FRCAMB: An f(R)f(R) Code for Anisotropies in the Microwave Background

An $f(R)$ gravity model is proposed to realize a late time accelerated expansion of our Universe. To test the viability of an $f(R)$ gravity model through cosmic observations, the background evolution and the Einstein-Boltzmann equation should be solved for studying the effects on the cosmic microwave background power spectrum and on the matter power spectrum. In the market, we already have the modified versions of {\bf CAMB} code, for instance {\bf EFTCAMB} and {\bf MGCAMB}. However, in these publicly available Einstein-Boltzmann codes, a specific background cosmology, for example the $\Lambda$CDM or $w$CDM, is assumed. This assumption would be non-proper for a specific $f(R)$ model where the background evolution may be different from a $\Lambda$CDM cosmology. Therefore the main task for this paper is to present a code to calculate the anisotropies in the microwave background for any $f(R)$ gravity model based on {\bf CAMB} code, i.e. {\bf FRCAMB}, where the background and perturbation evolutions are included consistently. As results, one can treat {\bf FRCAMB} as a blackbox to output the CMB power spectrum and matter power spectrum, once an $f(R)$ function, its first two derivative with respect to $R$, i.e. $f_R\equiv df/dR$, $f_{RR}\equiv d^2f/dR^2$ and the reasonable values of the model parameters are inputted properly. As by-products, one can also output the effective equation of state of $f(R)$ model, the evolution of the dimensionless energy densities and other interesting cosmological quantities.Comment: 8 pages, 7 figure

Deep Epitome for Unravelling Generalized Hamming Network: A Fuzzy Logic Interpretation of Deep Learning

This paper gives a rigorous analysis of trained Generalized Hamming Networks(GHN) proposed by Fan (2017) and discloses an interesting finding about GHNs, i.e., stacked convolution layers in a GHN is equivalent to a single yet wide convolution layer. The revealed equivalence, on the theoretical side, can be regarded as a constructive manifestation of the universal approximation theorem Cybenko(1989); Hornik (1991). In practice, it has profound and multi-fold implications. For network visualization, the constructed deep epitomes at each layer provide a visualization of network internal representation that does not rely on the input data. Moreover, deep epitomes allows the direct extraction of features in just one step, without resorting to regularized optimizations used in existing visualization tools.Comment: 25 pages, 14 figure

A New Unified Dark Fluid Model and Its Cosmic Constraint

In this paper, we propose a new unified dark fluid (UDF) model with equation of state (EoS) $w(a)=-\alpha/(\beta a^{-n}+1)$, which includes the generalized Chaplygin gas model (gGg) as its special case, where $\alpha$, $\beta$ and $n$ are three positive numbers. It is clear that this model reduces to the gCg model with EoS $w(a)=-B_s/(B_s+(1-B_s)a^{-3(1+\alpha)})$, when $\alpha=1$, $\beta=(1-B_s)/B_s$ and $n=3(1+\alpha)$. By combination the cold dark matter and the cosmological constant, one can coin a EoS of unified dark fluid in the form of $w(a)=-1/(1+(1-\Omega_{\Lambda})a^{-3}/\Omega_{\Lambda})$. With this observations, our proposed EoS provides a possible deviation from $\Lambda$CDM model when the model parameters $\alpha$ and $n$ deviate from 1 and 3 respectively. By using the currently available cosmic observations from type Ia supernovae (SN Ia) Union2.1, baryon acoustic oscillation (BAO) and cosmic microwave background radiation (CMB), we test the viability of this model and detect the possible devotion from the $\Lambda$CDM model. The results show that the new UDF model fits the cosmic observation as well as that of the $\Lambda$CDM model and no deviation is found from the $\Lambda$CDM model in $3\sigma$ confidence level. However, our new UDF model can give a non-zero sound speed, as a contrast, which is zero for the $\Lambda$CDM model. We expect the large structure formation information can distinct the new UDF model from the $\Lambda$CDM model.Comment: 7 pages, 3 figures. arXiv admin note: substantial text overlap with arXiv:1204.5571, arXiv:1204.479

CVA and FVA to Derivatives Trades Collateralized by Cash

In this article, we combine replication pricing with expectation pricing for derivative trades that are partially collateralized by cash. The derivatives are replicated by underlying assets and cash, using repurchasing agreement (repo) and margining, which incur funding costs. We derive a partial differential equation (PDE) for the derivatives price, obtain and decompose its solution into the risk-free value of the derivative plus credit valuation adjustment (CVA) and funding valuation adjustment (FVA). For most derivatives, as we shall show, CVAs can be evaluated analytically or semi-analytically, while FVAs, as well as the derivatives values, will have to be solved recursively through numerical procedures due to their interdependence. In numerical demonstrations, continuous and discrete margin revisions are considered, respectively, for an equity call option and a vanilla interest-rate swaps.Comment: 29 pages, 4 figure

Constraints on the Holographic Dark Energy Model from Type Ia Supernovae, WMAP7, Baryon Acoustic Oscillation and Redshift-Space Distortion

In this paper, we use the joint measurement of geometry and growth rate from matter density perturbations to constrain the holographic dark energy model. The geometry measurement includes type Ia supernovae (SN Ia) Union2.1, full information of cosmic microwave background (CMB) from WMAP-7yr and baryon acoustic oscillation (BAO). For the growth rate of matter density perturbations, the results $f(z)\sigma_8(z)$ measured from the redshift-space distortion (RSD) in the galaxy power spectrum are employed. Via the Markov Chain Monte Carlo method, we try to constrain the model parameters space. The jointed constraint shows that $c=0.750_{- 0.0999- 0.173- 0.226}^{+ 0.0976+ 0.215+ 0.319}$ and $\sigma_8=0.763_{- 0.0465- 0.0826- 0.108}^{+ 0.0477+ 0.0910+ 0.120}$ with $1,2,3\sigma$ regions. After marginalizing the other irrelevant model parameters, we show the evolution of the equation of state of HDE with respect to the redshift $z$. Though the current cosmic data points favor a phantom like HDE Universe for the mean values of the model parameters in the future, it can behave like quintessence in $3\sigma$ regions.Comment: 10 pages, 4 figures, to appear in Phys. Rev.

Unified Dark Fluid with Constant Adiabatic Sound Speed: Including Entropic Perturbations

In this paper, we continue to study a unified dark fluid model with a constant adiabatic sound speed but with the entropic perturbations. When the entropic perturbations are included, an effective sound speed, which reduces to the adiabatic sound speed when the entropic perturbations are zero, has to be specified as an additional free model parameter. Due to the relations between the adiabatic sound speed and equations of state (EoS) $c^2_{s,ad}(a)=w(a)-d\ln(1+w(a))/3 d\ln a$, the equation of state can be determined up to an integration constant in principle when an adiabatic sound speed is given. Then there are two degrees of freedom to describe the linear perturbations for a fluid. Its micro-scale properties are characterized by its EoS or adiabatic sound speed and an effective sound speed. We take the effective sound speed and adiabatic sound speed as free model parameters and then use the currently available cosmic observational data sets, which include type Ia supernova Union 2.1, baryon acoustic oscillation and WMAP 7-year data of cosmic background radiation, to constrain the possible entropic perturbations and the adiabatic sound speed via the Markov Chain Monte Carlo method. The results show that the cosmic observations favor a small effective sound speed $c^2_{s,eff}=0.00155_{- 0.00155}^{+ 0.000319}$ in $1\sigma$ region.Comment: 7 pages, 5 figures, to appear in Phys. Rev.

Probing the Neutrino Mass through the Cross Correlation between the Rees-Sciama Effect and Weak Lensing

Cosmology plays a fundamental role to determine the neutrino mass, therefore also to determine its mass hierarchy, since the massive neutrino contributes to the total matter density in the Universe at the background and perturbation levels, once it becomes non-relativistic. After the non-relativistic transition the fluctuations are smashed out at the scales $k\gg k_{fs}$. Therefore, the missing fluctuation in the total matter is imprinted on the large scale structure, say the suppression of the matter power spectrum $\Delta P/P\approx -8f_{\nu}$ at the scales $k\gg k_{fs}$. In this paper, instead of considering the linear perturbation theory, which is well understood in the presence of neutrino, we propose to use the cross correlation between the Rees-Sciama effect and weak lensing to probe the neutrino mass. At the small scales, the density contrast grows faster than the background scale factor $\delta\sim a$, that makes a sign flipping on $\Phi' \propto \mathcal{H}\delta d\ln (\delta/a)/d\ln a$, which happens only in the non-linear regime. We show that the flipping scale in the cross power spectrum between the Rees-Sciama effect and weak lensing depends on the neutrino mass by assuming the shallow and deep weak lensing surveys. Our analysis shows that the Deep survey has larger signal-to-noise ratio $S/N\sim 160$. Finally, we use the Fisher information matrix to forecast constraint on the neutrino mass.Comment: 7 pages, 5 figures, to appear in JCA

Inflation-rate Derivatives: From Market Model to Foreign Currency Analogy

In this paper, we establish a market model for the term structure of forward inflation rates based on the risk-neutral dynamics of nominal and real zero-coupon bonds. Under the market model, we can price inflation caplets as well as inflation swaptions with a formula similar to the Black's formula, thus justify the current market practice. We demonstrate how to further extend the market model to cope with volatility smiles. Moreover, we establish a consistency condition on the volatility of real zero-coupon bonds using arbitrage arguments, and with that re-derive the model of Jarrow and Yildirim (2003) with real forward rates based on "foreign currency analogy", and thus interconnect the two modeling paradigms.Comment: 36 pages, 8 figure

Confronting DGP Braneworld Gravity with Cosmic Observations after Planck Data

The normal branch of Dvali-Gabadadze-Porrati braneworld gravity with brane tension is confronted by the currently available cosmic observations from the geometrical and dynamical perspectives. On the geometrical side, the type Ia supernova as standard candle, the baryon acoustic oscillation as standard ruler and the cosmic microwave background measurement from the first released 15.5 months data were used to fix the background evolutions. On the dynamical side, the redshift space distortion data will be used to determine the evolution of the matter perturbation. Through a Markov chain Monte Carlo analysis, we found the dimensionless crossover scale $\Omega_{r_c}=1/(4H^2_0r^2_{c})=0.00183_{-0.00183}^{+0.000338}$ in a spatially flat normal branch of Dvali-Gabadadze-Porrati braneworld. This result suggests that the crossover scale $r_c$ should be around $12H^{-1}_0$ which is consistent with the previous result $r_c>3H^{-1}_0$ and greater. It also implies that the five-dimensional gravity effect is weak to be observed in $H^{-1}_0$ scale.Comment: 6 pages, 3 figures, match the published versio

Detecting Primordial Gravitational Waves Signal from BICEP2 and {\it Planck} HFI 353353GHz Dust Polarization

The dust polarization is parameterized as a power law form of the multipole $l$: $D^{XX}_{l}=A^{XX}l(l+1)l^{\alpha_{XX}}/(2\pi)$ ($XX$ denotes $BB$ or $EE$), where $A^{XX}$ is its amplitude with the ratio $A^{BB}/A^{EE}=0.52\pm 0.02$ and $\alpha_{BB,EE}=-2.42\pm 0.02$. Extrapolating to $150$GHz from $353$GHz yields a value of $D^{BB}_{l=80}=(1.32\pm 0.29)\times 10^{-2}\mu K^2$ (and an additional uncertainty $(+0.28,-0.24)\times 10^{-2}\mu K^2$) over the range $40<l<120$. Based on these data, we report the tensor-to-scalar ratio $r=A_{t}/A_{s}$ defined at $k_0=0.05 \text{Mpc} ^{-1}$ by joining the BICEP2+{\it Planck}2013+WMAP9+BAO+HST and {\it Planck} HFI $353$GHz dust polarization and its implication to the detection of the primordial gravitational waves. Considering the $\Lambda$CDM+$r$ model, we found $r<0.108$ at $95\%$ confidence level with $\sigma_{stat}=0.29$ and $r<0.129$ at $95\%$ confidence level with $\sigma_{stat+extr}=0.29+0.28$. The results imply no significant evidence for the primordial gravitational waves in $1\sigma$ regions. However the post probability distribution of $r$ peaks at a small positive value. And $r$ moves to larger positive values when the extrapolation error bars are included. This might imply a very weak signal of the primordial gravitational waves. It also implies the crucial fact in calibrating the amplitude of the dust polarizations in detecting the primordial gravitational waves in the future. When the running of the scalar spectral tilt is included, we found $r<0.079$ at $95\%$ confidence level with $\sigma_{stat}=0.29$ and $r=0.091_{-0.069}^{+0.042}$ at $95\%$ confidence level with $\sigma_{stat+extr}=0.29+0.28$. The later one implies the detection of the primordial gravitational waves in $1\sigma$ regions at the cost of decreasing the value of $D^{BB}_{l=80}$ to $0.67_{-0.25}^{+0.25}$.Comment: 5 pages, 4 figures, title changed, n_{run} was include