted by resolved galaxies in deep surveys. In principle, the IGL should converge to the total EBL at $z=0$. However, because of the problem in the convergence of number counts and sensitivity of the surveys to resolve fainter galaxies, the IGL gives a lower limit to the EBL. Here, we use the IGL measurements in the UV from GALEX, in the optical from HST, and in the IR from Spitzer, ISO, and Herschel (Totani et al. 2001; Fazio et al. 2004; Xu et al. 2005; Levenson et al. 2007; Béthermin et al. 2010; Berta et al. 2010). The upper limits on the local EBL come from the direct measurements of the EBL. An important uncertainty in the direct measurements is the removal of strong foreground zodiacal light caused by the interplanetary dust and the stellar emission from the MW. Therefore, these measurements provide strict upper limits on the local EBL. In Figure 10, we take the absolute measurements of EBL in the optical from Pioneer 10/11 and in the IR from COBE (Dwek & Arendt 1998; Fixsen et al. 1998; Finkbeiner et al. 2000; Dole et al. 2006; Levenson et al. 2007; Matsuoka et al. 2011; Matsuura et al. 2011).

Our EBL estimate for the "smc" model goes below the lower limits at $\lambda \gt 1\;\mu {\rm m}$, whereas for all other models, the estimated EBL is within the allowed range. If we strictly follow the shaded region, we can rule out the "smc" model and conclude that the average extinction curve for galaxies is inconsistent with the SMC type of dust extinction. However, the allowed range is too large to distinguish between other models. All our EBL models in the UV, optical, and NIR ($\lambda \lt 2\;\mu {\rm m}$) follow the lower limits of the EBL where we use the emissivity consistent with the observed multiwavelength galaxy luminosity functions. The "lmc2" model, which also provides ${{A}_{{\rm FUV}}}(z)$ and SFRD$(z)$ consistent with different independent measurements, produces the IR background ($1\lt \lambda \lt 100\;\mu {\rm m}$) consistent with the lower limits, whereas other models produce slightly higher background intensity but are well within the allowed range and closer to the lower limits. In the FIR regime ($\lambda \gt 100\;\mu {\rm m},$) the estimated background for the "lmc2" and "lmc" model goes through the observed points. The EBL for the "mw" and "cal" models is just consistent or slightly higher than the observed upper limits at $\lambda \gt 100\;\mu {\rm m}$. In summary, the available local EBL measurements in the NIR to FIR regime do not support the average dust extinction observed in the case of SMC. However, these measurements cannot discriminate between the EBL obtained with other extinction curves.

Recently, using the rocket-borne instrument Cosmic Infrared Background Experiment, Zemcov et al. (2014) measured the fluctuation amplitude of the IR background at $1.1$ and $3.6\;\mu {\rm m}$. One of the plausible explanations for the large fluctuation found at these wavelengths is that it arises from the intrahalo light (IHL) produced by the tidally striped old stars in the halos of the galaxy (Cooray et al. 2004; Thacker et al. 2014). Using these fluctuations measured over large scales, Zemcov et al. (2014) give the model-dependent total EBL contributed by IHL. In Figure 10, we show their computed values of the total EBL, which is a sum of their estimated IHL and the compiled measurements of IGL by Franceschini et al. (2008). As expected, because we do not include the additional contributions such as the IHL in our emissivities, we find that our estimated EBL for all models is lower than predicted by Zemcov et al. (2014). Except for the "smc" model, the EBL obtained for all other models at $\lambda \leqslant 2.4\;\mu {\rm m}$ are within 2$\sigma$ lower than the total EBL predicted by Zemcov et al. (2014). We find that only for the "smc" model is it more than 2.5$\sigma$ lower at all wavelengths. At $\lambda \geqslant 3.6\;\mu {\rm m}$ our "lmc," "mw," and "cal" models match with the EBL predicted by Zemcov et al. (2014) within 1$\sigma$. In light of these recent developments, it will be interesting to consider the IHL contribution to IR, the signal arising from the epoch of reionization (Cooray et al. 2004; Kashlinsky et al. 2004) and the FIR light from dusty galaxies (Amblard et al. 2010; Thacker et al. 2013; Viero et al. 2013), which we will attempt in the near future.

From the very good agreement with the local emissivity (see Figure 9) and with different independent measurements of the ${{A}_{{\rm FUV}}}(z)$ and SFRD$(z)$ (see Figures 6 and 7), for the EBL calculations we prefer our "lmc2" model over other models.

In Figure 11, we plot the EBL at redshifts 0, 0.5, 1, and 1.5 for our preferred "lmc2" model. We also show the range covered by the high and low "lmc2" model by a gray shaded region and the range covered by all five median models by a vertical striped region. The fact that we made sure the SFRD$(z)$ and ${{A}_{{\rm FUV}}}(z)$ determined for different extinction curves should give the same observed emissivity has resulted in the narrow spread in the striped region for $\lambda \lt 3\;\mu {\rm m}$, especially at high redshifts. For comparison, in Figure 11, we also show the previous estimates of the EBL reported in the literature (Franceschini et al. 2008; Finke et al. 2010; Kneiske & Dole 2010; Domínguez et al. 2011; Gilmore et al. 2012; Helgason & Kashlinsky 2012; Inoue et al. 2013; Scully et al. 2014, HM12). Below we compare our EBL predictions with the previous EBL estimates that use observational data such as galaxy number counts and galaxy luminosity functions to obtain the EBL directly.

Zoom In Zoom Out Reset image size

Helgason & Kashlinsky (2012) and Stecker et al. (2012) reconstructed the EBL using the multiwavelength and multiepoch luminosity functions. We also use a similar compilation of the luminosity functions but up to the K band. Therefore, our EBL matches very well with the EBL predictions of Helgason & Kashlinsky (2012) up to $\lambda \sim 3\;\mu {\rm m}$, as shown in Figure 11. For $\lambda \gt 3\;\mu {\rm m}$, Helgason & Kashlinsky (2012) predicts a higher EBL intensity than do we. The EBL model of Scully et al. (2014) extends the model of Stecker et al. (2012) up to 5 μm and provides 1$\sigma$ upper and lower limits on the EBL as well as ${{\tau }_{\gamma }}$. Our EBL predictions are consistent with their lower and upper limit EBL for $\lambda \gt 1\;\mu {\rm m}$. The EBL estimated by Domínguez et al. (2011) is based on the observed K-band luminosity function with the galaxy SEDs based on the multiwavelength observations from the SWIRE library. For $z\lt 1$ and $\lambda \lt 3\;\mu {\rm m}$, our EBL is consistent with the predictions of Domínguez et al. (2011), which show slightly lower EBL intensities in the UV and slightly higher EBL intensities in the NIR. At high $z$ this difference is more prominent. Franceschini et al. (2008) used different multiwavelength survey data, which include the luminosity functions, number counts, and the redshift distribution of different galaxy types, and the relevant data are fitted and interpolated to obtain the EBL. Our estimated EBL matches well with the EBL of Franceschini et al. (2008) up to the NIR wavelengths. At $z\gt 1$, in the UV regime their EBL gives a factor of $\sim 1.5$ lower intensity than our EBL. However, in the FIR wavelengths, our EBL gives around a factor of ∼2 smaller EBL intensity as compared to the EBL of Franceschini et al. (2008) and Domínguez et al. (2011). In summary, as expected, our EBL at $\lambda \lt 3\;\mu {\rm m}$ is consistent with the models that use direct observations to obtain the EBL but gives a lower EBL in the NIR to FIR regime. Below we mention some of the general trends in different EBL estimates that can be seen from Figure 11.

Given that there are more observations of the EBL as well as the luminosity functions of galaxies in the local universe, almost all the different independent models including our "lmc2" model converge very well, span narrower range at $z=0$ in the UV to NIR regime, and pass through observed lower limits of EBL. However, at the NIR and FIR wavelengths the spread between different estimates is relatively higher. All the EBL estimates differ from each other at high $z$ and the difference is as high as a factor of ∼4. Our local EBL passes very well through the lower limit EBL data compiled by Kneiske & Dole (2010). Most of the EBL models for $0.4\lt \lambda \lt 2\;\mu {\rm m}$ give similar intensity up to $z\lt 1.5$.

Our local EBL is very much similar to the estimates of HM12. However, at higher $z$, the UV background ($\lambda \lt 0.4\;\mu {\rm m}$) intensity of HM12 is higher than our EBL. This will have implications on the values of the escape fraction for H i-ionizing photons, which indirectly plays an important role in interpreting the He ii Lyman-$\alpha$ effective optical depth measurements near the epoch of helium reionization (see Khaire & Srianand 2013).

Having obtained the EBL at different $z$ from UV to FIR, we calculate its effect on the transmission of high-energy $\gamma$-rays through the IGM and compare it with the different observations in the following section.

The optical depth encountered by $\gamma$-rays while traveling from the emission redshifts ${{z}_{0}}$ to Earth (i.e., $z=0$) and observed at $\gamma$-ray energies from the GeV to TeV range is plotted in Figure 12 for different emission redshifts. We plot ${{\tau }_{\gamma }}$ calculated for our median "lmc2" model along with the ${{\tau }_{\gamma }}$ for its low and high counterparts in a dark gray shaded region. We also show the range of ${{\tau }_{\gamma }}$ covered by all five median models by a light gray shaded region. The extent of the light gray shaded region for different $z$ up to the $\gamma$-ray energy of 0.6 TeV shows that the ${{\tau }_{\gamma }}$ is indistinguishable for all five EBL models. This is because the maximum effective wavelength of the EBL photons that interact with $\gamma$-rays of energy less than 0.6 TeV is less than $3\;\mu {\rm m}$ where, by construct, our different EBL models predict similar intensities (see Figure 11). The spread of the shaded region for $\gamma$-ray energy >0.6 TeV points to the fact that our EBL intensity at FIR wavelengths is different for different models.

Zoom In Zoom Out Reset image size

For comparison, in Figure 12, we also plot the ${{\tau }_{\gamma }}$ obtained by other estimates of EBL given in the literature. To clearly show the differences between ${{\tau }_{\gamma }}$ calculated for other EBL estimates and for our "lmc2" EBL model, we plot the ratio of the former to the latter in the bottom panel of Figure 12. Differences in the ${{\tau }_{\gamma }}$ obtained for various EBL estimates can be directly understood from the differences in the EBLs, as shown in Figure 11. Here, we mention some of the general trends seen in the various ${{\tau }_{\gamma }}$ estimates and then compare our ${{\tau }_{\gamma }}$ with the models that used direct observations of galaxy properties to construct the EBL.

The difference between the ${{\tau }_{\gamma }}$ at energies from GeV to TeV for various EBL estimates increases with $z$. The difference is more in the TeV energy range where the EBL photons that effectively attenuate the $\gamma$-rays are from the FIR part of the EBL; the large scatter in different EBL estimates in the FIR wavelengths (see Figure 11) is responsible for that. However, for the observationally relevant $\gamma$-rays, which are the ones with 0.1 < τ γ < 2 in the corresponding energy range, the differences between ${{\tau }_{\gamma }}$ estimates are quite small. The difference in ${{\tau }_{\gamma }}$ for $\gamma$-ray energies from 0.1 to 3 TeV is small and within 30% for various estimates for emission redshifts $z\lt 1.5$. In general, our EBL gives lower ${{\tau }_{\gamma }}$ compared to most of the other estimates. The ${{\tau }_{\gamma }}$ obtained for the lower limit EBL of Scully et al. (2014) at $z=0.01$ is a factor  of $\sim 2$ higher at ${{E}_{\gamma }}\lt 0.2$ TeV. The ${{\tau }_{\gamma }}$ obtained for the EBL by Inoue et al. (2013) at $z\geqslant 1$ is within 10% of our estimate at $0.2\lt {{E}_{\gamma }}\lt 40$ TeV. However, note that the ${{\tau }_{\gamma }}$ at ${{E}_{\gamma }}\gt 10$ TeV in our model is quite small because we do not consider the CMBR in our EBL model (see Stecker et al. 2006). The ${{\tau }_{\gamma }}$ in the energy range $0.05\lt {{E}_{\gamma }}\lt 2$ TeV estimated by Helgason & Kashlinsky (2012) up to $25\;\mu {\rm m}$ is within 15% of our ${{\tau }_{\gamma }}$. The ${{\tau }_{\gamma }}$ calculated using the EBL generated by Franceschini et al. (2008) and Domínguez et al. (2011) for $0.1\lt {{E}_{\gamma }}\lt 4$ TeV is within 30% of our ${{\tau }_{\gamma }}$. However, at high energies, ${{\tau }_{\gamma }}$ differs more than this, which is evident from the differences in the FIR part of the EBL inbetween their models and our "lmc2" model (see Figure 11).

The $\gamma$-ray horizon is defined as the maximum redshift of the $\gamma$-ray source that can be detected through $\gamma$-rays of the observed energy, ${{E}_{\gamma }}$, with ${{\tau }_{\gamma }}\leqslant 1$. This is nothing but the $\gamma$-ray source redshift ${{z}_{0}}$ for $\gamma$-rays of the observed energy, ${{E}_{\gamma }}$, on Earth corresponding to ${{\tau }_{\gamma }}\left( {{E}_{\gamma }},{{z}_{0}} \right)=1$. In Figure 13, we plot the $\gamma$-ray horizon for different source redshifts for our different EBL models where the gray shaded area shows the range spanned by the high and low "lmc2" model.

Zoom In Zoom Out Reset image size

It can be directly seen from Figure 13 that our universe is transparent for the $\gamma$-rays with energies less than 10 GeV. The $\gamma$-rays with energy ${{E}_{\gamma }}\;\lt$ 30 GeV can be observed from sources at $z\sim 3$ without significant attenuation. Due to the differences in the IR part of the EBL, the $\gamma$-ray horizon redshift for our models differs in TeV energies. The well-measured ${{\tau }_{\gamma }}$ at TeV energies at low $z$ can, in principle, distinguish between the different EBL models presented here.

In Figure 13, we also plot the measurements of the $\gamma$-ray horizon by Domínguez et al. (2013) where they model the multiwavelength observations of blazars to determine the $\gamma$-ray horizon, which is EBL-independent. Apart from the "smc" model, all of our other models show good agreement with these $\gamma$-ray horizon measurements. This is evident from the ${{\chi }^{2}}$ statistics. The reduced ${{\chi }^{2}}$ values are 13.1, 1.7, 0.5, 0.4, and 0.4 for our "smc," "lmc2," "lmc," "mw," and "cal" models, respectively. This indicates that for the "smc" model, the predicted NIR to FIR part of the EBL intensity is less and inconsistent with the available $\gamma$-ray horizon measurements.

Note that the reduced ${{\chi }^{2}}$ quoted above includes only observational and systematic errors. We notice that even when we allow for the EBL obtained using high and low models for "smc," it does not give $\gamma$-ray horizon consistent with the measurements. If we include the average deviation in the $\gamma$-ray horizon in the case of the "lmc2" model, due to its higher and lower limits of the EBL obtained by using the high and low models as the errors in the prediction of the $\gamma$-ray horizon, the reduced ${{\chi }^{2}}$ for "lmc2" becomes 0.92. It is evident from Figure 13 that, even though we could rule out the "smc" model, purely based on the available $\gamma$-ray horizon measurements alone, we cannot distinguish between the other four models. Given the uncertainties involved in modeling the intrinsic SEDs of the $\gamma$-ray sources, the contribution of IHL to the galaxy emissivity in the NIR, and the small spread of the $\gamma$-ray horizon predicted in our remaining four models, it may be challenging to distinguish them based on more such measurements.

Recently, Ackermann et al. (2012) reported the average measurements of ${{\tau }_{\gamma }}$ over a large redshift range using a sample of 150 blazars observed with the Fermi-LAT. Because it is difficult to determine ${{\tau }_{\gamma }}$ in an individual blazar spectrum, they divided their blazar sample into three redshift bins, $z\lt 0.2,0.2\lt z\lt 0.5$ and $0.5\lt z\lt 1.6$. Then they stacked spectra in each redshift bin and determined the intrinsic SED of blazars by extrapolating the stacked spectrum from the lower energies where various EBL estimates suggests that ${{\tau }_{\gamma }}$ is negligible. Using these stacked spectra, Ackermann et al. (2012) reported the measurements of ${{\tau }_{\gamma }}$ for the observed $\gamma$-ray energies from 10 to 500 GeV in the redshift range of $0\lt z\leqslant 1.6$. Based on the good agreement between the EBL measured by Fermi and that expected from the lower limits determined from the IGL measurements, they argued that there is negligible room for residual emission from other sources. The $\gamma$-rays in this observed energy range from 10 to 500 GeV will be attenuated effectively by the EBL photons of the rest wavelength, $\lambda \lt 3.1\;\mu {\rm m}$ for $z\lt 1.6$, λ < 1.8 μ m, and $\lambda \lt 1.4\;\mu {\rm m}$ for $z\lt 0.2$, where the cross-section of pair production is maximum (at $\theta =\pi$). This is the wavelength range where, by construct, all five of our models give similar EBL.

For comparison with the measurements of Ackermann et al. (2012), we calculate the ${{\tau }_{\gamma }}$ in a way that mimics stacking of individual blazar spectra as done by them. We take the number distribution of blazars as a function of redshift for the 150 blazars used by Ackermann et al. (2012) and calculate ${{e}^{-{{\tau }_{\gamma }}}}$ for each blazar at a corresponding $z$ and at different energies, ${{E}_{\gamma }}$. Then we take the average of ${{e}^{-{{\tau }_{\gamma }}}}$ over the same number of blazars in the redshift bins. This average is the equivalent of obtaining ${{e}^{-{{\tau }_{\gamma }}}}$ by stacking the individual blazar spectrum in a redshift bin. In Figure 14, we plot our estimates of the $\gamma$-ray optical depth, along with the measurements of Ackermann et al. (2012), in terms of the transmission, ${{e}^{-{{\tau }_{\gamma }}}}$, for our "lmc2" EBL model. The range in ${{e}^{-{{\tau }_{\gamma }}}}$ covered by all five EBL models with their high and low counterparts is shown by the gray shaded region in Figure 14. For all of our EBL models, the ${{e}^{-{{\tau }_{\gamma }}}}$ fits well in low ($z\;\leqslant$ 0.2) and high ($0.5\lt z\leqslant 1.6$) redshift bins. However, in the intermediate redshifts, our estimated ${{\tau }_{\gamma }}$ is slightly higher than that reported in Ackermann et al. (2012). This excess optical depth is not statistically significant, given the large uncertainties in the measurements of ${{\tau }_{\gamma }}$. However, if these measurements are indeed very accurate, then to account for this we need the EBL intensity to be a factor of 2 lower than predicted by our models at $z\lt 0.5$ in the optical to NIR regime. This requires a factor of 2 reduction in ${{\rho }_{\nu }}$ at $\lambda \lt 1.8\;\mu {\rm m}$ for $z\lt 0.5$. One can, in principle, reduce the ${{\rho }_{\nu }}$ by increasing ${{L}_{{\rm min} }}$ (in Equation (5)). However, how much ${{\rho }_{\nu }}$ can be reduced depends upon the $\alpha$ and the luminosity of the faintest galaxy detected to determine the luminosity functions. It can be seen from Table 5 that the values of $\alpha$ at $z\lt 0.5$ are high, and therefore to reduce ${{\rho }_{\nu }}$ by a factor of 2 one needs to take ${{L}_{{\rm min} }}\gt 0.2{{L}^{*}}$. However, in this wavelength range (UV to NIR), given the fact that our EBL models are consistent with the observational lower limits on local EBL, there is not much room available to reduce the EBL intensity. This reiterates the findings of Ackermann et al. (2012) that the observed luminosity densities of galaxies are just sufficient to reproduce the ${{\tau }_{\gamma }}$ measurements.

Zoom In Zoom Out Reset image size

Table 5.  Observations of the Galaxy Luminosity Function

Reference	Band	z	$\alpha$	${{L}_{{\rm min} }}$	${{{\Delta }}_{1}}$	${{{\Delta }}_{2}}$
				(L$^{*}$)	%	%
Cucciati et al. (2012)	FUV	0.125	−1.05	0	1	3
Tresse et al. (2007)	FUV	0.14	−1.13	0	1	4
Cucciati et al. (2012)	FUV	0.3	−1.17	0	2	5
Schiminovich et al. (2005)	FUV	0.3	−1.19	0	2	6
Tresse et al. (2007)	FUV	0.3	−1.6	0	15	25
Cucciati et al. (2012)	FUV	0.5	−1.07	0	1	3
Schiminovich et al. (2005)	FUV	0.5	−1.55	0	12	22
Tresse et al. (2007)	FUV	0.51	−1.6	0	15	25
Tresse et al. (2007)	FUV	0.69	−1.6	0	15	25
Cucciati et al. (2012)	FUV	0.7	−0.9	0	0	1
Schiminovich et al. (2005)	FUV	0.7	−1.6	0	15	25
Cucciati et al. (2012)	FUV	0.9	−0.85	0	0	1
Tresse et al. (2007)	FUV	0.9	−1.6	0	15	25
Schiminovich et al. (2005)	FUV	1	−1.6	0	15	25
Tresse et al. (2007)	FUV	1.09	−1.6	0	15	25
Cucciati et al. (2012)	FUV	1.1	−0.91	0	0	2
Dahlen et al. (2007)	FUV	1.125	−1.48	0	9	17
Tresse et al. (2007)	FUV	1.29	−1.6	0	15	25
Cucciati et al. (2012)	FUV	1.45	−1.09	0	1	4
Tresse et al. (2007)	FUV	1.55	−1.6	0	15	25
Dahlen et al. (2007)	FUV	1.75	−1.48	0	9	17
Schiminovich et al. (2005)	FUV	2.0	−1.49	0	10	18
Cucciati et al. (2012)	FUV	2.1	−1.3	0	4	9
Dahlen et al. (2007)	FUV	2.22	−1.48	0	9	17
Reddy & Steidel (2009)a	FUV	2.3	−1.73	0.01	0	15
Schiminovich et al. (2005)	FUV	2.9	−1.47	0	9	16
Cucciati et al. (2012)	FUV	3	−1.5	0	10	18
Reddy & Steidel (2009)a	FUV	3.05	−1.73	0.01	0	15
Bouwens et al. (2011)b	FUV	3.8	−1.73	0.01	0	15
Cucciati et al. (2012)	FUV	4	−1.5	0	10	18
Bouwens et al. (2011)b	FUV	5	−1.66	0.01	0	13
Bouwens et al. (2011)c	FUV	5.9	−1.74	0.01	0	16
Bouwens et al. (2011)b	FUV	6.8	−2.01	0.01	0	27
Bouwens et al. (2011)d	FUV	8	−1.91	0.01	0	22
Wyder & Treyer (2005)	NUV	0.055	−1.16	0	2	5
Cucciati et al. (2012)	NUV	0.125	−1.08	0	1	4
Tresse et al. (2007)	NUV	0.14	−1.32	0	4	9
Cucciati et al. (2012)	NUV	0.3	−1.02	0	1	3
Tresse et al. (2007)	NUV	0.3	−1.32	0	4	9
Dahlen et al. (2007)	NUV	0.33	−1.39	0	6	12
Cucciati et al. (2012)	NUV	0.5	−1.08	0	1	4
Tresse et al. (2007)	NUV	0.51	−1.32	0	4	9
Dahlen et al. (2007)	NUV	0.545	−1.39	0	6	12
Tresse et al. (2007)	NUV	0.69	−1.32	0	4	9
Cucciati et al. (2012)	NUV	0.7	−0.95	0	0	2
Cucciati et al. (2012)	NUV	0.9	−0.81	0	0	1
Tresse et al. (2007)	NUV	0.9	−1.32	0	4	9
Tresse et al. (2007)	NUV	1.09	−1.32	0	4	9
Cucciati et al. (2012)	NUV	1.1	−0.88	0	0	1
Dahlen et al. (2007)	NUV	1.125	−1.39	0	6	12
Tresse et al. (2007)	NUV	1.29	−1.32	0	4	9
Cucciati et al. (2012)	NUV	1.45	−1.05	0	1	3
Tresse et al. (2007)	NUV	1.55	−1.32	0	4	9
Dahlen et al. (2007)	NUV	1.75	−1.39	0	6	12
Cucciati et al. (2012)	NUV	2.1	−1.16	0	2	5
Dahlen et al. (2007)	NUV	2.225	−1.39	0	6	12
Cucciati et al. (2012)	NUV	3	−1.15	0	2	5
Tresse et al. (2007)	U	0.14	−1.05	0	1	3
Dahlen et al. (2005)	U	0.3	−1.31	0	4	9
Tresse et al. (2007)	U	0.3	−1.17	0	2	5
Dahlen et al. (2005)	U	0.46	−1.2	0	2	6
Tresse et al. (2007)	U	0.51	−1.17	0	2	5
Dahlen et al. (2005)	U	0.625	−1.31	0	4	9
Tresse et al. (2007)	U	0.69	−1.27	0	3	8
Dahlen et al. (2005)	U	0.875	−1.31	0	4	9
Tresse et al. (2007)	U	0.9	−1.44	0	8	15
Dahlen et al. (2005)	U	0.97	−1.2	0	2	6
Tresse et al. (2007)	U	1.09	−1.51	0	10	19
Dahlen et al. (2007)	U	1.245	−1.2	0	2	6
Tresse et al. (2007)	U	1.29	−1.59	0	15	25
Dahlen et al. (2005)	U	1.48	−1.2	0	2	6
Tresse et al. (2007)	U	1.55	−1.68	0	21	32
Dahlen et al. (2005)	U	1.695	−1.2	0	2	6
Dahlen et al. (2005)	U	1.9	−1.2	0	2	6
Tresse et al. (2007)	B	0.14	−1.09	0	1	4
Dahlen et al. (2005)	B	0.3	−1.37	0	5	11
Tresse et al. (2007)	B	0.3	−1.15	0	2	5
Dahlen et al. (2005)	B	0.46	−1.28	0	3	8
Tresse et al. (2007)	B	0.51	−1.22	0	2	6
Dahlen et al. (2005)	B	0.625	−1.37	0	5	11
Tresse et al. (2007)	B	0.69	−1.12	0	1	4
Dahlen et al. (2005)	B	0.875	−1.37	0	5	11
Tresse et al. (2007)	B	0.9	−1.33	0	4	10
Dahlen et al. (2005)	B	0.97	−1.28	0	3	8
Tresse et al. (2007)	B	1.09	−1.40	0	6	13
Dahlen et al. (2005)	B	1.245	−1.28	0	3	8
Tresse et al. (2007)	B	1.29	−1.48	0	9	17
Dahlen et al. (2005)	B	1.48	−1.28	0	3	8
Tresse et al. (2007)	B	1.55	−1.57	0	14	23
Dahlen et al. (2005)	B	1.695	−1.28	0	3	8
Dahlen et al. (2005)	B	1.9	−1.28	0	3	8
Faber et al. (2007)	B	0.3	−1.3	0.01	0	5
Faber et al. (2007)	B	0.5	−1.3	0.01	0	5
Faber et al. (2007)	B	0.7	−1.3	0.01	0	5
Faber et al. (2007)	B	0.9	−1.3	0.01	0	5
Faber et al. (2007)	B	1.1	−1.3	0.01	0	5
Tresse et al. (2007)	V	0.14	−1.15	0	2	5
Tresse et al. (2007)	V	0.3	−1.21	0	2	6
Tresse et al. (2007)	V	0.51	−1.35	0	5	11
Tresse et al. (2007)	V	0.69	−1.35	0	5	11
Tresse et al. (2007)	V	0.9	−1.50	0	10	18
Tresse et al. (2007)	V	1.09	−1.57	0	14	23
Tresse et al. (2007)	V	1.29	−1.65	0	19	30
Tresse et al. (2007)	V	1.55	−1.74	0	27	38
Tresse et al. (2007)	R	0.14	−1.16	0	2	5
Tresse et al. (2007)	R	0.3	−1.27	0	3	8
Dahlen et al. (2005)	R	0.46	−1.30	0	4	9
Tresse et al. (2007)	R	0.51	−1.42	0	7	14
Tresse et al. (2007)	R	0.69	−1.41	0	7	13
Tresse et al. (2007)	R	0.9	−1.53	0	11	20
Dahlen et al. (2005)	R	0.97	−1.30	0	4	9
Tresse et al. (2007)	R	1.09	−1.60	0	15	25
Dahlen et al. (2005)	R	1.245	−1.30	0	4	9
Tresse et al. (2007)	R	1.29	−1.68	0	21	32
Dahlen et al. (2005)	R	1.48	−1.30	0	4	9
Tresse et al. (2007)	R	1.55	−1.77	0	29	42
Dahlen et al. (2005)	R	1.695	−1.30	0	4	9
Dahlen et al. (2005)	R	1.9	−1.30	0	4	9
Tresse et al. (2007)	I	0.14	−1.19	0	2	6
Tresse et al. (2007)	I	0.3	−1.32	0	4	9
Tresse et al. (2007)	I	0.51	−1.47	0	9	16
Tresse et al. (2007)	I	0.69	−1.41	0	7	13
Tresse et al. (2007)	I	0.9	−1.52	0	11	19
Tresse et al. (2007)	I	1.09	−1.59	0	15	25
Tresse et al. (2007)	I	1.29	−1.67	0	20	31
Tresse et al. (2007)	I	1.55	−1.76	0	28	41
Dahlen et al. (2005)	J	0.3	−1.48	0	9	17
Pozzetti et al. (2003)	J	0.5	−1.22	0.02	−2	1
Dahlen et al. (2005)	J	0.625	−1.48	0	9	17
Dahlen et al. (2005)	J	0.875	−1.48	0	9	17
Pozzetti et al. (2003)	J	1	−0.86	0.02	0	0
SM13d	J	1.75	−1.24	0.01	0	4
SM13	J	2.25	−1.12	0.01	0	2
SM13	J	2.75	−1.17	0.01	0	3
SM13	J	3.35	−0.92	0.01	0	1
SM13	H	1.75	−1.30	0.01	0	5
SM13	H	2.25	−1.23	0.01	0	4
SM13	H	2.75	−1.11	0.01	0	2
SM13	H	3.35	−1.30	0.01	0	5
Smith et al. (2009)	K	0.155	−0.81	0.01	0	1
Arnouts et al. (2007)	K	0.30	−1.1	0.01	0	2
Arnouts et al. (2007)	K	0.50	−1.1	0.01	0	2
Cirasuolo et al. (2007)	K	0.5	−0.99	0.01	0	1
Pozzetti et al. (2003)	K	0.5	−1.25	0.02	−2	2
Arnouts et al. (2007)	K	0.70	−1.1	0.01	0	2
Cirasuolo et al. (2007)	K	0.875	−1.00	0.01	0	1
Arnouts et al. (2007)	K	0.90	−1.1	0.01	0	2
Pozzetti et al. (2003)	K	1	−0.98	0.02	0	0
Cirasuolo et al. (2007)	K	0.875	−1.00	0.01	0	1
Arnouts et al. (2007)	K	1.10	−1.1	0.01	0	2
Cirasuolo et al. (2007)	K	1.125	−0.94	0.01	0	1
Arnouts et al. (2007)	K	1.35	−1.1	0.01	0	2
Cirasuolo et al. (2007)	K	1.375	−0.92	0.01	0	1
Cirasuolo et al. (2007)	K	1.625	−1	0.01	0	1
Arnouts et al. (2007)	K	1.75	−1.1	0.01	0	2
Cirasuolo et al. (2007)	K	2	−1	0.01	0	1

Notes. ${{{\Delta }}_{1}}=\left( {{a}_{0}}-{{a}_{1}} \right)/{{a}_{0}}$, and ${{{\Delta }}_{2}}=\left( {{a}_{0}}-{{a}_{2}} \right)/{{a}_{0}}$, where ${{a}_{0}}={\Gamma }\left( \alpha +2,{{L}_{{\rm min} }}/{{L}^{*}} \right)$, ${{a}_{1}}={\Gamma }(\alpha +2,0.01)$, and ${{a}_{2}}={\Gamma }(\alpha +2,0.03).$

^aThe corresponding wavelength is 1700 Å. ^bThe corresponding wavelength is 1600 Å. ^cThe corresponding wavelength is 1350 Å. ^dThe corresponding wavelength is 1750 Å.

Download table as:  ASCIITypeset images: 1 2 3 4

Note that to obtain intrinsic blazar spectra, the continuum extrapolation from the lower energy to higher energy is a practical solution, but may not be the valid one. Therefore, the minor discrepancy mentioned above does not conclude anything significantly. However, it will be more interesting for constraining various EBL models if the errors on ${{\tau }_{\gamma }}$ are reduced significantly.

The convolution integral (Equation (7)) gives the ${{\rho }_{\nu }}\left( {{z}_{0}} \right)$ by convolving the SFRD$(z)$ with the intensity output of an instantaneous burst of star formation, which has occurred at epoch $t$, corresponding to redshift $z$. The population synthesis models give the specific intensity, ${{l}_{\nu }}(\tau )$, where $\tau$ is an age of the stellar population that goes through the starburst at $z$ and is shining at ${{z}_{0}}$. In the calculations discussed above we have taken the maximum redshift, ${{z}_{{\rm max} }}=\infty$, which means, in principle, the ${{\rho }_{\nu }}\left( {{z}_{0}} \right)$ will have the contribution from very old stars up to the ages of ${{t}_{{\rm max} }}$, equal to the light travel time from the Big Bang to redshift $z={{z}_{0}}$, which is less than 13.6 Gyr, depending on ${{z}_{0}}$. Note that the actual contribution from very old stars (age >10 Gyr) that went through the starburst at time $t$ is negligible because the SFRD$(z)$ at $z$ corresponding to these earlier epochs $t$ is negligible. Here, we are investigating the validity of the ${{z}_{{\rm max} }}=\infty$ assumption and the effect of taking different values of ${{z}_{{\rm max} }}$ (or corresponding maximum age of galaxy, ${{t}_{{\rm max} }}$) on our derived quantities such as ${{A}_{{\rm FUV}}}$ and SFRD$(z)$, mainly at low redshifts where the age of the universe is large. If we take sufficiently small values of ${{t}_{{\rm max} }}$, a contribution from the old stellar population, which shines at the optical and NIR wavelengths, will be less. Therefore, galaxies will be bluer than one expects. It requires a large dust attenuation to make them red and match the ${{\rho }_{\nu }}$ measurements at the NIR wavelengths. With this small ${{t}_{{\rm max} }}$, if we follow the progressive fitting method described in Section 3.2, we obtain large ${{A}_{{\rm FUV}}}(z)$ and hence large SFRD$(z)$ at low redshifts. This is demonstrated in the case of the "cal" and "lmc2" model in Figure 15 where we show the SFRD$(z)$ and ${{A}_{{\rm FUV}}}(z)$ determined by stopping the convolution integral after a time, ${{t}_{{\rm max} }}={{t}_{0}}$, for different values of ${{t}_{0}}$, ranging from 1 to 10 Gyr. We take ${{t}_{{\rm max} }}$ equal to the age of the universe when the age is smaller than ${{t}_{0}}.$ We see similar trends for all five models, but for the sake of presentation we show results in Figure 15 only for the "cal" and "lmc2" models. It is clear from Figure 15 that the lower the value of ${{t}_{{\rm max} }}$, the higher the values will be of the inferred SFRD($z$ ) and ${{A}_{{\rm FUV}}}(z)$.

Zoom In Zoom Out Reset image size

In Figure 15, we also plot the independent measurements shown in Figures 6 and 7. These measurements of the ${{A}_{{\rm FUV}}}$ show that it increases from $z=0$ to a peak at $z\sim 1$ and then decreases at higher $z$ (Takeuchi et al. 2005; Cucciati et al. 2012; Burgarella et al. 2013). To obtain such a shape of ${{A}_{{\rm FUV}}}$, as shown in Figure 15, one needs ${{t}_{{\rm max} }}\gt 5$ Gyr. We find that the SFRD$(z)$ and ${{A}_{{\rm FUV}}}(z)$ are insensitive to ${{t}_{{\rm max} }}$ when it is $\geqslant 10$ Gyr. Therefore, to obtain the SFRD$(z)$ and ${{A}_{{\rm FUV}}}(z)$ consistent with the trend seen in different independent observations one needs to consider the stellar population ages $\geqslant$ 10 Gyr, which is consistent with ${{z}_{{\rm max} }}=\infty$ and the estimated age 11 Gyr of MW (Krauss & Chaboyer 2003).

Another source of possible uncertainty in determining the SFRD$(z)$ and ${{A}_{{\rm FUV}}}(z)$ can be related to the redshift evolution of metallicity, which we do not consider. We use constant $Z=0.008$ metallicity for all redshifts. HM12 and Madau & Dickinson (2014) use the metallicity evolution as $Z(z)={{Z}_{\odot }}{{10}^{-0.15z}}$ suggested by Kewley & Kobulnicky (2007), where ${{Z}_{\odot }}=0.020$. This evolution gives $Z=0.008$ at $z=2.6$. To see the effect of using different metallicity, we determine the SFRD$(z)$ and ${{A}_{{\rm FUV}}}(z)$ for metallicity $Z=0.020$ and $Z=0.004$. Our results are plotted in Figure 16 for the "cal" and "lmc2" models. We also show the range covered by the ${{A}_{{\rm FUV}}}$ and SFRD when obtained for the respective high and low models for our fiducial metallicity, $Z=0.008$ (the same as in Figures 6 and 7). Note that the low and high models use 1$\sigma$ low and high fits through the ${{\rho }_{{\rm FUV}}}$ measurements used to determine the ${{A}_{{\rm FUV}}}$ and SFRD. It is clear from Figure 16 that the higher (lower) metallicity gives higher (lower) ${{A}_{{\rm FUV}}}(z)$ and SFRD$(z).$ We see similar trends for all five models, but for the sake of presentation we show them only for the "cal" and "lmc2" models. The difference between the ${{A}_{{\rm FUV}}}(z)$ obtained for these metallicities (as high as a factor of 5 in $Z$) is well within the allowed range of the ${{A}_{{\rm FUV}}}$ and SFRD obtained using our fiducial $Z=0.008.$ This suggests that the scatter in the ${{A}_{{\rm FUV}}}(z)$ and SFRD$(z)$ arising due to a change in metallicity (as high as a factor of 5 ) is smaller than the scatter one obtains in the ${{A}_{{\rm FUV}}}(z)$ and SFRD$(z)$ due to scatter in the ${{\rho }_{{\rm FUV}}}$ measurements at a constant metallicity.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The analysis presented here shows that the effect of the metallicity evolution is subdominant as compared to those arising from uncertainties in the ${{\rho }_{{\rm FUV}}}$ measurements. Therefore, our assumption of constant metallicity is valid and compatible with the current ${{\rho }_{\nu }}$ measurements.

In this paper, we consider the Salpeter (1955) IMF in the population synthesis model with stellar mass range from 0.1 to $100{{M}_{\odot }}.$ Even though there are other IMFs such as  Kroupa & Weidner (2003), Chabrier (2003), and Baldry & Glazebrook (2003), because most of the star formation rates reported in the literature use Salpeter IMF, we also prefer it for our work. Note that the different IMFs can change the combination of ${{A}_{{\rm FUV}}}(z)$ and SFRD$(z)$, but the fact that we use the progressive fitting method to obtain these combinations will ensure that the emissivities and EBL will remain the same in the UV to NIR wavelengths (the wavelengths where we have multiwavelength, multiepoch luminosity functions). However, the different IMFs, and, hence, different ${{A}_{{\rm FUV}}}(z)$ and SFRD$(z)$ will give a different FIR emissivity and FIR part of the EBL (see, e.g., Primack et al. 2001).

The luminosity densities calculated from the observed luminosity function depend on the values of ${{L}_{{\rm min} }}$ and the faint end slope, $\alpha .$ For most of the ${{\rho }_{\nu }}$ used here, we use ${{L}_{{\rm min} }}=0.$ In Table 5 of the Appendix, we list the $\alpha$ and ${{L}_{{\rm min} }}$ values used to obtain ${{\rho }_{\nu }}.$ We also list the percentage decrease in the ${{\rho }_{\nu }}$ if we use ${{L}_{{\rm min} }}=0.01{{L}^{*}}$ and ${{L}_{{\rm min} }}=0.03{{L}^{*}}$, instead of the fiducial ${{L}_{{\rm min} }}$ (see the columns labeled as ${{{\Delta }}_{1}}$ and ${{{\Delta }}_{2}}$ in Table 5). The difference is large for small $\alpha$ (i.e., $\alpha \lt -1.3$). The value ${{L}_{{\rm min} }}=0.01{{L}^{*}}$ is used by HM12 in their UV background calculations. The value of ${{L}_{{\rm min} }}=0.03{{L}^{*}}$ is used by Madau & Dickinson (2014) to obtain the SFRD. It can be directly seen from Table 5 that the choice of ${{L}_{{\rm min} }}$ between 0 to 0.03 ${{L}^{*}}$ can change the ${{\rho }_{\nu }}$ up to 30%. This difference is large for the ${{\rho }_{\nu }}$ measurements of Tresse et al. (2007) at high $z$ and higher wavebands, where $\alpha$ is small. Because of the sensitivity limit of the survey, Tresse et al. (2007) could not determine $\alpha$ for $z\gt 1.2.$ Therefore, at high $z$, the $\alpha$ is extrapolated from the low redshift measurements. However, note that the errors estimated on the ${{\rho }_{\nu }}$ values by Tresse et al. (2007) include uncertainties arising from the different values of $\alpha$, which is larger than the difference due to the ${{L}_{{\rm min} }}$ values mentioned here.

By increasing the values of ${{L}_{{\rm min} }}$, we can obtain lower ${{\rho }_{\nu }}$, which will give rise to the EBL with lower intensity. Because our EBL generated using the ${{\rho }_{\nu }}$ with ${{L}_{{\rm min} }}=0$ passes through the lower limits on the local EBL obtained from the IGL measurements in the FUV to NIR bands (see Figure 10), we do not consider the higher values of ${{L}_{{\rm min} }}.$

In Figure 17, we show our estimated ρ_ν, along with the compiled measurements for the high z and all other wavebands.

Table 5 lists the faint end slopes of the luminosity functions with the rest waveband and redshift, along with the L_min values we used to determine ρ_ν.