Using the Extragalactic Gamma-Ray Background to Constrain the Hubble Constant and Matter Density of the Universe

The attenuation produced by extragalactic background light (EBL) in $\gamma$-ray spectra of blazars has been used to constrain the Hubble constant ($H_0$) and matter density ($\Omega_{\rm m}$) of the Universe. We propose to estimate $H_0$ and $\Omega_{\rm m}$ using the well measured $>$10 GeV extragalactic $\gamma$-ray background (EGB). This suggestion is based on the facts that the $>$10 GeV EGB is totally explained by the emissions from blazars, and an EBL-absorption cutoff occurs at $\sim$50 GeV in the EGB spectrum. We fit the $>$10 GeV EGB data with modeled EGB spectrum. This results in $H_0=64.9^{+4.6}_{-4.3}\rm \ km\ s^{-1}\ Mpc^{-1}$ and $\Omega_{\rm m}=0.31^{+0.13}_{-0.14}$. Note that the uncertainties may be underestimated due to the limit of our realization for EBL model. $H_0$ and $\Omega_{\rm m}$ are degenerate in our method. Independent determination of $\Omega_{\rm m}$ by other methods would improve the constraint on $H_0$.


INTRODUCTION
A precise and accurate measurement of the Hubble constant (H 0 ) would provide deep understanding of fundamental physics questions. Multiple paths to independent estimates of H 0 are needed in order to access and control its systematic uncertainties (Suyu et al. 2012).
Gamma-ray astronomy provides a new approach to estimate H 0 (Salamon et al. 1994;Mannheim 1996). The optical depth of the γ-ray photons emitted by extragalactic objects, τ γγ , scales as n EBL σ T l, where n EBL is the photon density of the extragalactic background light (EBL), σ T is the Thomson cross section, and l is the distance from the γ-ray source to Earth. l is inversely proportional to H 0 , and n EBL also depends on H 0 . Therefore, through determining the optical depth τ γγ , one can estimate H 0 .
The above constraints on H 0 are all derived from point sources. Here, we propose to constrain H 0 and Ω m using the extragalactic γ-ray background (EGB). The EGB spectrum has been well measured from 0.1 GeV to ∼800 GeV by the Fermi-LAT. This spectrum can be described by a power law with a photon index of 2.32 that is exponentially cut off at ∼50 GeV (Ackermann et al. 2015). The cutoff is caused by the EBL absorption . Similar to the idea proposed by Salamon et al. (1994), the γ-ray absorption in the EGB spectrum could also be used to constrain the cosmological parameters.
EGB is dominated by the emission of γ-ray blazars Ackermann et al. 2016). With the source count distribution of hard-spectrum blazars, Ackermann et al. (2016) estimated that blazars can explain almost the totality (86 +16 −14 %) of the >50 GeV EGB. In particular, the calculation performed with improved luminosity function (LF) and modeling of the spectral energy distributions (SEDs) of blazars showed that blazars account for the totality of the ≥10 GeV EGB . Besides, modeling of the EGB spectrum also depends on H 0 . Therefore, we can use the above information to constrain H 0 and Ω m .

Calculation of the EGB spectrum
We follow Ajello et al. (2015) to compute the EGB spectrum contributed by blazars, where the LF, Φ(L γ , z, Γ) (at redshift z, for sources of γ-ray luminosity L γ ), is described as a broken power law multiplied by the photon index distribution dN dΓ (Equation (1) in Ajello et al. 2015). The γ-ray spectrum of each blazar, dNγ dE , is modeled as a broken power law (Equation (11) in Ajello et al. 2015). dV dzdΩ is the comoving volume element per unit redshift and unit solid angle, which is written as, where ΛCDM cosmology, and d L is the luminosity distance.

Absorption of γ-rays
The optical depth of the γ-ray photons emitted at redshift z as a function of observed γ-ray photon energy, E γ , is calculated by (e.g., Razzaque et al. 2009) , andφ(s 0 ) is adopted from Gould & Shréder (1967). We use the model of Razzaque et al. (2009) to calculate the comoving EBL density, where ψ(z) is the star formation rate (SFR) in unit of M yr −1 Mpc −3 , dN dM is the initial mass function (IMF), f esc ( ) is the escape fraction of photons from the host galaxy, and dN ( ,M ) d dt is the total number of photons emitted from a star. The normalization is determined by is the redshift of the star (born at redshift z) that had evolved off the main sequence. See Razzaque et al. (2009) for more details.
The uncertainties in modeling the EBL density primarily come from SFR and IMF. We adopt the Models B and C in Razzaque et al. (2009). Both models use the same SFR (Cole et al. 2001;Hopkins & Beacom 2006), but different IMFs. Model B uses Salpeter A IMF (Salpeter 1955), and Model C uses Baldry-Glazebrook IMF (Baldry & Glazebrook 2003).

Verification of our calculations
We calculate the contribution to the EGB from blazars with the pure luminosity evolution (PLE) LF in  and the EBL Models B and C. The parameters in Table 1 in Ajello et al. (2015) are used. Here we adopt H 0 = 67 km s −1 Mpc −1 and Ω m = 1 − Ω Λ = 0.3, same as that in Ajello et al. (2015). The results are shown in Fig. 1. One can see that EGB above 100 GeV can be explained by the emission from the blazars below the redshift of 0.8, whereas EGB between 10 GeV and 100 GeV can be explained by the blazars below the redshift of 1.5.
We compare our results with Ajello et al. (2015) who adopted the EBL model of Finke et al. (2010). We found that our results are almost the same as that in  (see their Fig. 3) below 300 GeV. Above 300 GeV, the intensity that we calculated with EBL Model C is higher than the one in Ajello et al. (2015). This is due to our underestimation of the EBL intensity. However, we note that above 300 GeV, the intensity in  agrees with ours within the errors of the data points.
The results in Fig. 1 show two points: (1) the emission from blazars could be used to explain the EGB above ∼ 10 GeV; (2) the difference between EBL models of Razzaque et al. (2009) andFinke et al. (2010) has little impact on explaining the origins of EGB.

RESULTS
Calculations of LF and SFR depend on the measurements of H 0 and Ω m . Ajello et al. (2015) constructed the LF with H 0 = 67 km s −1 Mpc −1 and Ω m = 0.3. In our purpose, the LF should be modified with different cosmological parameters. Therefore, the LF in Equation (1) is, The SFR in Equation (4) is modified as (e.g., Domínguez et al. 2019), The primed quantities are computed with H 0 = 67 km s −1 Mpc −1 for the LF, and H 0 = 70 km s −1 Mpc −1 for the SFR, and Ω m = 0.3.

Dependence on H 0
Calculations of both the intrinsic EGB spectrum and τ γγ (E, z) depend on H 0 and Ω m . In Fig. 2, we can see that the intrinsic spectrum strongly relies on H 0 , especially at the energies below 100 GeV (left panel; H 0 is fixed to 67 km s −1 Mpc −1 in the calculation of the optical depth); and the dependence of τ γγ (E, z) on H 0 occurs at the energies above 100 GeV (right panel; H 0 is fixed to 67 km s −1 Mpc −1 in the calculation of the intrinsic EGB spectrum).

Fitting results
We use the modeled EGB spectrum to fit the >10 GeV observed data. H 0 and Ω m are set to free, and the other parameters are fixed to those in Ajello et al. (2015) and in Razzaque et al. (2009). The Markov Chain Monte Carlo (MCMC) technique is used to perform our fitting. More details of our MCMC method can be found in Yan et al. (2013). Fig. 3 shows the best-fitting results with EBL Model B. We obtain H 0 = 72 +10 −9 km s −1 Mpc −1 and Ω m = 0.23 +0.14 −0.13 1 . In the fitting, H 0 is anti-correlated with Ω m (see the 2D confidence contours of the parameters in the right panel), which is consistent with the result obtained by using the EBL model of Finke et al. (2010) in Domínguez et al. (2019). We note that the calculated EGB spectrum below 5 GeV is more sensitive to H 0 and Ω m (see the solid and dashed lines in the left panel of Fig. 3). This effect is brought by the LF. Fig. 4 shows the best-fitting results with EBL Model C. We obtain H 0 = 63.1 +6.2 −4.7 km s −1 Mpc −1 and Ω m = 0.44 +0.13 −0.19 . The uncertainties on H 0 are at the 9% level. Again, there is a strong degeneracy between H 0 and Ω m in this model. The EGB spectrum calculated with H 0 = 67 km s −1 Mpc −1 and Ω m = 0.3 is almost same with the best-fitting EGB spectrum.
The uncertainties on H 0 are comparable with those obtained by Domínguez et al. (2019). There is a clear degeneracy between H 0 and Ω m in our calculation. Measurement of Ω m using other independent methods would improve the constraint on H 0 .
We choose the two easily calculated EBL models to examine the uncertainties introduced by the EBL models. Actually, these two models belong to the same method-  ology, i.e., the physically motivated model. These two models use the same assumption for SFR, and only differ in IMFs. Different assumptions for SFR may introduce extra uncertainties on H 0 . In addition, we cannot examine the uncertainties introduced by different methodologies of building EBL models (e.g., Domínguez et al. 2019). The uncertainties in our results mainly come from EBL models. Therefore, we may underestimate the uncertainties in our results. Currently, the values of H 0 measured from type Ia supernovae and from cosmic microwave background radiation (CMB) are discrepant at 3σ (Riess et al. 2018). Alternative methods of measuring the Hubble constant, like the method presented here, is helpful to understand this discrepancy.