The Lyman-$α$ power spectrum - CMB lensing convergence cross-correlation
Chi-Ting Chiang Anže Slosar

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bi-spectrum



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        [(Phys. Rev. DPhys. Rep.MNRASJHEPa]Chi-Ting Chiang,b]and Anze Slosar[a]C.N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY 11794, U.S.A.[b]Brookhaven National Laboratory, Blgd 510, Upton, NY 11375, U.S.A.chi-ting.chiang@stonybrook.edu,anze@bnl.govWe investigate the three-point correlation between the 
          forest and the CMB weak lensing (δ_F δ_F κ) expressed
          as the cross-correlation between the CMB weak lensing field
          and local variations in the forest power spectrum. In addition to
          the standard gravitational bispectrum term, we note the existence of
          a non-standard systematic term coming from mis-estimation of the mean flux
          over the finite length of  skewers. We numerically calculate the
          angular cross-power spectrum and discuss its features. We integrate
          it into zero-lag correlation function and compare our predictions
          with recent results by Doux et al.. We find that our predictions are
          statistically consistent with the measurement, and including the systematic
          term improves the agreement with the measurement. We comment on the
          implication of the response of the  forest power spectrum to the
          long-wavelength density perturbations. YITP-SB-17-30The Lyman-α power spectrum - CMB lensing convergence cross-correlation
            [
            Accepted ?.
              Received ?;
              in original form ?.
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        § INTRODUCTION
        
        The three-point function of the large-scale structure contains ample information
        that cannot be probed by the two-point function, hence it is sensitive to the
        nonlinear gravitational evolution <cit.>, tracer bias
        <cit.>, and even the inflationary physics <cit.>.
        Measurement of the three-point function, or its Fourier counterpart bispectrum,
        by auto- or cross-correlation is thus one of the biggest scientific goals for ongoing
        and future surveys.
        
        In this paper, we propose a new bispectrum formed by the  forest power spectrum
        (hereafter forest power spectrum) and the CMB lensing convergence. Since the forest power
        spectrum is sensitive to the small-scale matter fluctuation at 2≤ z≤4 whereas
        CMB lensing convergence measures the total matter fluctuation along the line-of-sight,
        this combination of bispectrum allows us to probe the nonlinear structure formation
        that has not been explored using galaxies as tracers. Note that a similar observable
        has been measured in Ref. <cit.> by cross-correlating the one-dimensional
        forest power spectrum and the CMB lensing at  the same angular position.
        We shall generalize the bispectrum to include the complete angular scale between
        the  forest skewers and the CMB lensing, and demonstrate that their observable
        is an integral over the angular scale of the total bispectrum.
        
        The rest of this paper is organized as follows.
        In true_signal, we compute the underlying signal generated by the gravitational evolution.
        In biased_signal, we calculate the signal due to the continuum fitting when
        measuring the  forest, which we refer to as the “continuum-misestimation bias”.
        In observation, we discuss the contamination from damped  absorbers
        and compare our prediction with the measurement in Ref. <cit.>.
        We conclude in conclusion.
        Throughout the paper we adopt the Planck cosmology <cit.>,
        i.e. h=0.6803, Ω_bh^2=0.0226, Ω_ch^2=0.1186, A_s=2.137×10^-9,
        and n_s=0.9667.
        
        
        
        
        § UNDERLYING SIGNAL DUE TO GRAVITATIONAL EVOLUTION
        
        The mean of the forest power spectrum at redshift z with width ,
        corresponding to the comoving distance r_ and width Δ r,
        can be written as <cit.>
         P_FF(_s)=P_FF(k_s,μ_s)
         =b_F^2(1+β_Fμ_s^2)^2P_̣̣(k_s)(k_s,μ_s)  ,
        
        where _s is the small-scale wavevector (opposed to the large-scale mode
        that we introduce later), μ_s is the cosine of _s along the line-of-sight,
        b_F is the flux bias, β_F is the redshift-space distortion parameter
        for the flux, P_̣̣ is the linear power spectrum, and  is the fitting
        formula for the small-scale nonlinearity. In this paper we shall use the fitting
        function provided in Ref. <cit.> as
        (k_s,μ_s)=exp{
            q_1Δ^2(k_s)+q_2Δ^4(k_s)
        
            1-k_s/k_v^a_vμ_s^b_v
        -k_s/k_p^2} ,
        
        where Δ^2(k)=k^3P_̣̣(k)/(2π^2) and (q_1,q_2,k_v,a_v,b_v,k_p)
        are fitting parameters obtained from simulations. Note that the linear power
        spectrum as well as all biases and fitting parameters (provided in Table 4
        and 5 of Ref. <cit.>) depend on redshift, but we do
        not write the dependence explicitly when no confusion occurs.
        
        If there is a large-scale fluctuation on the sky, the forest power spectrum would be modulated
        and at the leading order the forest power spectrum at angular position _⊥
        becomes <cit.>
         P_FF(_s,_⊥)=P_FF(_s)+dP_FF(_s)/ḍ̣̅̅(_⊥)  ,
        
        where ̣̅ is the two-dimensional projected large-scale density fluctuation,
        and dP_FF(_s)/ḍ̅ is the response of the forest power spectrum with
        respect to ̣̅ due to the gravitational evolution, which can be measured from
        the separate universe simulations <cit.>. In the
        flat-sky approximation, the three-dimensional position can be decomposed into
        parallel r_ and transverse _⊥ components, and ̣̅ is given by
        ̣̅(_⊥)
         =1/Δ r∫_r_-Δ r/2^r_+Δ r/2
         dr'_δ(r'_,_⊥)
         =1/Δ r∫ dr'_δ(r'_,_⊥)
         ΘΔ r/2-|r'_-r_| ,
         where δ is the underlying matter density perturbation and
        Θ is the heaviside step function. At the leading order, i.e in
        the limit where the wavevector of the large-scale mode ̣̅ is much
        smaller than _s, the response depends only on the small-scale mode
        _s and is independent of the wavelength of ̣̅.
        
        Consider measuring the correlation between the forest power spectrum with
        the two-dimensional projected CMB lensing convergence in the same redshift
        bin
        κ(_⊥)    =∫_r_-Δ r/2^r_+Δ r/2dr'_∫ d^2r'_⊥δ(')W_κ(r'_)Λ_κ('_⊥-_⊥)     =∫ d^3r'δ(')W_κ(r'_)ΘΔ r/2-|r'_-r_|Λ_κ('_⊥-_⊥)  ,
        
        where W_κ is the lensing kernel and Λ_κ is the
        weighting filter in the transverse direction (which we will later set
        to be the Wiener filter). The lensing kernel is given by
        
         W_κ(r_)=3H_0^2Ω_m/2c^2r_/a(r_)r_*-r_/r_* ,
        
        where H_0 is the present-day Hubble, Ω_m is the fractional energy
        density of matter, c is the speed of light, a is the scale factor,
        and r_* is the comoving distance to the last-scattering surface. The
        three-point correlation between P_FF and κ in Fourier space
        is thus
        ⟨ P_FF(_s,_l⊥)κ('_l⊥)⟩'
         =dP_FF(_s)/ḍ̅⟨̣̅(_l⊥)κ('_l⊥)⟩'
         =dP_FF(_s)/ḍ̅P^ 2d_̣̅κ(_l⊥)  ,
        
        where ⟨⟩' is the ensemble average without the Dirac delta function.
        Using bardkappa, we can compute P^ 2d_̣̅κ as
        
         P^ 2d_̣̅κ(_l⊥)    =
         ∫_r_-Δ r/2^r_+Δ r/2dr'_1W_κ(r'_1)
         ∫ dr'_21/Δ rΘΔ r/2-|r'_2-r_|    ×∫dq_/2πP_̣̣(q_,_l⊥)
         Λ_κ(-_l⊥)e^iq_(r'_1-r'_2)    =∫_r_-Δ r/2^r_+Δ r/2dr'_ W_κ(r'_)
         ∫dq_/2πP_̣̣(q_,_l⊥)Λ_κ(-_l⊥)
         q_Δ r/2e^iq_(r'_-r_) ,
        
        and  the only assumption we make is that P_̣̣ is independent of
        z within Δ z. Because of the sinc function, the contribution to
        P^ 2d_̣̅κ from the parallel mode is mainly from q_≲Δ r^-1.
        Physically, this means that P^ 2d_̣̅κ is sensitive to
        the largest parallel mode, which is set by the redshift bin size. Following
        Ref. <cit.>, in the flat-sky approximation we can convert the
        two-dimensional power spectrum to the angular power spectrum as
        
         l(l+1)C_l^̣̅κ≈ k^2_l⊥P^ 2d_̣̅κ(_l⊥)  ,
        
        where l+1/2≈ r_ k_l⊥. Combining p2d_pffkappacl_bardkappa,
        we have the most general correlation between forest power spectrum and CMB
        lensing convergence, which includes both the dependence on the small-scale
        mode _s and the large-scale mode _l⊥, or equivalently the
        angular scale l. This calculation is general and can be used for other
        fields in place of Lyman-α, most notably 21-cm
        emission from intensity mapping.
        
        
        
        The left panel of corr_bardkappa shows C_l^̣̅κ with
        Δ z=0.2 as a function of the angular scale l at various redshifts.
        We set l≥100 so that the flat-sky approximation works well. To numerically
        evaluate C_l^̣̅κ, we set q_, max=1000 and
        confirm that the result is insensitive to the choice of q_, max.
        We also take W_κ to be the Wiener filter from Ref. <cit.> with l=r'_ k_l⊥.
        The power spectrum is larger at lower redshift, and this is the result of
        the gravitational evolution. To check the approximation that the linear power
        spectrum does not vary within the redshift bin Δ z, we include the
        redshift evolution and find that the fractional difference is less than 1%.
        The agreement is slightly better at higher redshift since Δ r is smaller
        at larger z when Δ z is fixed.
        
        
        
        Since the forest power spectrum is frequently measured as the one-dimensional
        power spectrum, we can Fourier transform in the small-scale transverse direction:
        
         P^ X_FF(k_s,_s⊥)=∫d^2k_s⊥/(2π)^2P_FF(k_s,_s⊥)
         e^i_s⊥·_s⊥,
        
        with the one-dimensional power spectrum given by
        P_FF^ 1d(k_s)=P^ X_FF(k_s,,_s⊥=0).
        Similarly one can Fourier transform the cross-correlation of p2d_pffkappa
        in the small-scale transverse wavevector _s⊥ to get a cross-correlation
        as a function of (k_s,_s⊥,k_l⊥). This tells us how a
        cross-power spectrum between two close skewers as a function of the small-scale
        parallel wavevector k_s and small-scale perpendicular separation _s⊥
        responds to a large-scale mode in κ field fluctuation. For this we need
        dP^ X_FF(k_s,_s⊥)/ḍ̅    =∫d^2k_s⊥/(2π)^2dP_FF(k_s,_s⊥)/ḍ̅
         e^i_s⊥·_s⊥    =∫d^2k_s⊥/(2π)^2P_FF(k_s,_s⊥)
         dln P_FF(k_s,_s⊥)/ḍ̅ e^i_s⊥·_s⊥ .
        
        As before, the one-dimensional specialization is just the case when we set
        _s⊥=0, i.e.
        dP^ 1d_FF(k_s)/ḍ̅=dP^ X_FF(k_s,_s⊥=0)/ḍ̅.
        
        To numerically evaluate the one-dimensional forest power spectrum, we use
        pff_fiddnl and set k_s⊥, max=1000. To compute
        the response of the one-dimensional forest power spectrum, we first measure
        and smooth dln P_FF(k_s,_s⊥)/d from the separate 
        universe simulations as described in Refs. <cit.>
        and then use the analytic form of P_FF(k_s,_s⊥), i.e.
        pff_fiddnl. To minimize
        the possible bias on the integration from the smoothing and interpolation, we
        set k_s⊥, max=10. pff1d shows the one-dimensional forest
        power spectrum (left) and its response to ̣̅ (right). The one-dimensional
        forest power spectrum is larger at higher redshift due to the increase of |b_F|.
        Conversely, the response of the one-dimensional forest power spectrum, which
        contains the responses from linear power spectrum, flux bias, and the nonlinear
        fitting function, is larger at lower redshift for k≲1.
        
        
        
        To evaluate the correlation between the one-dimensional forest power spectrum
        and the CMB lensing convergence, we take the product of C_l^̣̅κ from
        the left panel of corr_bardkappa and dP^ 1d_FF(k_s)/ḍ̅
        from the right panel of pff1d. true_sig shows the result at
        z=2.4 (left) and 2.8 (right) as a function of ℓ (the angular scale between
        the skewer and the lensing convergence) and k_s (the scale of the one-dimensional
        forest power spectrum), assuming Δ z=0.2, which corresponds respectively
        to Δ r=166.63 and 142.08. We find that at both redshifts the scale
        dependence on l and k_s is similar, and the more apparent difference
        is the amplitude of the signal. Specifically, at lower redshift the signal is
        larger, which is likely due to the stronger gravitational evolution.
        
        Lastly, if the correlation between the one-dimensional forest power spectrum
        (_s⊥=0) and the lensing convergence is measured in configuration
        space, then we can also Fourier transform the large-scale_l⊥:
        ⟨ P^ 1d_FF(k_s,_⊥)κ('_⊥)⟩    =∫d^2 _l⊥/(2π)^2⟨ P^ 1d_FF(k_s,_l⊥)κ(_l⊥)⟩
         e^i_l⊥·(_⊥-'_⊥)    =dP^ 1d_FF(k_s)/ḍ̅∫d^2 _l⊥/(2π)^2
         P^ 2d_̣̅κ(_l⊥)e^i_l⊥·(_⊥-'_⊥) .
        
        Furthermore, if the correlations are only measured at the same angular position
        (_⊥='_⊥), as done in Ref. <cit.>, then the relevant
        quantity in this case is the Fourier transform of P^ 2d_δ̅κ
        evaluated at zero lag, i.e.
        σ^2_̣̅κ=∫d^2 _l⊥/(2π)^2
         P^ 2d_̣̅κ(_l⊥)  ,
        
        which the variance of the δ̅κ field. Note that while this is the
        region where majority of signal to noise is concentrated, there is in principle
        more information available in the data.
        
        We show the correlation at the same angular position in the right panel of
        corr_bardkappa as a function of redshift for various Δ z.
        We confirm that the results are insensitive to the choice of k_ max
        for the integration and the redshift evolution of P_̣̣. As for
        C_l^̣̅κ, σ^2_̣̅κ is larger at lower redshift
        due to the gravitational evolution. Interestingly, while the comoving distances
        between Δ z=0.1 and 0.3 differ by a factor of three, the change in
        σ^2_̣̅κ is only from 12 to 18%. Should we compute
        σ^2_̣̣̅̅=∫d^3k_l/(2π)^3P_̣̣(k_l)
         ^2k_lΔ r/2 ,
        
        we have σ^2_̣̣̅̅(Δ z=0.1)/σ^2_̣̣̅̅(Δ z=0.3)≈2.9
        between 2.2≤ z≤3.0. This is because σ^2_̣̣̅̅ is the
        variance of ̣̅ for a parallel scale Δ r, and the larger the
        Δ r the smaller the variance. On the other hand, σ^2_̣̅κ
        includes the integral of lensing kernel along the line-of-sight, and the
        larger the Δ r the larger the lensing contribution. These two effects
        roughly cancel each other, as a result σ^2_̣̅κ is much less
        affected by the width in the line-of-sight direction.
        
        
        
        
        § CONTINUUM-MISESTIMATION BIAS DUE TO LYMAN-Α FOREST MEASUREMENT
        
        In the previous section we compute the gravitational bispectrum of flux-flux-lensing.
        In this section we shall discuss a non-standard systematic term coming from
        the mis-estimation of the mean flux over the finite length of  forest skewers,
        which we shall refer to as the “continuum-misestimation bias”.
        
        Intuitively, this effect comes from the fact that over the length of
        the forest, one cannot distinguish between a large-scale absorption
        that would lower the measured flux in all pixels and the change in the
        brightness of the quasars that would do exactly the same. Of course,
        using information outside the quasar allows one to, in principle,
        recover some of this information, but in practice this introduces
        large amounts of noise in the continuum calibration. Therefore, most
        analysis just fit for the quasar brightness within the forest,
        in effect nulling large-scale flux fluctuations modes, which leads to
        two effects: i) distortion of measured large-scale power spectra or
        correlation functions (see e.g. <cit.>)
        and ii) modulation of the local small scale power spectrum normalization
        by the large scale mode <cit.>. The second effect is
        the one that affects the calculation in this paper as we discuss next.
        
        The observed  flux can be modeled as
        
         F(r_,_⊥)=A(_⊥)C̅F̅(r_)
            1+δ_F(r_,_⊥)
        +ϵ ,
        
        where A describes the brightness of the quasar so depends on the angular
        position, C̅ is the mean continuum shape obtained from stacking all
        skewers in the survey, F̅ is the true mean flux which is independent
        of the angular position on the sky, δ_F is the true flux fluctuation
        around F̅, and ϵ is the noise. To measure δ_F, one
        performs the continuum fitting on a skewer by adjusting A until the integral
        of δ_F becomes zero across this skewer. In other words, for a skewer
        at _⊥ we estimate the quasar brightness as
        Â(_⊥)    =1/Δ∫_r_-Δ/2^r_+Δ/2dr'_F(r'_,_⊥)/C̅(r'_)F̅(r'_)
         =A(_⊥)
            1+1/Δ∫_r_-Δ/2^r_+Δ/2
             dr'_δ_F(r'_,_⊥)
            =A(_⊥)
            1+Δ_F(r_,_⊥,Δ)
         ,
        
        where r_ and Δ are the center and the width of the skewer.
        Note that the typical size of Δ is 280 to 420<cit.>,
        so Δ is generally larger than the size of the redshift bin Δ r.
        Plugging the estimated quasar luminosity Â into the flux, we find the
        estimated flux perturbation at _⊥ to be (ignoring the r_ dependence
        in Δ_F as it only describes the center of the forest)
        _F(r'_,_⊥)
             =F(r'_,_⊥)/Â(_⊥)C̅(r'_)F̅(r'_)-1
         =1+δ_F(r'_,_⊥)/1+Δ_F(_⊥,Δ)-1     ≈δ_F(r'_,_⊥)
            1-Δ_F(_⊥,Δ)
        
         -Δ_F(_⊥,Δ)  .
        
        Note that the last term in hat_dF is a constant at this angular position
        of _⊥, hence it contributes only to _s=0 mode of _F in
        Fourier space. We shall ignore it as we are interested in the forest power spectrum
        with k_s>0, and to the leading order the estimated forest power spectrum of this
        skewer becomes
        P̂_FF(_s,_⊥)=⟨_F(_s,_⊥)_F('_s,_⊥)⟩'
         =P_FF(_s)
            1-2Δ_F(_⊥,Δ)
         ,
        
        where the ensemble average is taken for the small-scale mode _s while keeping
        the large-scale Δ_F fixed. Physically, the correction term appears because
        we use the  local instead of the  global mean flux (which is independent
        of _⊥) to compute the fluctuation. The same effect is also discussed in
        Ref. <cit.> for the measurement of the position-dependent correlation
        function. Moreover, the larger the Δ the smaller the correction.
        
        
        
        Though for each skewer the estimated forest power spectrum is biased,
        the total estimated forest power spectrum is still unbiased at the first
        order since <Δ_F>=0. It is well known that there are
        corrections at the second order <cit.>. However, if we
        measure the correlation between the forest power spectrum of each skewer
        and the CMB lensing convergence, the signal would be biased due to the
        correlation between Δ_F and κ at the first order. Specifically,
        if we consider the angular power spectrum between the forest power spectrum
        and the lensing convergence, then the continuum-misestimation bias is
        ⟨P̂_FF(_s,_l⊥)κ('_l⊥)⟩'
         =-2P_FF(_s)⟨Δ_F(_l⊥,Δ)κ('_l⊥)⟩'
         =-2P_FF(_s)P^ 2d_Δ_Fκ(_l⊥)  ,
        
        where
        
         P^ 2d_Δ_Fκ(_l⊥)
         =∫_r_-Δ/2^r_+Δ/2dr'_ W_κ(r'_)
         ∫dq_/2πP_F̣(q_,_l⊥)Λ_κ(-_l⊥)
         q_Δ/2e^iq_(r'_-r_) ,
        
        with P_F̣()=
            P_̣̣()P_FF()
        ^1/2 in linear theory.
        Note that p2d_DFkappa
        is basically the same as p2d_bardkappa, excpet for different power
        spectra as well as the lengths appearing in the integration boundary and
        the sinc function. The left panel of corr_DFkappa shows the angular
        power spectrum of the continuum-misestimation bias C_l^Δ_Fκ
        under the flat-sky approximation with Δ=350 as a function
        of the angular scale l at various redshifts. Unlike C_l^̣̅κ
        shown in corr_bardkappa, we find that the continuum-misestimation
        bias is larger at higher redshift, which is due to the fact that C_l^Δ_Fκ
        is proportional to b_F and |b_F| is larger at higher redshift.
        
        
        
        For the correlation between the one-dimensional forest power spectrum and the
        lensing convergence, the continuum-misestimation bias becomes -2P^ 1d_FF(_s)P^ 2d_Δ_Fκ(_l⊥).
        To evaluate the signal, we take the one-dimensional forest power spectrum
        from the left panel of pff1d and C_l^Δ_Fκ from the right
        panel of corr_DFkappa, and the result is shown in bias_sig at
        z=2.4 (left) and 2.8 (right), assuming the length of the forest to be 350.
        We find that the scale dependences of the true signal and the continuum-misestimation bias are similar.
        This is because the scale dependences are predominantly determined by C_l^̣̅κ
        and C_l^Δ_Fκ, and the primary difference between the two angular
        power spectra is the flux bias as on large scale →1. More interestingly,
        the relative contribution of the continuum-misestimation bias becomes larger at higher redshift,
        which is the result of increasing |b_F| at higher redshift.
        
        Finally, for the correlation measured in configuration space at the same angular
        position, the continuum-misestimation bias is -2P^ 1d_FF(k_s)σ^2_Δ_Fκ,
        where σ^2_Δ_Fκ is the Fourier transform of P^ 2d_Δ_Fκ(_l⊥)
        at zero lag, i.e. 
        σ^2_Δ_Fκ=∫d^2k_l⊥/(2π)^2P^ 2d_Δ_Fκ(_l⊥)  .
        
        The right panel of corr_DFkappa shows σ^2_Δ_Fκ as
        a function of redshift for various Δ. We find that as the left panel
        the continuum-misestimation bias is larger at high redshift due to the larger bias. Also,
        σ^2_Δ_Fκ is almost unchanged between Δ=280 to
        420. This is not surprising because the integration over the lensing
        kernel cancels the effect from the variance of the density perturbation
        σ^2_̣̣̅̅, as we have discussed in true_signal.
        It is, however, interesting in the sense that in a  survey the lengths
        of the forest may vary a lot, but to model the continuum-misestimation bias
        of the lensing-flux-flux bispectrum we only need one typical length for the
        forest, hence the modeling is simpler.
        
        
        
        
        § COMPARISON WITH OBSERVATION
        
        How does our prediction compare to the measurement? In this section we shall
        first discuss the contamination from damped  systems that cannot be
        removed perfectly in observation, and then compare our prediction to the
        measured correlation between the one-dimensional forest power spectrum and
        CMB lensing convergence done in Ref. <cit.>.
        
        
        
         §.§ Contamination from Damped Lyman-α systems
        
        In the  forest measurement, the largely broadened damping wings produced by
        high neutral hydrogen column density systems with N_HI>2×10^20  cm^-2,
        which is known as the damped  absorbers (DLAs), are discarded to avoid contamination
        on the forest power spectrum. However, systems such as Lyman-limit systems and sub-DLAs
        with less column density than DLAs but still higher than that of the  forest
        have less prominent features hence are difficult to remove. The presence of these
        systems may affect the forest power spectrum and so its correlation with the
        lensing convergence, and we shall quantify the effect.
        
        As pointed out in Ref. <cit.>, the dominant effect of the high
        column density systems is to add power to the forest power spectrum due to their
        random distribution. In the limit that the number of the systems are small (typically
        in one skewer one in 100-1000 pixels is in the high column density system), the
        power is proportional to the number density of such systems. As a result, in the
        presence of a large-scale fluctuation ̣̅, the leading order correction to
        the one-dimensional forest power spectrum due to the high column density systems
        is given by
        
         P^ 1d_ DLA(k_s,_⊥)=
         
            1+b_ DLẠ̅(_⊥)
        P^ 1d_ DLA(k_s)  ,
        
        where P^ 1d_ DLA and b_ DLA are the one-dimensional power
        spectrum and the bias of the high column density systems. Correlating this with
        the lensing convergence, the general three-point function containing the angular
        scale between the small-scale one-dimensional DLA power spectrum and the lensing
        convergence is
        ⟨ P^ 1d_ DLA(k_s,_l⊥)κ('_l⊥)⟩'
         =b_ DLAP^ 1d_ DLA(k_s)P^ 2d_̣̅κ(_l⊥)  .
        
        If the correlation is computed as the same angular position, then the signal
        becomes
        ⟨ P^ 1d_ DLA(k_s,_⊥)κ(_⊥)⟩
         =b_ DLAP^ 1d_ DLA(k_s)σ^2_̣̅κ .
        
        
        To evaluate this signal, we need both the bias and the one-dimensional power
        spectrum of DLAs. Ref. <cit.> has measured b_ DLA
        by cross-correlating DLAs with  forest, and we should take their central
        value b_ DLA=2.17, assuming that the redshift-space distortion parameter
        of DLAs is subdominant. For the DLA power spectrum, we use the fitting function
        provided in Ref. <cit.> describing the ratio between P^ 1d_ DLA
        and P^ 1d_FF, and we utilize our fiducial forest power spectrum. Note
        that we only consider the contributions from Lyman-limit systems and sub-DLAs
        (i.e. the red line in figure 5 of Ref. <cit.>), as the prominent
        DLAs are discarded in Ref. <cit.>.
        
        
        
         §.§ Measurement of Doux et al.
        
        In Ref. <cit.>, the first detection of the correlation between the
        one-dimensional forest power spectrum from SDSS-III Baryon Oscillation Spectroscopic
        Survey Data Release 12 <cit.> and the CMB lensing
        convergence from Planck <cit.> at the same line-of-sight is reported
        with a significance of 5 σ. The authors interpreted this signal to be the
        response of the forest power spectrum to a large-scale overdensity, and subtracting
        the contribution from the leading-order squeezed-limit matter bispectrum they
        measured the “effective nonlinear bias”, which contains the nonlinearity such
        as redshift-space distortion and nonlinear clustering of gas. We shall compare the
        measurement of ⟨ P^ 1d_FF(k_s,_⊥)κ(_⊥)⟩
        with our prediction.
        
        Let us first compute all components of the signal. The true signal is produced by
        the nonlinear gravitational evolution and given by
        dP^ 1d_FF(k_s)/ḍ̅
        σ^2_̣̅κ, where the
        dP^ 1d_FF(k_s)/ḍ̅ and σ^2_̣̅κ are taken from
        the right panel of pff1d and the right panel of corr_bardkappa,
        respectively. The continuum-misestimation bias is due to the  forest continuum
        fitting and given by -2P^ 1d_FF(k_s)σ^2_Δ_Fκ, where
        P^ 1d_FF(k_s) and σ^2_Δ_Fκ are taken from the left
        panel of pff1d and the right panel of corr_DFkappa. Finally, the DLA
        contamination is given by b_ DLAP^ 1d_ DLA(k_s)σ^2_̣̅κ,
        where b_ DLA=2.17 and P^ 1d_ DLA(k_s) is taken from the fitting
        function in Ref. <cit.>. Thus, we predict the total signal of the
        correlation between the one-dimensional forest power spectrum and the lensing convergence
        from the same line-of-sight to be
        dP^ 1d_FF(k_s)/ḍ̅σ^2_̣̅κ
         -2P^ 1d_FF(k_s)σ^2_Δ_Fκ
         +b_ DLAP^ 1d_ DLA(k_s)σ^2_̣̅κ .
        sig_tot shows the different components of the cross-correlation between
        lensing convergence and the forest power spectrum at the same angular position
        for various redshifts. We first find that the DLA contamination is only important
        on large scales, and for k_s≳0.6 the contribution becomes
        negligible. This is due to the shape of the DLA power spectrum, which can be seen
        in figure 5 of Ref. <cit.>. Next, we find that true signal dominates
        over the continuum-misestimation bias at lower redshift, but the trend reverses at
        high redshift. This is due to the increase of the  flux bias, which enhances
        both P^ 1d_FF(k_s) and σ^2_Δ_Fκ, as well as the
        decrease of the forest power spectrum response. The fact that the continuum-misestimation
        bias is larger than the true signal at z≥2.8 means that a large portion of the
        measured signal is coming from the mis-estimation of mean flux over the finite length
        of  forest skewers, which needs to be removed for unbiasedly extracting the
        three-point function due to the gravitational evolution. Interestingly, the total signal
        shows a mild redshift evolution between z=2.2 and 3.0. However, due to the lack
        of snapshots of separate universe simulations at z≥3 we cannot examine this
        correlation at higher redshift where some of the forests are observed.
        
        
        
        Since Ref. <cit.> measures the correlation from all
        redshifts between 2.1≤ z≤3.6, we shall combine the total signals
        at different redshifts in sig_tot into one line in order to
        compare with the measurement. We use the weight given in eq. (18) of
        Ref. <cit.>, assuming that the number of pixels of the
        i^ th forest is a constant and noise of the forest is zero.  We
        also include the observed  forest distribution obtained from the
        sample in Ref. <cit.>, hence the total weight is
        
         w(k,z)=N_(z)/
            P^ 1d_FF(k,z)
        ^2 ,
        
        where N_(z) is the number of  forest at z.
        The joint true and total signals from 2.2≤ z≤3 are shown
        respectively as the blue dashed and red solid lines in meas,
        whereas the measurement in Ref. <cit.> is shown as the
        black data points with error bar.  We find that the total signal is
        closer to the measurement than the true signal alone, but still
        smaller than the measurement on all scales. To better quantify the
        difference, we compute χ^2 using the full covariance matrix
        estimated in Ref. <cit.>, and the values are shown in the
        legend. As there are 12 data points and without any fitting
        parameters, the degrees of freedom is 12. While the measurement is
        consistently larger than the total signal, we still obtain a
        reasonable reduced χ^2. This is due to the high correlation
        between measurement points, which is shown in figure 5 of
        Ref. <cit.> and can be noticed by the lack of scatter
        between neighboring points. We find that the total signal gives a
        χ^2=9.1 and that neglecting the biased signal and DLA
        contamination worsens the fit to χ^2=14.19. Therefore, even in
        the current data, it is important to include new effects discussed in
        this paper. We also note that Doux et al report best-fit χ^2 of
        5.4 (which, while low, is not anomalously low with p=0.9 with 11
        degrees of freedom). Their model would roughly correspond to an
        artificial increase in bias for our true signal, but the improvement
        in fit is not significant, so we cannot exclude our model as a
        complete model for the data.  One would be able to examine this more
        critically with future surveys that have more and better  forest
        measurements.
        
        
        
        
        § CONCLUSION
        
        In this paper, we investigate the three-point function form by the cross-correlation
        between forest power spectrum and the CMB lensing convergence. We find that
        not only the gravitational evolution produces the flux-flux-lensing bispectrum,
        but the mis-estimation of the mean flux over the finite length of  forest skewers
        would also generate non-zero correlation. In particular, this systematic effect
        dominates the underlying signal at z≳2.8, which has to be taken into
        account for unbiasedly probing the gravitational effect at high redshift using
        this specific bispectrum.
        
        We demonstrate that integrating the flux-flux-lensing bispectrum with full
        angular and scale dependences over the angular information between the forest
        power spectrum and the CMB lensing is equivalently to the cross-correlation
        between the forest power spectrum and the CMB lensing at the same angular
        positions, which is measured in Ref. <cit.>. We show that our
        predictions are consistent with the signal measured by Ref. <cit.>,
        and the inclusion of the systematic effects from  forest continuum
        fitting and the DLA contamination improves the agreement. However, for
        both with and without the systematic effects, we obtain acceptable χ^2
        values. Furthermore, the reported best-fit χ^2 with one fitting parameter
        in Ref. <cit.> is low but not anomalous small.
        
        If our prediction is correct, then one interesting implication is that while
        Ref. <cit.> shows that the effective nonlinear bias, which encodes
        the excess of the forest power spectrum response with respect to that of the
        linear field, is b_2^ eff=1.16±0.53, our simulations suggest it to be
        negative as the forest power spectrum response dln P^ 1d_FF(k_s)/dδ
        is always smaller than the linear power spectrum response dln P_l(k)/dδ=68/21-(1/3)[dln k^3P_l(k)/dln k]
        for all redshifts and lines-of-sight (see figure 2 of Ref. <cit.>).
        This is likely due to the missing of the systematic term, hence to interpret
        b_2^ eff measured in Ref. <cit.> as the responses of flux
        bias, redshift-space distortion, and the nonlinear small-scale forest power
        spectrum one has to account for the continuum-misestimation bias. However,
        given the large uncertainty from current measurement, it is challenging to
        draw any concrete conclusions. With future surveys such as DESI <cit.>
        that have less uncertainty, one can more critically examine this correlation
        as well as how the forest power spectrum responds to the overdensity gravitationally.
        
         We would like to thank Cyrille Doux for providing the
        data points and the covariance matrix of the measurement in Ref. <cit.>.
        We would also like to thank Emmanuel Schaan and the referees for useful
        comments on the draft. AS acknowledges insightful conversations
        with Andreu Font-Ribera and hospitality of the University College
        London where parts of this work were performed. Results in this paper
        were obtained using the high-performance computing system at the
        Institute for Advanced Computational Science at Stony Brook
        University. CC is supported by grant NSF PHY-1620628.
