# A Determination of the Intergalactic Redshift Dependent UV-Optical-NIR
Photon Density Using Deep Galaxy Survey Data and the Gamma-ray Opacity of the
Universe

Floyd W. Stecker  Astrophysics Science Division, NASA/Goddard Space Flight
Center  Greenbelt, MD 20771<EMAIL_ADDRESS>Department of Physics
and Astronomy, University of California, Los Angeles  Los Angeles,
CA90095-1547  Matthew A. Malkan  Department of Physics and Astronomy,
University of California, Los Angeles  Los Angeles, CA90095-1547;
<EMAIL_ADDRESS>Sean T. Scully  Department of Physics, James Madison
University  Harrisonburg, VA 22807<EMAIL_ADDRESS>
###### Abstract

We calculate the intensity and photon spectrum of the intergalactic background
light (IBL) as a function of redshift using an approach based on observational
data obtained in many different wavelength bands from local to deep galaxy
surveys. This allows us to obtain an empirical determination of the IBL and to
quantify its observationally based uncertainties. Using our results on the
IBL, we then place 68% confidence upper and lower limits on the opacity of the
universe to $\gamma$-rays, free of the theoretical assumptions that were
needed for past calculations. We compare our results with measurements of the
extragalactic background light and upper limits obtained from observations
made by the Fermi Gamma-ray Space Telescope.

diffuse radiation – galaxies:observations – gamma rays:theory

##  1 Introduction

###  1.1 Empirical Approach to Determining the Intergalactic Background
Radiation

The purpose of this paper is to present the results of a new, fully empirical
approach to calculating the intergalactic background light (IBL) as well as
the $\gamma$-ray opacity of the Universe. This methodology, hitherto
unavailable, is now enabled by very recent data from deep galaxy surveys
spanning the electromagnetic spectrum from millimeter to UV wavelengths and
using galaxy luminosity functions for redshifts $0\leq z\leq 8$ in the UV and
for redshifts up to 2 or 3 in other wavelength ranges. We stress that this
approach is both capable of delineating empirically based uncertainties on the
determination of the IBL, and the $\gamma$-ray opacity of the Universe.

In this paper (Paper I) we specifically consider the frequency range from the
far ultraviolet (FUV) to the near infrared $I$ band (NIR), as this range is of
particular relevance to the $\gamma$-ray opacity studies in the $\sim$0.1-200
GeV energy range being made by the Fermi $\gamma$-ray space telescope. A
follow-up paper (Paper II) will address the frequency range from the NIR to
the far-IR (FIR). That range has particular relevance for opacity studies by
ground-based air Čerenkov telescopes.

Previous calculations the IBL at different redshifts have been based on
various theoretical models and assumptions. These include backward evolution
models (Malkan & Stecker 1998, 2001; Stecker, Malkan & Scully 2006;
Franceschini et al. 2008), semi-analytical forward evolution models (e.g.,
Gilmore et al. 2009; Somerville et al. 2011) and other models based on the
evolution of galaxy parameters such as star formation rate and stellar
population synthesis models (Salamon & Stecker 1998 (hereafter SS98); Kneiske
et al. 2004). Kneiske & Dole (2010) have recently used a forward evolution
model to derive lower limits on the EBL. Finke, Razzaque & Dermer (2010)
employed a triple blackbody approximation to extimate the EBL. Domínguez et
al. (2011) used an approach based on the redshift evolution of the $K$-band
galaxy luminosity functions (LFs) derived by Cirasuolo et al. (2010), together
with model templates based on Spitzer-based $0.2\leq z\leq 1$ infrared galaxy
SEDs and AEGIS data. To obtain $K$-band LFs for $1<z<4$, Cirasuolo et al.
(2010) used 8 $\mu$m Spitzer/IRAC (Infrared Array Camera) channels combined
with population synthesis models of Bruzual & Charlot (2003), including a
correction for dust obscuration. Most recently, a semi-analytic model of the
EBL has been published by Gilmore et el. (2012). The earlier exploration of
the EBL using direct measurements, galaxy counts, and indirect constraints was
reviewed some time ago by Hauser & Dwek (2001).

We note that previous studies had to adopt at least some assumptions about how
galaxy LFs evolves with cosmic time, starting either at the present (well-
measured epoch) and going back in time, or starting with the simulations of
the galaxy formation epoch using semi-analytic models (see above) or modeled
galaxy SEDs. However, the latest observations have become sufficiently
extensive and accurate to allow direct integration of observational data on
galaxy LFs from the deep galaxy surveys at many wavelengths, where we can
interpolate between observationally determined LFs at many wavelengths from
the far UV to near infrared and the redshift range extending in the UV from
$z=0$ to $z\geq 8$. Thus, the first goal of our paper is to determine the IBL
based on empirical data from deep survey galaxy observations. This avoids the
complications entailed by theoretical calculations that have need of making
various assumptions for stellar population synthesis models, stellar initial
mass functions, unknown amounts of dust extinction, and poorly known stellar
metallicity-age modeling for different evolving galaxy types (e.g., Wilkins et
al. 2012). This is because the observational data are the direct result of all
of the physical processes involved in producing galactic emission. Thus our
treatment only involves uncertainties inherent in the analyses discussed in
the observational survey papers that we used.

###  1.2 Gamma Ray Opacity and the IBL

The second goal of our paper is to use our results on the IBL to determine the
$\gamma$-ray opacity of the universe as a function of energy and redshift. It
was first suggested by Stecker, De Jager & Salamon (1992) that $\gamma$-ray
observations from high redshift sources such as blazars (and later
$\gamma$-ray bursts) could be used to probe the IBL. Such studies make use of
the opacity caused by the annihilation of $\gamma$-rays owing to interactions
with low energy photons that produce $e^{+}e^{-}$ pairs. The Fermi Gamma-Ray
Space Telescope (Fermi) is now being used to probe the high redshift IBL at
optical and UV wavelengths by constraining the opacity of the universe to
multi-GeV $\gamma$-rays (Abdo et al. 2010). This is accomplished by measuring
the energy of the highest energy photons observed by Fermi that have been
emitted by GRBs and blazars at known redshifts.

Observations of TeV $\gamma$-ray emitting blazars utilizing modern air
Čerenkov telescope arrays also probe, or at least constrain, the nearby
(redshift $z\sim 0-0.5$) intergalactic infrared background radiation. Attempts
to constrain the IBL have been made by various authors (Stecker & de Jager
1993; Aharonian et al. 2006 (but see Stecker, Baring & Summerlin 2009); Mazin
& Raue 2007; Georganopoulos, Fincke & Reyes 2010; Abdo et al. 2010; Orr,
Krennrich & Dwek 2011, but see Stecker, Baring & Summerlin 2009).

Our methodology will also be used to define secure upper and lower limits on
the opacity of the universe to high energy $\gamma$-rays based on the
observational uncertainties in the deep survey data. We then compare the
opacity range defined by these limits with the upper limits derived using the
Fermi observations of multi-GeV $\gamma$-rays from high redshift sources Abdo,
et al (2010).

##  2 Intergalactic Photon Energy Densities and Emissivities from Galaxies

The co-moving radiation energy density $u_{\nu}(z)$ is derived from the co-
moving specific emissivity ${\cal E}_{\nu}(z)$, which, in turn is derived from
the galaxy luminosity function (LF). The galaxy luminosity function,
$\Phi_{\nu}(L)$, is defined as the distribution function of galaxy
luminosities at a specific frequency or wavelength. The specific emissivity at
frequency $\nu$ and redshift $z$ (also referred to in the literature as the
luminosity density, $\rho_{L_{\nu}}$) , is the integral over the luminosity
function

${\cal E}_{\nu}(z)=\int_{L_{min}}^{L_{max}}dL_{\nu}\,L_{\nu}\Phi(L_{\nu};z)$
(1)

There are many references in the literature where the LF is given and fit to
Schechter parameters, but where $\rho_{L_{\nu}}$ is not given. In those cases,
we could not determine the covariance of the errors in the Schechter
parameters used to determine the dominant statistical errors in their
analyses. Thus, we could not ourselves accurately determine the error on the
emissivity from equation (1). We therefore chose to use only the papers that
gave values for $\rho_{L_{\nu}}(z)={\cal E}_{\nu}(z)$ with errors. We did not
consider cosmic variance, but this uncertainly should be minimized since we
used data from many surveys.

In compiling the observational data on ${\cal E}_{\nu}(z)$, we scaled all of
the results to a value of h = 0.7. Thus results using h = 0.5 were scaled by a
factor of (7/5)111Using the most recent and accurate value of 0.74 (Riess et
al. 2011) would increase all of our results by $\sim 6\%$.

The co-moving radiation energy density $u_{\nu}(z)$ is the time integral of
the co-moving specific emissivity ${\cal E}_{\nu}(z)$,

$u_{\nu}(z)=\int_{z}^{z_{\rm max}}dz^{\prime}\,{\cal
E}_{\nu^{\prime}}(z^{\prime})\frac{dt}{dz}(z^{\prime})e^{-\tau_{\rm
eff}(\nu,z,z^{\prime})},$ (2)

where $\nu^{\prime}=\nu(1+z^{\prime})/(1+z)$ and $z_{\rm max}$ is the redshift
corresponding to initial galaxy formation (Salamon & Stecker 1998, hereafter
SS98), and

$\frac{dt}{dz}{(z)}={[H_{0}(1+z)\sqrt{\Omega_{\Lambda}+\Omega_{m}(1+z)^{3}}}]^{-1},$
(3)

with $\Omega_{\Lambda}=0.72$ and $\Omega_{m}=0.28$.

The opacity factor for frequencies below the Lyman limit is dominated by dust
extinction. In the model of SS98, which relied on the population synthesis
studies of Bruzual & Charlot (1993), dust absorption was not included. Our
earlier paper (Stecker, Malkan & Scully 2006) used a rough approximation of
the results obtained by Salamon & Stecker (1998) (SS98) and therefore, also
did not take dust absorption into account. However, since we are here using
actual observations of galaxies rather than models, dust absorption is
implicitly included. The remaining opacity $\tau_{\nu}$ refers to the
extinction of ionizing photons with frequencies above the rest frame Lyman
limit of $\nu_{LyL}\equiv 3.29\times 10^{15}$ Hz by interstellar and
intergalactic hydrogen and helium. It has been shown that this opacity is very
high, corresponding to the expectation of very small fraction of ionizing
radiation in intergalactic space compared with radiation below the Lyman limit
(Lytherer et al.1995; SS98). In fact, the Lyman limit cutoff is used as a
tool; when galaxies disappear when using a filter at a given waveband (e.g.,
”$U$-dropouts”, ”$V$-dropouts”) it is an indication of the redshift of the
Lyman limit. We thus replace equation (2) with the following expression

$u_{\nu}(z)=\int_{z}^{z_{\rm max}}dz^{\prime}\,{\cal
E}_{\nu^{\prime}}(z^{\prime})\frac{dt}{dz}(z^{\prime}){{\cal
H}(\nu(z^{\prime})-\nu^{\prime}_{LyL})},$ (4)

where ${\cal H}(x)$ is the Heavyside step function.

###  2.1 Empirical Specific Emissivities

####  2.1.1 Luminosity Densities

We have used the results of many galaxy surveys to compile a set of luminosity
densities, $\rho_{L_{\nu}}(z)={\cal E}_{\nu}(z)$ (LDs), at all observed
redshifts, and at rest-frame wavelengths from the far-ultraviolet, FUV = 150
nm to the $I$ band, $I$ = 800 nm. Figure 1 shows the redshift evolution of the
luminosity ${\cal E}_{\nu}(z)$ for the various wavebands based on those
published in the literature.222Table 1 references used to construct Figure 1
are as follows: Bouwens et al. (2007)(BO07), Bouwens et al. (2010)(BO10),
Budavári et al.(2005)(BU05), Burgarella et al. (2007)(BU07), Chen et al.(2003)
(CH03), Cucciati et al. (2012)(CU12), Dahlen et al. (2007)(DA07), Faber et al.
(2007)(FA07) and references therein, Iwata et al. (2007)(IW07), Ly et al.
(2009)(LY09), Reddy & Steidel (2009)(RE09), Marchesini et al. (2007)(MA07),
Marchesini & Van Dokkum 2007 (MAV07), Marchesini et al. (2012)(MA12), Oesch et
al. (2010)(OE10), Paltani et al. (2007)(PA07), Reddy et al. (2008)(RE08),
Sawicki & Thompson (2006)(SA06), Schiminovich et al. (2005)(SC05), Steidel et
al. (1999)(ST99), Tresse et al. (2007) (TR07), Wolf et al.(2003) (WO03), Wyder
et al. (2005)(WY05), Yoshida et al. (2006)(YO06). The lower right panel shows
all of the observational determinations of galaxy LDs from the references in
footnote 2. The specific waveband and mean redshift identifications for these
data are listed in Table 1 using the key abbreviations indicated in footnote
2. This table reflects the fact that direct determinations of galaxy LDs are
only available out to an observed wavelength of about 2.2 $\mu$m (rest
wavelength $2.2/(1+z)~{}\mu$m). This is because any attempt to survey large
areas of the sky with ground-based telescopes in wavebands longer than 2$\mu$m
is prevented by the sudden increases in background noise.333This 2$\mu$m
barrier is only circumvented by using space-based mid-infrared (3 to 8$\mu$m)
telescopes such as AKARI (with its Infrared Camera, IRC), and Spitzer (with
its Infrared Array Camera, IRAC). These telescopes have only conducted multi-
band imaging and redshift surveys with the necessary sensitivity to measure
the high-redshift ($z\geq 2$) galaxy population in a few, relatively small
deep fields.

Thus, at redshifts above 1.6, the longest rest-wavelengths under consideration
no longer have well measured LDs. At these longer wavelengths, we are obliged
to fall back on a secondary method for estimating galaxy luminosities: we use
the closest available LDs, and extrapolate them using the average observed
color of galaxies from measurements at that redshift. This ’minimal
extrapolation’ should be reliable because the average galaxy colors,
especially at long wavelengths, change only gradually with redshift. For
example, the galaxies that are included in the rest-frame $R$ band LD at
$z=2.2$ by Marchesini et al. are very similar to those of the galaxies that
would have been included in an $I$-band LD at that redshift. Since we are only
extrapolating by a small step in wavelength ($\Delta\lambda/\lambda\sim
0.15$), it is quite reasonable to shift the $R$-band LD using the average
$R-I$ colors observed at that redshift. The incremental color shifts we apply
become large only at $z\geq 4$, where, as we show in Section 4, the overall
contributions to the IBL $\gamma$-ray opacity are not very substantial. Our
color relations, which are also used to interpolate between the closely spaced
wavebands, are given the next subsection. They are given as a function of
redshift, $z$, since galaxies tend to be bluer on average at higher redshifts.

####  2.1.2 Average Colors

It is hardly surprising that there are often large apparent jumps, or changes,
in the shape and the normalization of the LDs going from one waveband to an
immediately adjacent one. We therefore applied an independent test of the
consistency of these LDs, by comparing the integrated ratios of LDs at
adjacent wavebands to the published average colors measured by observers. This
test has the great advantage of not requiring accurate estimates of volume
incompleteness or even very accurate redshifts. Broadband colors (i.e., local
continuum slopes) are easier to measure than LDs. The main problem is that all
galaxy samples at all redshifts show a wide observed range of broadband
colors. The typical $1\sigma$ scatter we found in published color
distributions was $\pm~{}0.5$ mag. A few rest-frame colors that are very
sensitive to stellar population, such as $U-B$, often show even larger
variation.

In order to determine the redshift evolution of the LD in each of the bands
out to a redshift of $\sim$ 8, we utilized color relations to transform data
from other bands. We have chosen to include all data possible in excess of
$z=1.5$ to fill in the gaps for various wavebands mostly at higher
redshifts.444The most comprehensive observations of galaxies in the best
observed Deep Fields include extremely sensitive Spitzer/IRAC photometry. The
IRAC data are most complete in its Band 1 (3.6 $\mu$m observed) wavelength,
and gradually become less sensitive out to the reddest IRAC band at 8 $\mu$m
observed wavelength corresponding to a rest wavelength of $8/(1+z)~{}\mu$m..
This also provides both an overlap to existing data and multiple sources of
data as a check for consistency of our color relations.

Published estimates of average colors from galaxy surveys at various wavebands
and redshifts tend to be bluer at shorter wavelengths, and redder at longer
wavelengths. This is due to the composite nature of stellar populations in
galaxies, with hot young stars making a stronger contribution in the UV
portion of the spectrum while red giants dominate the long wavelengths. Thus,
the galaxies that are included in a UV LF and not all the same galaxies as
those included in an LF in the $R$ band.

There is a clear trend with redshift over all wavelengths, which is well
known. Redder galaxies (e.g., local E and S0 galaxies) are more and more
outnumbered by blue, actively star-forming galaxies, at higher redshifts. The
average characteristic age of stellar populations decreases with redshift. Our
color relations agree with this trend. At the highest redshifts most known
galaxies are dominated by young starburst populations of O and B stars. This
tends to produce very blue overall spectral energy distribution without very
much sensitivity to the exact details of the star formation. These factors are
automatically taken into account when one uses the actual observational data
on the LDs at various wavelengths and redshifts.

Defining the average wavelengths of the various bands in $nm$ as follows:

FUV = 150, NUV = 280, $U$ = 365, $B$ = 445, $V$ = 551, $R$ = 658, $I$ = 806 nm

We then use the commonly measured astronomical parameter $\beta$, which is
defined by the relation between the differential flux and wavelength of a
galaxy, $f_{\lambda}\propto\lambda^{\beta}$. We have adopted the following
relations (colors) for $\beta_{\Delta\lambda}(z)$:

$\beta(FUV-NUV)=-1.0-1.25log(1+z),\ log(1+z)\leq 0.8$

derived from Bouwens, et al. (2009); Budavári et al.(2005); Castellano et al.
(2012); Cucciati, et al. (2012); Dunlop et al. (2012); Willott, et al. (2012);
Wyder et al.(2005),

$\beta(B-V)=+0.3-1.6log(1+z),\ log(1+z)\leq 0.6$

derived from Arnouts et al.(2007); Brammer (2011),

$\beta(NUV-U)=+0.5-1.2log(1+z),\ log(1+z)\leq 0.6$

derived from Tresse et al. (2007),

$\beta(NUV-R)=+2.5-6.0log(1+z),\ log(1+z)\leq 0.6$

$\beta(U-V)=+1.3-3.0log(1+z),\ log(1+z)\leq 0.6$

derived from Arnouts, et al. (2007); Brammer (2011): Ly et al. (2009),

$\beta(U-B)=+3.0-5.0log(1+z),\ log(1+z)\leq 0.6$

derived from Marchesini et al. (2007); González et al. (2011),

For the FUV-NUV relation we set $\beta[log(1+z)>0.8]=\beta[log(0.8)].$ For all
of the other relations we set $\beta[log(1+z)>0.6]=\beta[log(0.6)].$

We used the above redshift-dependent relations where appropriate in our
analysis. We stress that in the redshift ranges where they overlap, the
colored (observational) data points shown for the various wavelength bands in
Figure 1 agree quite well, within the uncertainties, with the black data
points that were extrapolated from the shorter wavelength bands using our
color relations. Also, where there is no overlap at the higher redshifts, the
uncertainty bands in photon density (see next section) show no
discontinuities.

###  2.2 Photon Density Calculations

The observationally determined LDs, combined with the color relations, extend
our coverage of galaxy photon production from the FUV to the NIR in the galaxy
rest frame. We have at least one or two determinations at each wavelength
across the most crucial redshift range $0\leq z\leq 2.5$. However, to
calculate the opacity for photons at energies higher than $\sim 250/(1+z)$ GeV
(see next section), requires the determination of galaxy LDs at longer rest
wavelengths and higher redshifts. These regimes are less well constrained by
observations, since they require measurement of very faint galaxies at long
wavelengths (mid-IR observed frame.) We will address this topic further in
Paper II. We have assumed a constant color at high redshift at the longer
wavelengths as stated above. However, we stress that our final results are not
very sensitive to errors in our average color relations because the
interpolations that we make cover very small fractional wavelength intervals,
$\Delta\lambda(z)$. We have directly tested this by numerical trial.

The second goal of our paper is to place upper and lower limits (within a 68%
confidence band) on the opacity of the universe to $\gamma$-rays . These
limits are a direct result of the 68% confidence band upper and lower limits
of the IBL determined from the observational data on $\rho_{L_{\nu}}$ . In
order to determine these limits, we make no assumptions about the luminosity
density evolution. We derive a luminosity confidence band in each waveband by
using a robust rational fitting function characterized by

$\rho_{L_{\nu}}={\cal E}_{\nu}(z)={{ax+b}\over{cx^{2}+dx+e}}$ (5)

where $x=\log(1+z)$ and $a$,$b$,$c$,$d$,and $e$ are free parameters.

The 68% confidence band is then computed from Monte Carlo simulation by
finding 100,000 realizations of the data and then fitting the rational
function. In order to best represent the tolerated confidence band,
particularly at the highest redshifts, we have chosen to equally weight all
FUV points in excess of a redshift of 2. Our goal is not to find the best fit
to the data but rather the limits tolerated by the current observational data.
In order to perform the Monte Carlo of the fitting function, a likelihood is
determined at each redshift containing data. The shape of the function is
taken to be Gaussian (or the sum of Gaussians where multiple points exist) for
symmetric errors quoted in the literature. Where symmetric errors are not
quoted it is impossible to know what the actual shape of the likelihood
functions is. We have chosen to utilize a skew normal distribution to model
asymmetric errors. This assumption has very little impact on the determination
of the confidence bands. The resulting bands are shown along with the
luminosity density data in Figure 1.

With the confidence bands established, we take the upper and lower limits of
the bands to be our high and low IBL respectively. We then interpolate each of
these cases separately between the various wavebands to find the upper and
lower limit rest frame luminosity densities. The calculation is extended to
the Lyman limit using the slope derived from our color relationship between
the near and far UV bands.

The specific emissivity is then the derived high and low IBL luminosity
densities ${\cal E}_{\nu}(z)=\rho_{L_{\nu}}(z)$. The co-moving radiation
energy density is determined from equation 4. Figure 2 shows the resulting
photon density determined by dividing the energy density by the energy in each
frequency for high and low IBL. This result is used as input for the
determination of the optical depth of the universe to $\gamma$-rays .

The photon densities

$\epsilon n(\epsilon,z)=u(\epsilon,z)/\epsilon\ \ ,$ (6)

with $\epsilon=h\nu$, as calculated using equation (2), are shown in Figure 2.

##  3 Comparison of z = 0 IBL with Data and Constraints

As a byproduct of our determination of the IBL as a function of redshift using
LDs from galaxy surveys, we have also determined the local ($z=0$) IBL, also
known as the extragalactic background light (EBL). Determining the EBL
directly has been the object of intense observational effort, although the
various estimates and limits in the published literature are far from
consistent with each other. Nonetheless, since these observations provide a
potential consistency check on our calculations, we consider them here.

Using equation (2), together with our empirically based determinations given
the confidence band derived for our specific emissivities, ${\cal
E}_{\nu}(z)$, we have evaluated the EBL within the 68% confidence band upper
and lower limits within the wavelength range of our calculations. This band is
indicated by the gray zone in Figure 3. We also show recent measurements using
the Hubble Wide-field Planetary Camera 2 (Bernstein 2007), the dark field from
Pioneer 10/11 (Matsuoka et al. 2011) and the preliminary analysis of Mattila
et al. (2011) using differential measurements using the ESO VLT (very large
telescope array). Figure 3 also shows the various lower limits from galaxy
counts obtained by Gardner et al. (2000) from the ST Imaging Spectrograph
data, by Madau & Pozzetti (2000) using Hubble Deep Field South data, and by Xu
et al. (2005) from GALEX (Galaxy Evolution Explorer) data, all indicated by
upward-pointing arrows.

##  4 The Optical Depth from Interactions with Intergalactic Low Energy
Photons

The cross section for photon-photon scattering to electron-positron pairs can
be calculated using quantum electrodynamics (Breit & Wheeler 1934). The
threshold for this interaction is determined from the frame invariance of the
square of the four-momentum vector that reduces to the square of the threshold
energy, $s$, required to produce twice the electron rest mass in the c.m.s.:

$s=2\epsilon E_{\gamma}(1-\cos\theta)=4m_{e}^{2}$ (7)

This invariance is known to hold to within one part in $10^{15}$ (Stecker &
Glashow 2001; Jacobson, Liberati, Mattingly & Stecker 2004).

With the co-moving energy density $u_{\nu}(z)$ evaluated, the optical depth
for $\gamma$-rays owing to electron-positron pair production interactions with
photons of the stellar radiation background can be determined from the
expression (Stecker, De Jager, & Salamon 1992)

$\tau(E_{0},z_{e})=c\int_{0}^{z_{e}}dz\,\frac{dt}{dz}\int_{0}^{2}dx\,\frac{x}{2}\int_{0}^{\infty}d\nu\,(1+z)^{3}\left[\frac{u_{\nu}(z)}{h\nu}\right]\sigma_{\gamma\gamma}[s=2E_{0}h\nu
x(1+z)],$ (8)

In equations (7) and (8), $E_{0}$ is the observed $\gamma$-ray energy at
redshift zero, $\nu$ is the frequency at redshift $z$, $z_{e}$ is the redshift
of the $\gamma$-ray source at emission, $x=(1-\cos\theta)$,
$\theta$ being the angle between the $\gamma$-ray and the soft background
photon, $h$ is Planck’s constant, and the pair production cross section
$\sigma_{\gamma\gamma}$ is zero for center-of-mass energy
$\sqrt{s}<2m_{e}c^{2}$, $m_{e}$ being the electron mass. Above this threshold,
the pair production cross section is given by

$\sigma_{\gamma\gamma}(s)=\frac{3}{16}\sigma_{\rm
T}(1-\beta^{2})\left[2\beta(\beta^{2}-2)+(3-\beta^{4})\ln\left(\frac{1+\beta}{1-\beta}\right)\right],$
(9)

where $\sigma_{T}$ is the Thompson scattering cross section and
$\beta=(1-4m_{e}^{2}c^{4}/s)^{1/2}$ (Jauch & Rohrlich 1955).

It follows from equation (7) that the pair-production cross section energy has
a threshold at $\lambda=4.75\ \mu{\rm m}\cdot E_{\gamma}({\rm TeV})$. Since
the maximum $\lambda$ that we consider here is in the rest frame I band at 800
nm at redshift $z$, and we observe $E_{\gamma}$ at redshift 0, so that its
energy at interaction in the rest frame is $(1+z)E_{\gamma}$, we then get a
conservative upper limit on $E_{\gamma}$ of $\sim 200(1+z)^{-1}$ GeV as the
maximum $\gamma$-ray energy affected by the photon range considered here.
Allowing for a small error, our opacities are good to $\sim 250(1+z)^{-1}$
GeV. The 68% opacity ranges for $z=0.1,0.5,1,3~{}$and $5$, calculated using
equation (8) are plotted in Figure 4.

The widths of the grey uncertainty ranges in the LDs shown in Figure 1
increase towards higher redshifts, especially at the longest rest wavelengths.
This reflects the decreasing amount of long-wavelength data and the
corresponding increase in uncertainties about the galaxies in those regimes.
However, these uncertainties do not greatly influence the opacity
calculations. Because of the short time interval of the emission from galaxies
at high redshifts their photons do not contribute greatly to the opacity at
lower redshifts. Indeed, Figure 4 shows that the opacities determined for
redshifts of 3 and 5 overlap within the uncertainties.

##  5 Results and Implications

We have determined the IBL using local and deep galaxy survey data, together
with observationally produced uncertainties, for wavelengths from 150 nm to
800 nm and redshifts out to $z>5$. We have presented our results in terms of
68% confidence band upper and lower limits. As expected, our $z=0$ (EBL) 68%
lower limits are higher than those obtained by galaxy counts alone, since the
EBL from galaxies is not completely resolved. Our results are also above the
theoretical lower limits given recently by Kneiske and Dole (2010). In Figure
3, we compare our $z=0$ result with both published and preliminary
measurements and limits.

Figure 5 shows our 68% confidence band for $\tau=1$ on an energy-redshift plot
compared with the Fermi data on the highest energy photons from extragalactic
sources at various redshifts as given by Abdo et al. (2010). It can be seen
that none of the photons from these sources would be expected to be
significantly annihilated by pair production interactions with the IBL. This
point is brought out further in Figure 6. This figure compares the 68%
confidence band of our opacity results with the 95% confidence upper limits on
the opacity derived for specific blazars by Abdo et al. (2010).

For purposes of discussion, we mention some points of comparison with previous
work. Our EBL results for $z=0$, while lower than the fast evolution model of
our previous work, are generally higher than those modeled more recently. As
an example, at a wavelength of 200 nm in the FUV range our uncertainty range
is a factor of 1.8 - 4.2 higher than the recent fiducial semi-analytic model
of Gilmore et al. (2012) and similarly higher than the previous model result
of Dominguez et al. (2011). Our opacity results at $z\simeq 1$ are comparable
to, or lower than, the models of Kneiske et al. (2004). They are also
consistent with the results of the non-metallicity corrected model of SS98.
However, they are higher than the models of Franceschini et al. (2008),
Gilmore et al. (2009), and Finke et al. (2010), as indicated by comparing
Figure 3 of Abdo et al. (2010) with our Figure 5. We stress that these
comparisons are for illustrative purposes only. Because our new methodology is
based on the direct use of luminosity densities derived directly from
observations, we take the position that they stand by themselves and should be
compared primarily with the observational data as shown in our Figures 3, 5
and 6. In that regard, we find full consistency within our observationally
determined uncertainties.555While we were preparing our revised manuscript for
publication a similar empirically based calculation by Helgason & Kashlinsky
(2012) appeared on the arXiv. These authors calculated the EBL and
$\gamma$-ray opacity based on galaxy luminosity functions compiled by
Helgason, Ricotti & Kashlinsky (2012) extrapolated to $z\geq 2$ using an
exponential cutoff in $z$. Their opacity results are generally consistent with
the results presented here.

Our result bears on questions regarding the possible modification of the pair-
production opacity effect on the $\gamma$-ray flux from distant extragalactic
sources, either by line-of-sight photon-axion oscillations during propagation
(e.g., De Angelis et al. 2009) or by the addition of a component of secondary
$\gamma$-rays from interactions of blazar-produced cosmic-rays with photons
along the line-of-sight to the blazar (e.g., Essey et al. 2010; Essey &
Kusenko 2012). Future theoretical studies and future $\gamma$-ray observations
of extragalactic sources with Fermi and the Čerenkov Telescope Array, which
will be sensitive to extragalactic sources at energies above 10 GeV (Gernot
2011), should help to clarify these important aspects of high energy
astrophysics.

##  6 Our Results Online

Our results in numerical form are available at the following link:

http://csma31.csm.jmu.edu/physics/scully/opacities.html

## Acknowledgments

We would like to thank Luis Reyes and Anita Reimer for supplying us with the
Fermi results shown in Figure 5. We thank Richard Henry for a helpful
discussion of the UV background data. We also thank Tonia Venters for helpful
discussions. This research was partially supported by a NASA Astrophysics
Theory Grant and a NASA Fermi Guest Investigator Grant.

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## Table 1. Identification of References for Fig. 1 Data by Waveband and
Redshift

z | FUV | NUV | U | B | V | R | I
---|---|---|---|---|---|---|---
.05 | SC05, WY05 | WY05 |  |  |  |  |
.1 | BU05,CU12 | BU05,CU12 |  |  |  |  |
.15 | TR07 | TR07 | TR07 | TR07 | TR07 | TR07 | TR07
.20 | BU05 | BU05 |  |  |  |  |
.25 | WO03 | WO03 |  |  |  | WO03 |
.3 | SC05,CU12,SC05,TR07 | TR07,CU12 | TR07,DA05 | TR07,DA05,FA07 | TR07 | TR07 | TR07
.35 |  | DA07, WO03 |  | WO03 |  | WO03 |
.45 |  | WO03 | DA05 | DA05, WO03 |  | WO03 |
.5 | SC05, CU12, TR07 | TR07 | TR07 | TR07, FA07 | TR07 | TR07 | TR07
.55 |  | DA07, WO03 |  | WO03 |  | WO03 |
.6 |  |  | DA05 | DA05 |  | CH03 |
.65 |  | WO03 |  | WO03 | MA12 | WO03 |
.7 |  | TR07,CU12 | TR07 | TR07, FA07 | TR07 | TR07 | TR07
.75 |  | WO03 |  | WO03 |  | WO03 |
.85 |  | WO03 |  | WO03 |  | WO03 |
.9 | TR07,CU12 | TR07,CU12 | TR07, DA05 | TR07, DA05, FA07 | TR07 | TR07 | TR07
.95 |  | WO03 | DA05 | WO03, DA05 | MA12 | WO03, DA05 |
1.0 | SC05 | WO03 |  | WO03 |  | WO03 |
1.1 | CU12, TR07, DA07, BU07 | DA07,TR07,CU12, WO03 | TR07 | TR07, FA07, WO03 | TR07 | TR07, WO03 | TR07
1.2 |  |  | DA05 | DA05 |  | CH03, DA05 |
1.3 | CU12, TR07 | TR07 | TR07 | TR07 | TR07 | TR07 | TR07
1.4 | CU12 | CU12 |  |  |  |  |
1.5 |  |  | DA05 | DA05 |  | DA05 |
1.6 | TR07 | TR07 | TR07 | TR07 | TR07 | TR07 | TR07
1.7 |  |  | DA05 | DA05 |  | DA05 |
1.8 | DA07 | DA07 |  |  | MA12 |  |
1.9 |  |  | DA05 | DA05 |  | DA05 |
2.0 | SC05 |  |  |  |  |  |
2.1 | CU12 | CU12 |  |  |  |  |
2.2 | RE08, SA06 |  |  | MA07 | MA07 | MA07 |
2.3 | LY09 |  |  |  |  |  |
2.4 |  |  |  |  | MA12 |  |
2.9 | SC05 |  |  |  |  |  |
3.0 | CU12 | CU12 |  | MA07 | MA07, MA12 |  |
3.5 | PA07 |  |  |  |  |  |
3.8 | BO07 |  |  |  | MA12 |  |
4.0 | YO06,CU12 |  |  |  |  |  |
4.1 | SA06 |  |  |  |  |  |
4.8 | IW07 |  |  |  |  |  |
5.0 | BO07 |  |  |  |  |  |
5.9 | BO07 |  |  |  |  |  |
6.8 | BO11 |  |  |  |  |  |
7.0 | OE10 |  |  |  |  |  |
8.2 | BO10 |  |  |  |  |  |

Figure Captions

Figure 1: The observed specific emissivities in our fiducial wavebands. The
lower right panel shows all of the observational data from the references in
footnote 1. In the other panels, non-band data have been shifted using the
color relations given in the text in order to fully determine the specific
emissivities in each waveband. The symbol designations are FUV: black filled
circles, NUV: magenta open circles, $U$: green filled squares, $B$: blue open
squares, $V$: brown filled triangles, $R$: orange open triangles, $I$: yellow
open diamonds. Grey shading: 68% confidence bands (see text).

Figure 2: The photon densities $\epsilon n(\epsilon)$ shown as a continuous
function of photon energy and redshift for both the high (upper panel) and low
(lower panel) IBL.

Figure 3: Our empirically-based determination of the EBL together with lower
limits and data as described in the text. The legend is as follows: Madau &
Pozzetti(2000):Black Cicles, Xu et al.(2005):Crosses, Gardner et
al.(2000):Open Squares, Matsuoka et al.(2011):Open Circles, Mattilla et
al.(2011)(preliminary):Black Squares, Bernstein(2007):Black Diamonds. The
upper limit from Mattilla et al.(2011) is thickened for clarity.

Figure 4: Our empirically determined opacities for redshifts of 0.1, 0.5, 1,
3, 5. The dashed lines are for $\tau=1$ and $\tau=3$.

Figure 5: A $\tau=1$ energy-redshift plot (Fazio & Stecker 1970) showing our
uncertainty band results compared with the Fermi plot of their highest energy
photons from FSRQs (red), BL Lacs (black) and and GRBs (blue) vs. redshift
(from Abdo et al. 2010).

Figure 6: Our opacity results for the redshifts of the blazars compared with
95% confidence opacity upper limits (red arrows) and 99% confidence limits
(blue arrows) as given by the Fermi analysis of Abdo, et al. (2010).

Figure 1.

Figure 2.

Figure 3.

Figure 4.

Figure 5.

Figure 6.