Cross-correlations of the Lyman-\alpha forest with weak lensing convergence I:Analytical Estimates of S/N and Implications for Neutrino Mass and Dark Energy

Cross-correlations of the Lyman- forest with weak lensing convergence I:
Analytical Estimates of S/N and Implications for Neutrino Mass and Dark Energy

Alberto Vallinotto avalli@fnal.gov Center for Particle Astrophysics, Fermi National Accelerator Laboratory, P.O. Box 500, Kirk Rd. & Pine St., Batavia, IL 60510-0500 USA    Matteo Viel viel@oats.inaf.it INAF - Osservatorio Astronomico di Trieste, Via G.B. Tiepolo 11, I-34131 Trieste, Italy
INFN - National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, Italy
   Sudeep Das sudeep@astro.princeton.edu Princeton University Observatory, Peyton Hall, Ivy Lane, Princeton, NJ 08544 USA
Berkeley Center for Cosmological Physics, LBNL and Department of Physics, University of California, Berkeley, CA 94720.
   David N. Spergel dns@astro.princeton.edu Princeton University Observatory, Peyton Hall, Ivy Lane, Princeton, NJ 08544 USA
July 6, 2019
Abstract

We expect a detectable correlation between two seemingly unrelated quantities: the four point function of the cosmic microwave background (CMB) and the amplitude of flux decrements in quasar (QSO) spectra. The amplitude of CMB convergence in a given direction measures the projected surface density of matter. Measurements of QSO flux decrements trace the small-scale distribution of gas along a given line-of-sight. While the cross-correlation between these two measurements is small for a single line-of-sight, upcoming large surveys should enable its detection. This paper presents analytical estimates for the signal to noise (S/N) for measurements of the cross-correlation between the flux decrement and the convergence, , and for measurements of the cross-correlation between the variance in flux decrement and the convergence, . For the ongoing BOSS (SDSS III) and Planck surveys, we estimate an S/N of 30 and 9.6 for these two correlations. For the proposed BigBOSS and ACTPOL surveys, we estimate an S/N of 130 and 50 respectively. Since , the amplitude of these cross-correlations can potentially be used to measure the amplitude of at to 2.5% with BOSS and Planck and even better with future data sets. These measurements have the potential to test alternative theories for dark energy and to constrain the mass of the neutrino. The large potential signal estimated in our analytical calculations motivate tests with non-linear hydrodynamical simulations and analyses of upcoming data sets.

pacs:
98.62.Ra, 98.70.Vc, 95.30.Sf
preprint: FERMILAB-PUB-09-508-T

I Introduction

The confluence of high resolution Cosmic Microwave Background (CMB) experiments and large-scale spectroscopic surveys in the near future is expected to sharpen our view of the Universe. Arcminute scale CMB experiments such as Planck PLA (), the Atacama Cosmology Telescope ACT (); Hincks et al. (2009), the South Pole Telescope SPT (); Staniszewski et al. (2009), QUIET QUI () and PolarBeaR POL (), will chart out the small scale anisotropies in the CMB. This will shed new light on the primordial physics of inflation, as well as the astrophysics of the low redshift Universe through the signatures of the interactions of the CMB photons with large scale structure. Spectroscopic surveys like BOSS McDonald et al. (2005); Seljak et al. (2005) and BigBOSS Schlegel et al. (2009) will trace the large scale structure of neutral gas, probing the distribution and dynamics of matter in the Universe. While these two datasets will be rich on their own, they will also complement and constrain each other. An interesting avenue for using the two datasets would be to utilize the fact that the arcminute-scale secondary anisotropies in the CMB are signatures of the same large scale structure that is traced by the spectroscopic surveys, and study them in cross-correlation with each other. In this paper, we present the analytic estimates for one such cross correlation candidate - that between the gravitational lensing of the CMB and the flux fluctuations in the Lyman- forest.

The gravitational lensing of the CMB, or CMB lensing in short, is caused by the deflection of the CMB photons by the large scale structure potentials (see Lewis and Challinor, 2006, for a review). On large scales, WMAP measurements imply that the primoridal CMB is well described as an isotropic Gaussian random field Komatsu et al. (2009). On small scales, lensing breaks this isotropy and introduces a specific form of non-Gaussianity. These properties of the lensed CMB sky can be used to construct estimators of the deflection field that lensed the CMB. Therefore, CMB lensing provides us with a way of reconstructing a line-of-sight (los) projected density field from zero redshift to the last scattering surface, with a broad geometrical weighting kernel that gets most of its contribution from the range Hu and Okamoto (2002); Hirata and Seljak (2003); Yoo and Zaldarriaga (2008). While CMB lensing is mainly sensitive to the geometry and large scale projected density fluctuations, the Lyman- forest, the absorption in quasar (QSO) spectra caused by intervening neutral hydrogen in the intergalactic medium, primarily traces the small-scale distribution of gas (and hence, also matter) along the line of sight.

A cross-correlation between these two effects gives us a unique way to study how small scale fluctuations in the density field evolve on top of large scale over and under-densities, and how gas traces the underlying dark matter. This signal is therefore a useful tool to test to what extent the fluctuations in the Lyman- flux relate to the underlying dark matter. Once that relationship is understood, it can also become a powerful probe of the growth of structure on a wide range of scales. Since both massive neutrinos and dark energy alter the growth rate of structure at , these measurements can probe their effects. This new cross-correlation signal, should also be compared with other existing cross-correlations between CMB and LSS that have already been observed and that are sensitive to different redshift regimes Peiris and Spergel (2000); Giannantonio et al. (2008); Hirata et al. (2008); Croft et al. (2006); Xia et al. (2009).

In this work, we build an analytic framework based on simplifying assumptions to estimate the cross-correlation of the first two moments of the Lyman- flux fluctuation with the weak lensing convergence , obtained from CMB lensing reconstruction, measured along the same line of sight. The finite resolution of the spectrogram limits the range of parallel -modes probed by the absorption spectra and the finite resolution of the CMB experiments limits the range of perpendicular -modes probed by the convergence measurements. These two effect break the spherical symmetry of the -space integration. However, we show that by resorting to a power series expansion it is still possible to obtain computationally efficient expressions for the evaluation of the signal.

We then investigate the detectability of the signal in upcoming CMB and LSS surveys, and the extent to which such a signal can be used as a probe of neutrino masses and early dark energy scenarios. A highlight of our results is that the estimated cross-correlation signal seems to have significant sensitivity to the normalization of the matter power spectrum . Consistency with CMB measurements – linking power spectrum normalization and the sum of the neutrino masses – allows to use this cross-correlation to put additional constrain on the latter.

The structure of the paper is as follows. In Section II we introduce the two physical observables, the Lyman- flux and the CMB convergence (II.1), the cross-correlation estimators (II.2) and their variances (II.3). Our main result is presented in section II.4 where the signal-to-noise ratios are computed. Section II.5 contains a spectral analysis of the observables that aims at finding the Lyman- wavenumbers that contribute most to such a signal. We focus on two cosmologically relevant applications in sections III.1 and III.2, for massive neutrinos and early dark energy models, respectively. We conclude with a discussion in section IV.

Ii Analytical Results

ii.1 Physical Observables

Fluctuations in the Lyman- flux

Using the fluctuating Gunn–Peterson approximation Gunn and Peterson (1965), the transmitted flux along a los is related to the density fluctuations of the intergalactic medium (IGM) by

(1)

where and are two functions relating the flux fluctuation to the dark matter overdensities. These two functions depend on the redshift considered: is of order unity and is related to the mean flux level, baryon fraction, IGM temperature, cosmological parameters and the photoionization rate of hydrogen. A good approximation for its redshift dependence is (see Kim et al. (2007)). on the other hand depends on to the so-called IGM temperature-density relation and in particular on the power-law index of this relation (e.g. Hui and Gnedin (1997); McDonald (2003)) and should be less dependent on redshift (unless temperature fluctuations due for example to reionization play a role, see McQuinn et al. (2009)). For the calculation of signal/noise in the paper, we neglect the evolution of and with redshift. While the value of the correlators considered will depend on and , their signal-to-noise (S/N) ratio will not.

On scales larger than about (comoving), which is about the Jeans length at , the relative fluctuations in the Lyman- flux are proportional to the fluctuations in the IGM density field Bi and Davidsen (1997); Croft et al. (1998, 2002); Viel et al. (2002); Saitta et al. (2008). We assume that the IGM traces the dark matter on large scales,

(2)

The (variance of the) flux fluctuation in the redshift range covered by the Lyman- spectrum is then proportional to (the variance of) the fluctuations in dark matter

(3)

where the range of comoving distances probed by the Lyman- spectrum extends from to . The case corresponds to the fluctuations in the flux and the case corresponds to their variance. We stress that the above approximation is valid in linear theory neglecting not only the non-linearities produced by gravitational collapse but also those introduced by the definition of the flux and those produced by the thermal broadening and peculiar velocities. Note that while the assumption of “tracing” between gas and dark matter distribution above the Jeans length is expected in the standard linear perturbation theory Eisenstein and Hu (1998), the one between the flux and the matter has been verified a-posteriori using semi-analytical methods (Bi and Davidsen (1997); Zaroubi et al. (2006)) and numerical simulations (Gnedin and Hui (1998); Croft et al. (1998); Viel et al. (2006)) that successfully reproduce most of the observed Lyman- properties. Furthemore, non-gravitational processes such as temperature and/or ultra-violet fluctuations in the IGM should alter the Lyman- forest flux power and correlations in a distinct way as compared to the gravitational instability process and to linear evolution (e.g. Fang and White (2004); Croft (2004); Slosar et al. (2009)).

Cosmic Microwave Background convergence field

The effective weak lensing convergence measured along a los in the direction is proportional to the dark matter overdensity through

(4)

where the integral along the los extends up to a comoving distance and where is the lensing window function. In what follows we consider the cross-correlation of Lyman- spectra with the convergence field measured from the CMB, as in Vallinotto et al. Vallinotto et al. (2009), in which case is the comoving distance to the last scattering surface. Note however that it is straightforward to extend the present treatment to consider the cross-correlation of the Lyman- flux fluctuations with convergence maps constructed from other data sets, like optical galaxy surveys.

It is necessary to stress here that Eq. (1) above depends on the density fluctuations in the IGM, which in principle are distinct from the ones in the dark matter, whereas depends on the dark matter overdensities . If the IGM and dark matter overdensity fields were completely independent, the cross-correlation between them would inevitably yield zero. If however the fluctuations in the IGM and in the dark matter are related to one another, then cross-correlating and will yield a non-zero result. The measurement of these cross-correlations tests whether the IGM is tracing the underlying dark matter field and quantifies the bias between flux and matter.

ii.2 The Correlators

Physical Interpretation

The two correlators and have substantially different physical meaning: is proportional to the over(under)density integrated along the los and is dominated by long wavelength modes with . Intuitively therefore measures whether a specific los is probing an overall over(under)dense region. If the IGM traces the dark matter field, then by Eq. (3) is expected to measure the dark matter overdensity along the same los extending over the redshift range spanned by the QSO spectrum. This implies that

  • quantifies whether and how much the overdensities traced by the Lyman- flux contribute to the overall overdensity measured all the way to the last scattering surface. Because both and are proportional to , it is reasonable to expect that this correlator will be dominated by modes with wavelengths of the order of hundreds of comoving Mpc. As such, this correlator may be difficult to measure as it may be more sensitive to the calibration of the Lyman- forest continuum.

  • measures the relationship between long wavelength modes in the density and the amplitude of the variance of the flux. The variance on small scales and the amplitude of fluctuations on large-scales are not coupled in linear theory. However, in non-linear gravitational theory regions of higher mean density have higher matter fluctuations. These lead to higher amplitude fluctuations in flux Zaldarriaga et al. (2001). Since is sensitive to this interplay between long and short wavelength modes, this correlator is much more sensitive than to the structure growth rate. Furthermore, because is sensitive to short wavelengths, this signal is dominated by modes with shorter wavelength than the ones dominating . As such, this signal should be less sensitive to the fitting of the continuum of the Lyman- forest.

Tree level approximation

In what follows we focus on obtaining analytic expressions for the correlations between the (variance of the) flux fluctuations in the Lyman- spectrum and the CMB convergence measured along the same los. From Eqs. (3, 4) above it is straightforward to obtain the general expression for the signal

(5)

Since the QSOs used to measure the Lyman- forest lie at , it is reasonable to expect that non-linearities induced by gravitational collapse will not have a large impact on the final results. In the following we therefore calculate the and correlators at tree-level in cosmological perturbation theory. While beyond the scope of the current calculation, we could include the effects of non-linearities induced by gravitational collapse by applying the Hyperextended Perturbation Theory of Ref. Scoccimarro and Couchman (2001) to the terms in Eq. (II.2).

At tree level in perturbation theory the redshift dependence of the matter power spectrum factorizes into , where denotes the zero-redshift linear power spectrum and the growth factor at comoving distance . Furthermore, the correlator appearing in the integrand of Eq. (II.2) depends on the separation between the two points running on the los and in general it will be significantly non-zero only when . Also, at tree level in perturbation theory these correlators carry factors of .111Notice in fact that even though in the case it would be reasonable to expect three factors of , the first non-zero contribution to the three-point function carries four factors of because the gaussian term vanishes exactly. Using the approximation

(6)
(7)
(8)

we can then write and trade the double integration (over and ) for the product of two single integrations over and . Equation (II.2) factorizes into

(9)

This is the expression used to evaluate the signal. The determination of an expression for and of an efficient way for evaluating it is the focus of the rest of the section.

Window Functions

The experiments that measure the convergence and the flux fluctuations have finite resolutions. We approximate the effective window functions of these experiments by analytically tractable Gaussian function.

These two window functions act differently: the finite resolution of the CMB convergence measurements limits the accessible range of modes perpendicular to the los, , and the finite resolution of the Lyman- spectrum limits the range of accessible modes parallel to the los. This separation of the modes into the ones parallel and perpendicular to the los is intrinsically dictated by the nature of the observables and it cannot be avoided once the finite resolution of the various observational campaigns is taken into account. Because of this symmetry, the calculation is most transparent in cylindrical coordinates: .

The high- (short wavelength) cutoff scales for the CMB and Lyman- modes are denoted by and respectively. Furthermore, we also add a low- (long wavelength) cutoff for the Lyman- forest, to take into account the fact that wavelengths longer than the spectrum will appear in the spectrum itself as a background. We denote this low- cutoff by . After defining the auxiliary quantities

(10)
(11)

the window functions acting on the Lyman- and on the CMB modes, denoted respectively by and , are defined through

(12)
(13)

where the direction dependence of the two window functions has been made explicit.

We determine the values of the cutoff scales as follows. For the Lyman- forest, we consider the limitations imposed by the spectrograph, adopting the two cutoff scales and according to the observational specifications. For the reconstruction of the CMB convergence map we compute the minimum variance lensing reconstruction noise following Hu and Okamoto Hu and Okamoto (2002). We then identify the multipole , where the signal power spectrum equals the noise power spectrum for the reconstructed deflection field (for the noise is higher than the signal). Finally, we translate the angular cutoff into a 3-D Fourier mode at the relevant redshift so to keep only modes with in the calculation. Note that if we had used the shape of the noise curve instead of this Gaussian cutoff, we would have effectively retained more Fourier modes, thereby increasing the signal. However, to keep the calculations simple and conservative we use the above Gaussian window. In what follows, we will present results for convergence map reconstructions from the datasets of two CMB experiments: Planck and an hypothetical CMB polarization experiment based on a proposed new camera for the Atacama Cosmology Telescope (ACTPOL). For the former, we adopt the sensitivity values of the 9 frequency channels from the Blue Book AA. (2006). For the latter we assume a hypothetical polarization based CMB experiment with a arcmin beam and detectors, each having a noise-equivalent-temperature (NET) of 300 K- over sq. deg., with an integration time of seconds. We further assume that both experiments will completely cover the sq. deg footprint of BOSS.

Figure 1: Absolute value of the correlator along a single line-of-sight as a function of the source redshift and of the length of the measured spectrum , for convergence maps recontructed from Planck (left panel, ) and ACTPOL (right panel, ). The value of the resolution of the QSO spectrum is the one predicted for SDSS-III, . To make the physics of structure formation apparent, we turn off the IGM physics by setting (it is straightforward to rescale the values of the correlator to reflect different values of and ).

Auxiliary Functions

Because the calculation has cylindrical rather than spherical symmetry, the evaluation of the correlators of Eq. (II.2) is more complicated, particularly for . As shown in the appendix, it is possible to step around this complication and to obtain results that are computationally efficient with the adoption of a few auxiliary functions that allow the integrations in -space to be carried out in two steps, first integrating on the modes perpendicular to the los, and subsequently on the ones parallel to the los. The perturbative results for the correlators are expressed as combinations of the following auxiliary functions:

(14)
(15)
(16)
(17)

Equations (14) and (15) above represent an intermediate step, where the integration on the modes perpendicular to the los is carried out. Equations (16) and (17) are then used to carry out the remaining integration over the modes that are parallel to the los.

The symmetry properties of the auxiliary functions are as follows. The functions are real and even in regardless of the actual value of . This in turn implies that are real and even (imaginary and odd) in when is even (odd). Furthermore, the coefficients are real and non-zero only if is even, thus ensuring that is always real-valued.

Figure 2: Absolute value of the correlator along a single line-of-sight as a function of the source redshift and of the length of the measured spectrum , for convergence maps recontructed from Planck (left panel, ) and ACTPOL (right panel, ). The value of the resolution of the QSO spectrum is the one predicted for SDSS-III, . As before, we set to make the physics of structure formation apparent.

The correlator

In the case it is straightforward to identify with a two point correlation function measured along the los. However, the intrinsic geometry of the problem and the inclusion of the window functions leads to evaluate this correlation function in a way that is different from the usual case, where the spherical symmetry in -space can be exploited. In the present case we have

(18)

It is then straightforward to plug Eq. (18) into Eq. (II.2) to obtain .222We checked that in the limit where , and the usual two point correlation function is recovered. Whereas one would naively expect that letting and would lead to recover the usual two point function calculated exploiting spherical symmetry in -space with a cutoff scale equal to the common , this is actually not the case. The reason for this is that the volume of - space over which the integration is carried out is different for the two choices of coordinate systems. In particular, the spherical case always includes fewer modes than the cylindrical one. The two results therefore coincide only in the limit. In Fig. 1 we show the absolute value of the cross-correlation of the convergence of the CMB with the Lyman- flux fluctuations observed for a quasar located at redshift and whose spectrum spans a range of redshift . The cosmological model used (and assumed throughout this work) is a flat CDM universe with , and consistent with the WMAP-5 cosmology Komatsu et al. (2009). The left and right panel show the results for the resolution of Planck and of the proposed ACTPOL experiment. We artificially set , effectively “turning off” the physics of IGM: this choice is not dictated by any physical argument but from the fact that it makes apparent the dynamics of structure formation.

The behavior of shown in Fig. 1 makes physical sense. Recall that this correlator is sensitive to the overdensity integrated along the redshift interval (spanned by the QSO spectrum) that contributes to the CMB convergence. It then increases almost linearly with the length of the QSO spectrum . It also increases if the resolution of the CMB experiment is increased. An increased value of corresponds to a longer Lyman- spectrum, carrying a larger amount of information and thus leading to a larger correlation. Similarly, an increased value of corresponds to a higher resolution of the reconstructed convergence map and therefore more modes – and information – being included in the correlation. Deepening the source’s redshift (while keeping and fixed) on the other hand results in a decrease in . This fact is related to the growth of structure: the spectrum of a higher redshift QSO is probing regions where structure is less clumpy and therefore the absolute value of the correlation is smaller. Finally, once the redshift dependence of is turned on ( is only mildly redshift dependent) the above result change, leading to a final signal that is increasing with redshift.

We stress here that values of the correlators will be different when and are different from unity. Ultimately these values should be recovered from a full non-linear study based on large scale-high resolution hydrodynamical simulations. However, numerical studies based on hydrodynamical simulations have shown convincingly that for both the flux power spectrum (2-pt function) and flux bispectrum (3-pt function) the shape is very similar to the matter power and bispectrum, while the amplitude is usually matched for values of and that are different from linear predictions (see discussion in Viel et al. (2004)). In this framework, non-linear hydrodynamical simulations should at the end provide the “effective” values for and that will match the observed correlators and our results can be recasted in terms of these new parameters in a straightforward way.

The correlator

The case, where the variance of the flux fluctuation integrated along the los is cross-correlated with , is more involved. Looking back at Eqs. (II.2, II.2) it is possible to realize that the cumulant correlator corresponds to a collapsed three-point correlation function, as two of the ’s refer to the same physical point. The evaluation of is complicated by the introduction of the window functions and . For sake of clarity, we report here only the final results at tree level in cosmological perturbation theory, relegating the lengthy derivation to the appendix. Letting

(19)

and using the auxiliary functions defined in Eqs. (14-17) above, it is possible to obtain the following series solution

(20)
(21)

In Fig. 2 we show the result obtained using the tree level expression for , Eqs. (19-21). As before, we focus on the physics of structure formation and we turn off the IGM physics by setting . First, it is necessary to keep in mind that is sensitive to the interplay of long and short wavelength modes and it probes the enhanced growth of short wavelength overdensities that lie in an environment characterized by long wavelength overdensities. The behavior of with respect to and is similar to that of : it increases if is increased or if the QSO redshift is decreased. However, the effect of the growth of structure is in this case stronger than in the previous case. This does not come as a surprise, as the growth of structure acts coherently in two ways on . Since in a CDM model all modes grow at the same rate, a lower redshift for the source QSO implies larger overdensities on large scales which in turn enhance even further the growth of overdensities on small scales. Thus by lowering the source’s redshift two factor play together to enhance the signal: first the fact that long and short wavelength modes have both grown independently, and second the fact that being coupled larger long-wavelength modes boost the growth of short wavelength modes by a larger amount. This dependence is also made explicit in Eq. (II.2), where we note that depends on four powers of the growth factor. Finally, as before, the higher the resolution of the CMB experiment the larger is . This too makes physical sense, as a larger resolution leads to more modes contributing to the signal and therefore to a larger cross-correlation.

ii.3 Variance of correlators

To assess whether the correlations between fluctuations in the flux and convergence are detectable we need to estimate the signal-to-noise ratio, which in turn requires the evaluation of the noise associated with the above observable. As mentioned above, both instrumental noise and cosmic variance are considered. We then move to estimate the variance of our correlator

(22)

Since is just the square of the signal, we aim here to obtain estimates for . From Eq. (II.2), we get:

(23)

where there are now two integrals running along the convergence los (on and ) and two running along the Lyman- spectrum (on and ). The correlator appearing in the integrand of Eq. (23) is characterized by an even () number of factors. This implies that an approximation to its value can be obtained using Wick’s theorem. When Wick’s theorem is applied, many different terms will in general appear. Adopting for sake of brevity the notation , terms characterized by the contraction of and will receive non-negligible contributions over the overlap of the respective los. The terms providing the largest contribution to are the ones where is contracted with : these terms in fact contain the value of the cosmic variance of the convergence and receive significant contributions from all points along the los from the observer all the way to the last scattering surface. On the other hand, whenever we consider the cross-correlation between a and a , this will acquire a non-negligible value only for those set of points where the los to the last scattering surface overlaps with the Lyman- spectrum. As such, these terms are only proportional to the length of the Lyman- spectrum, and thus sensibly smaller than the ones containing the variance of the convergence. We note in passing that the same argument should also apply to the connected part of the correlator, which should be significantly non-zero only along the Lyman- spectrum. Mathematically, these facts become apparent from Eq. (23) above, where terms containing are the only ones for which the integration over and can be traded for an integration over and an integration over that extends all the way to . If on the other hand is contracted with a factor, then the approximation scheme of Eqs. (6-8) leads to an integral over and to an integral over that extends only over the length probed by the Lyman- spectrum. It seems therefore possible to safely neglect terms where the ’s referring to the convergence are not contracted with each other.

The variance of

We start by considering the variance of . Setting in Eq.  (23) and using Wick’s theorem we obtain

(24)

We notice immediately that the first term is twice the square of , while the second term is proportional to two correlation function characterized by cutoffs acting either on the modes that are parallel or perpendicular to the los, but not on both. It is then possible to show that

(25)
(26)
(27)
(28)

where the last two equations have been added here for sake of completeness, as they will be useful in what follows. The variance of is then

(29)

In the upper panels of Fig. 3 we show the values obtained for the standard deviation of for two different CMB experiments’ resolution, again turning off the IGM physics evolution and focusing on the growth of structure.

Figure 3: Estimates of the standard deviation of the correlator (upper panels) and (lower panels) along a single line-of-sight as a function of the source redshift and of the length of the measured spectrum , for convergence maps recontructed from Planck (left panels) and ACTPOL (right panels). As before, we set , effectively turning off the physics of IGM, to make apparent the physics of structure formation.

The variance of

Setting in Eq. (23), we then apply Wick’s theorem to . Neglecting again terms where the ’s are not contracted with one another, we obtain

(30)

which then leads to the expression for

(31)

In the lower panels of Fig. 3 we show the estimates for the standard deviation along a single line-of-sight for the two different CMB experiment. We note in Fig. 3 the same trends that have been pointed out for the correlator itself in Fig. 1 and 2: the standard deviation of and of increase almost linearly with increasing length of the Lyman- spectrum and it decreases as the source redshift is increased because of the fact that the spectrum probes regions that are less clumpy. Also, by increasing the resolution of the CMB experiment used to reconstruct the convergence map, the deviation of and also increase: if on one hand more modes carry more information, on the other hand they also carry more cosmic variance.

One last aspect to note here is that while the signal for arises from a three point correlation function (which in the gaussian approximation would yield zero), the dominant terms contributing to its variance arise from products of two point correlation functions. In particular, it is possible to show that the terms appearing in the second line of Eq. (30) significantly outweight the square of the signal that appears in the first line.

Figure 4: Estimates for the signal-to-noise ratios for the observation of the correlators along a single line-of-sight as a function of the source redshift and of the length of the measured spectrum , for Planck (left panels) and ACTPOL (right panels). As long as the functions and can be assumed to be constant in the redshift range spanned by the Lyman- spectrum, these result do not depend on the specific value taken by the latter.

ii.4 Signal-to-Noise ratio

We now have all the pieces to assess to what extent the correlations will be detectable by future observational programs. Even before moving to plot the S/N ratios for and it is possible to point out a couple of features of these ratios. First, we note that the S/N ratio for and do present a radical difference in their dependence on the QSO source redshift. This is because the signal for is characterized by mode coupling, whereas the dominant contributions to the variance are not. Physically, the signal for is more sensitive to the growth of structure with respect to its variance: while for the former the growth of long wavelength modes enhances the growth of structure on small scales, for the latter long and short wavelength modes grow independently at the same rate. Mathematically, this is apparent when comparing Eq. (II.2) with Eq. (31): while the signal carries four powers of the growth factor, the dominant terms contributing to its variance carry only six. In this case then the S/N is characterized by four growth factors in the numerator and only three in the denominator, thus leading to a “linear” dependence of S/N on the redshift (modulo integration over the los and behaviour of the lensing window function). Note that this is in stark contrast with the case, where the signal is not characterized by mode coupling and the number of growth factors are equal for the signal and its standard deviation, thus leading to a S/N ratio with no dependence on the source’s redshift.

Second, we note that S/N does not depend on the value of any constant. In particular, regardless of their redshift dependence, the S/N ratio will not depend on the functions and used to describe the IGM. This is of course very important since in such a way, at least in linear theory and using the FGPA at first order, the dependence on the physics of the IGM cancels out when computing the S/N ratio.

In Fig. 4 we show the estimates for the S/N per los of the (upper panels) and (lower panels) measurements. As expected, while the S/N for does not show any strong redshift dependence, the S/N for decreases linearly with increasing source redshift: the growth of structure is indeed playing a role and shows that QSOs lying at lower redshift will yield a larger S/N. Also, in both cases an increase in the resolution of the experiment measuring the convergence field translates in a larger S/N and in a larger derivative of the S/N with respect to . This is not surprising, as it is reasonable to expect that a higher resolution convergence map will be carrying a larger amount of information about the density field.

All this suggests that depending on what is the correlator that one is interested in measuring, different strategies should be pursued. In case of increasing the length of the spectra will provide a better S/N. In case of , however, Fig. (4) suggests that an increase in the number of quasar will be more effective in producing a large S/N, whereas an increase in the redshift range spanned by the spectrum will increase the S/N only marginally.

Having obtained the S/N per los, we can then estimate the total S/N that will be obtained by cross-correlating the BOSS sample ( QSOs) and the proposed BigBOSS sample Schlegel et al. (2009) ( QSOs) with the convergence map measured by Planck or by the proposed ACTPOL experiment considered. Assuming a mean QSO redshift of and a mean Lyman- spectrum length of , a rough estimate of the S/N for the measurements of and of are given in Tab. 1 and 2.

CMB Exp. S/N Total S/N Total S/N
per los in BOSS in BigBOSS
Planck 0.075 30 75
ACTPOL 0.130 52 130
Table 1: Estimates of the total and per single los signal–to–noise (S/N) of the cross–correlation for different CMB experiments combined with BOSS and BigBOSS.

It is necessary to point out here that despite that the value of the S/N for is almost three times larger than the one for , the actual measurement of the former correlator strongly depends on the ability of fitting the continuum of the Lyman- spectrum. The correlator, on the other hand, is sensitive to the interplay between long and short wavelength modes and as such should be less sensitive to the continuum fitting procedure. Therefore, even if it is characterized by a lower S/N, it may actually be the easier to measure in practice. The numbers obtained above are particularly encouraging since the S/N values are typically very large and well above unity.

CMB Exp. S/N Total S/N Total S/N
per los in BOSS in BigBOSS
Planck 0.024 9.6 24
ACTPOL 0.05 20.0 50
Table 2: Estimates of the total and per single los signal–to–noise (S/N) of the cross–correlation for different CMB experiments combined with BOSS and BigBOSS.

ii.5 Analysis

Having developed a calculation framework for estimating and the S/N for their measurement, we turn to estimate what is the range of Lyman- wavelengths contributing to the signal and what is the effect of changing the parameters that control the experiments’ resolution.

Spectral Analysis

We investigate here how the different Lyman- modes contribute to the correlators. This should tell us whether long wavelength modes have any appreciable effect on our observables and what is the impact of short and very short wavelength modes (in particular the ones that are expected to have entered the non-linear regime).

Since the mean flux appearing in the definition of the flux fluctuation is a global quantity which is usually estimated from a statistically significant sample of high resolution QSO spectra (see the discussion in Seljak et al. (2003) for the impact that such quantity has on some derived cosmological parameters), is sensitive also to modes with wavelengths longer than the Lyman- spectrum. These modes appear as a “background” in each spectra but they still have to be accounted for when crosscorrelating with because the fluctuation in the flux is affected by them. More specifically, a QSO that is sitting in an overdense region that extends beyond the redshift range spanned by its spectrum will see its flux decremented by a factor that in its spectrum will appear as constant decrement. On the other hand, if the QSO spectrum extends beyond the edge of such overdensity, this mode would appear as a fluctuation (and not as a background) in the spectrum. This extreme scenario is somewhat mitigated by the fact that present and future QSO surveys will have many QSOs with los separated by few comoving Mpc Schlegel et al. (2009): as such, fluxes from neighboring QSO lying in large overdense regions should present similarities that should in principle allow to detect such large overdensities in 3D tomographical studies Saitta et al. (2008).

To measure the contributions of the different modes to the correlators, we vary and to build appropriate filters. As can be seen from Fig. 5, where three such filters are plotted for , and , the gaussian functional form assumed for the window function does not provide very sharp filters (hence this spectral analysis will not reach high resolution). Also, if then the filters add exactly to one. This allows us to measure the contributions of the different wavenumber decades to the correlators and its standard deviation.

Figure 5: Three filters used to calculate the contribution of the different modes to the correlators, their variance and the SN ratio. The filters have (solid curve), (dotted curve) and (dashed curve). Also shown is the sum of the filters (red dashed-dotted curve).
Ratio
1.00e-04 1.00e-03 1.66e-04 1.77e-04 9.39e-01
1.00e-03 1.00e-02 1.20e-03 1.21e-03 9.87e-01
1.00e-02 1.00e-01 2.12e-04 6.29e-04 3.37e-01
1.00e-01 1.00e+00 6.11e-07 1.42e-03 4.30e-04
1.00e+00 1.00e+01 7.26e-08 2.44e-03 2.97e-06
Table 3: Contribution of the different wavenumbers (split over decades) to the absolute value of the correlator , its standard deviation and ratio of the two quantities. In this calculation we took into account the evolution of with redshift.

Table 3 and 4 summarize the results for and respectively. Considering we note immediately that the signal and the S/N ratio both peaks around , as expected from the fact that this signal is proportional to the two point correlation function, which in turn receives its largest contribution from the wavelengths that dominate the power spectrum: isolating the long wavelength modes of the Lyman- flux would allow to increase the S/N. However, this procedure is sensibly complicated by the continuum fitting procedures that are needed to correctly reproduce the long wavelength fluctuations of the Lyman- flux. The behavior of the variance is interesting, as in the first three decades shows an oscillating behavior. This is due to the different weights of the two terms appearing in Eq. (29) for each range of wavelengths. In particular, for the variance of is dominated by the first term, that is just the square of the signal. However, as the signal gets smaller with increasing , for it is the second term that dominates the variance.

Ratio
1.00e-04 1.00e-03 1.08e-04 2.18e-02 4.99e-03
1.00e-03 1.00e-02 6.69e-03 1.96e-01 3.40e-02
1.00e-02 1.00e-01 5.92e-02 1.31e+00 4.52e-02
1.00e-01 1.00e+00 3.39e-01 7.06e+00 4.80e-02
1.00e+00 1.00e+01 9.92e-01 2.07e+01 4.79e-02
Table 4: Contribution of the different wavenumbers (split over decades) to the correlator , its standard deviation and ratio of the two quantities. In this calculation we took into account the evolution of with redshift.
Figure 6: Value of (left panel, red dashed contours), of its standard deviation (left panel, black solid contour) and of its S/N ratio (right) for a single QSO lying at and whose spectrum covers . Here we assume and .

Regarding , it is necessary to point out two aspects. First, short wavelengths (high-) modes provide the larger contribution to both the correlator and its standard deviation. Second, for the ratio of the contribution to the correlator and to its standard deviation remain almost constant. This means that above the different frequency ranges contribute roughly in the same proportion. This fact is both good news and bad news at the same time. It is bad news because it means that increasing the resolution of the Lyman- spectra does not automatically translate into increasing the precision with which the correlator will be measured, as the high- modes that are introduced will boost both the correlator and its variance in the same way. On the other, this appears also to be good news because it tells us that low resolution spectra which do not record non-linearities on small scales can be successfully used to measure this correlation. To increase the S/N ratio and to achieve a better precision for this measurement it is better to increase the number of QSO spectra than to increase the resolution of each single spectra. Finally, cutting off the long-wavelength modes with should not have a great impact on the S/N ratio or on the measured value of the correlator: if on one hand the contribution of the modes with are noisier due to cosmic variance, on the other hand the absolute value of such contributions to the correlator and to its variance are negligible compared to the ones arising from