On the baryon acoustic oscillation amplitude as a probe of radiation density
The baryon acoustic oscillation (BAO) feature in the distribution of galaxies has been widely studied as an excellent standard ruler for probing cosmic distances and expansion history, and hence dark energy. In contrast, the amplitude of the BAO feature has received relatively little study, mainly due to limited signal-to-noise, and complications due to galaxy biasing, effects of non-linear clustering and dependence on several cosmological parameters. As expected, the amplitude of the BAO feature is sensitive to the cosmic baryon fraction: for standard radiation content, the cosmic microwave background (CMB) acoustic peaks constrain this precisely and the BAO amplitude is largely a redundant cross-check. However, the CMB mainly constrains the redshift of matter-radiation equality, , and the baryon/photon ratio: if a non-standard radiation density is allowed, increasing while matching the CMB peaks leads to a reduced baryon fraction and a lower relative BAO amplitude. We construct an observable for the relative area of the BAO feature from the galaxy correlation function (Eq. 8); from linear-theory models, we find that this is mainly sensitive to and quite insensitive to other cosmological parameters. More detailed work from N-body simulations will be needed to constrain the effects of non-linearity and scale-dependent galaxy bias on this observable.
keywords:cosmic background radiation – cosmological parameters – cosmology:theory – large-scale structure of Universe.
The detection of baryon acoustic oscillations (BAOs) in the large-scale distribution of galaxies in both the SDSS (Eisenstein et al, 2005) and 2dFGRS (Cole et al, 2005) redshift surveys was a major milestone for cosmology, strongly supporting the standard paradigm for structure formation based on gravitational instability including cold (or warm) dark matter. Recently, there have been several new independent measurements of the BAO feature in galaxy redshift surveys, e.g. from SDSS-DR8 (Percival et al, 2010), WiggleZ (Blake et al, 2011), 6dFGRS (Beutler et al, 2011), an angular measurement from SDSS-DR9 (Seo et al, 2012), and a first measurement from BOSS (Anderson et al, 2012), which are all consistent with the concordance CDM model at the few-percent level.
The BAO feature in galaxy clustering (Peebles & Yu, 1970; Bond & Efstathiou, 1984; Eisenstein & Hu, 1998; Meiksin, White & Peacock, 1999) has a very similar origin to the acoustic peaks in the cosmic microwave background (CMB) temperature power spectrum. Most recent attention has focused on the length-scale of the BAO feature, used as a standard ruler to measure cosmic distances in units of the sound horizon at the baryon drag epoch. Many theoretical and computational studies (Seo et al, 2008, 2010) have concluded that the comoving length-scale of the BAO feature evolves by between the CMB era and the recent past due to the non-linear growth of structure, but this shift can be corrected down to the level using reconstruction methods (Eisenstein et al, 2007; Padmanabhan et al, 2012). Therefore, the BAO feature is probably the best-understood standard ruler in the moderate-redshift universe, and when combined with CMB observations it offers great power for probing the cosmic expansion history and therefore the properties of dark energy (Weinberg et al, 2013). These BAO distance measurements are complementary to those from type-Ia supernovae, have potentially smaller systematic errors, and can offer direct information on the time-variable without differentiation. On the downside, cosmic variance sets a floor on the BAO precision in the low-redshift universe, percent at and worsening below this (Seo & Eisenstein, 2007) .
However, in this paper we look at a different property, specifically the overall amplitude of the BAO feature, rather than the length-scale. As expected, the amplitude is mainly sensitive to the cosmic baryon fraction (relative to total matter), . Until now, the BAO amplitude has received much less attention than the length-scale, for two main reasons: firstly, recent CMB results from WMAP (Hinshaw et al, 2012), SPT (Keisler et al, 2011; Story et al, 2013) and ACT (Sievers et al, 2013) measure the baryon fraction to around 4 percent relative precision (given standard assumptions), while the strongest detection of the BAO peak (Anderson et al, 2012) gives significance or error in amplitude. Secondly, complications due to galaxy bias, the non-linear growth of structure, redshift-space distortions and the uncertain global shape of the power spectrum make it challenging to extract the baryon fraction from the BAO feature, even with very large future redshift surveys.
However, we note that parameter estimates from the CMB are subject to a significant degeneracy between and the total radiation density in the CMB era, usually parametrized by an effective number of neutrino species . Recent reviews of are given by e.g. Riemer-Sorensen, Parkinson & Davis (2013a) and Abazajian et al (2012).
In contrast, the BAO feature is sensitive to the baryon fraction rather directly: therefore, combining CMB measurements (primarily sensitive to the physical baryon density and the redshift of matter-radiation equality ) with a BAO measurement sensitive to may provide an interesting probe of the radiation density, . This may be less precise than other methods, but is largely complementary.
The plan of the paper is as follows: in § 2 we review the main effects of varying cosmological parameters, including and radiation density, on CMB and BAO observations. In § 3 we present numerical predictions of the BAO feature for a set of models (selected to give a good match to WMAP) with varying matter density and , and we derive a statistic based on the galaxy correlation function which is sensitive to and , but cancels galaxy bias and dark energy to leading order. We summarize our conclusions in § 4. Most of this work was completed before the Planck release in March 2013, so we mainly use WMAP-9 fit parameters (Hinshaw et al, 2012) as our baseline. The adjustments post-Planck are moderate, and we discuss the implications of recent Planck results (Planck Collaboration, Ade et al (2013)) in § 3.3.
Throughout the paper we use the standard notation that is the present-day density of species relative to the critical density; and the physical density , with .
2 BAOs, radiation density and the cosmic baryon fraction
2.1 Overview of BAOs
The BAO feature appears as a single hump in the galaxy correlation function , or equivalently a series of decaying wiggles in the power spectrum (see Eisenstein, Seo & White (2007) for a clear explanation in real space, and Bassett & Hlozek (2010) and Weinberg et al (2013) for reviews). The length-scale of the hump is very close to the comoving sound horizon at the baryon drag epoch ; this is commonly defined by the fitting formula Eq. 4 of Eisenstein & Hu (1998). (This formula is defined for standard ; however the dependence of on is weak, so the error from adopting the fitting formula is small). This comoving length is predicted precisely mainly from CMB constraints, and several very large redshift surveys are ongoing or planned to exploit this as a standard ruler to measure cosmic distances at and thus probe dark energy.
Standard cosmological models contain a density of collisionless dark matter larger than the baryon density. This explains naturally why the acoustic peaks in the CMB power spectrum have large relative amplitude, while the BAO feature is relatively weak in the late-time galaxy correlation function. Qualitatively, this occurs because the acoustic peaks at last scattering appeared only in the power spectrum of baryons, not dark matter: the peaks are prominent in the CMB because almost all CMB photons last scattered off a free electron. After decoupling, the distribution of baryons and dark matter became averaged together by gravitational growth of structure over the next few –foldings between the CMB era and redshift (Eisenstein, Seo & White, 2007), well before the formation of large galaxies. The dark matter dominates in this averaging, so the BAO signal in the galaxy correlation function becomes diluted by a factor . In the following we define the baryon fraction as
so that the denominator includes CDM and baryons, but excludes massive neutrinos.
Thus, the BAO peak amplitude provides a potential measure of the baryon fraction; this has been used as a simple and compelling argument against MOND-type modified gravity theories without non-baryonic dark matter (Dodelson, 2011).
We can make this more quantitative and use the BAO feature to directly estimate the cosmic baryon fraction. However, there are several reasons why this has received little attention to date:
Recent observations of the CMB power spectrum (Hinshaw et al, 2012) measure the physical baryon density to percent precision, and (assuming standard ), measure the physical matter density to 3 percent; the errors on these are weakly correlated, so this gives an estimate of the cosmic baryon fraction to percent relative precision. (Recent results from the Planck mission have improved this to the percent level, see § 3.3).
The BAO feature is affected by several effects: it is blurred by the non-linear growth of structure, and amplified by galaxy bias, which is challenging to measure and may also be scale-dependent.
The overall large-scale shape of the galaxy power spectrum also depends on other cosmological parameters, and this will also have some influence on the BAO peak shape.
Thus, at first sight it appears that BAOs cannot compete in precision with the CMB as a probe of . This is partly true, but with an important caveat: the CMB-based estimates of are significantly degenerate with the total radiation density in the CMB era.
2.2 Radiation density
At this point we define our notation on densities: as usual, we define the parameter such that the radiation density at matter-radiation equality is
where , are densities of (total) radiation and photons respectively. Here is an “effective” number of neutrinos, but in fact it is not specific to neutrinos and counts any species (except photons) which were relativistic until around matter-radiation equality. Assuming the standard population of only three very light neutrinos with the oscillation parameters given by solar, atmospheric and beam-based neutrino experiments, the value of can be accurately predicted as (Mangano et al, 2005); here the additional arises from a small residual coupling of neutrinos to baryons and photons at the epoch of electron/positron annihilation.
It is also convenient to define the scaled radiation density by
where the factor of is the bracket in Eq. 2 for ; therefore for standard radiation content, and for example for , i.e. the case of additional “dark radiation” with energy density equal to one standard neutrino flavour. Here and are equivalent, but the latter is convenient later since several parameters of interest scale almost as half-integer powers of .
There are now several known routes to probe from observations: historically it was first constrained by big bang nucleosynthesis, (Steigman, Schramm & Gunn, 1977; Mangano & Serpico, 2011). However, He is the nuclide with the main sensitivity to , and observational measurements of the primordial He abundance, , appear to be limited by systematic errors; over the past 25 years the estimates of have shifted significantly upwards, but the realistic error bars have not much improved. Unless a new better method of measuring can be found, we cannot expect dramatic progress from the Helium route. Recently, a constraint on has been derived from deuterium abundance (Pettini & Cooke, 2012), but this currently relies on only a single object, and also uses the baryon density derived from the CMB.
Secondly, the CMB damping tail at high multipoles is sensitive to (Jungman et al, 1996; Bashinsky & Seljak, 2003; Hou et al, 2013); and several recent measurements (Hou et al 2012;Riemer-Sorensen, Parkinson & Davis 2013a; Ade et al 2013) give tantalizing but not decisive hints for a value higher than the standard . However, the CMB damping tail method is significantly degenerate with other possible new parameters, including running of the spectral index and non-standard helium abundance (Hou et al, 2013; Joudaki, 2013), so other complementary probes of are desirable.
Thirdly, combining CMB data with a direct local measurement of can also probe ; however, using CMB+ alone is critically dependent on other assumptions such as and flatness. An improvement on the method is given by Sutherland (2012), who showed that a theory-free measurement of can be obtained by combining a low- BAO redshift survey and a suitable absolute distance measurement to a matched redshift (specifically , where is the characteristic redshift of the BAO survey). This almost cancels the distance effects from dark energy and curvature; comparing such a direct measurement to CMB data (which mainly constrains rather than alone) therefore probes . The above-mentioned method is less theory-dependent than the CMB damping tail, but requires a challenging measurement of a distance to to percent absolute accuracy.
We will demonstrate below that the amplitude of the BAO feature provides a fourth possible probe of : this is currently much less precise than the known methods above, but involves different assumptions and systematics; with future massive redshift surveys expected in the next decade, it may provide a useful complement to the better-known methods above.
2.3 Cosmological parameter set
The present-day photon density is very well constrained by the observed CMB temperature (Fixsen, 2009) and spectrum to be ; and we define to be the physical matter density today, specifically CDM and/or WDM plus baryons, excluding neutrinos. Defining as the redshift of matter-radiation equality, and simply assuming that the photon density scales with redshift as , and matter conservation so CDM and baryon densities scale (i.e. no decaying dark matter, dark energy to dark matter transitions, or other exotic effects) leads to the following identities:
The equations above are independent of assumptions about dark energy and flatness. They remain valid for the case of small non-zero neutrino mass, : since our definition of excludes the contribution from massive neutrinos today, while neutrinos this light were almost fully relativistic at the era of matter-radiation equality. This assumption is reasonable given recent upper limits on neutrino mass from CMB+galaxy clustering data (Ade et al 2013; Giusarma et al 2013; Riemer-Sorensen, Parkinson & Davis 2013b). Clearly, low-mass neutrinos are matter-like at and do contribute to in late-time observables, but we treat as a separate contribution.
We now choose a basic 6+1 set of cosmological parameters as
where the first three and are defined above, as usual is the scalar perturbation amplitude (which cancels in the following), is the scalar spectral index and is the optical depth to last scattering. Then, , and are derived parameters from Eqs. (4)–(6). We may add optional parameters, curvature , dark energy equation of state and present-day neutrino density defaulting to respectively (for minimal neutrino mass). Then , and the dark energy density is another derived parameter, via .
This parameter set (7)
including and in the basic six
looks unconventional compared to the more common
choice including as two of the basic parameters;
but for variable , our set links more naturally to
as we will see below; see also Appendix A,
and the discussion in Section 4.2 of Sutherland (2012).
To summarize the latter, dimensionless observables such as the CMB
acoustic wavenumber ,
It has been shown by several authors (Hu & Sugiyama 1996; Jungman et al 1996; Bashinsky & Seljak 2003; Jimenez et al 2004; Komatsu et al 2011)
that the heights of the first few acoustic peaks in the CMB primarily
constrain the redshift of
matter-radiation equality , not the
physical matter density .
The WMAP data also constrains the baryon density accurately. The effect of baryons on the CMB derives mainly from the baryon/photon ratio; given the photon density measured very accurately by COBE (Fixsen, 2009), the estimate from WMAP is only weakly dependent on or .
Measuring both and immediately gives us the product from Eq. 6; so, the key point from the above is that the first few CMB acoustic peaks provide an accurate constraint on the product , but and are significantly degenerate. Therefore, adding a non-CMB observable which is sensitive to can provide another probe of .
In the next section, we show that the relative amplitude of the BAO peak in galaxy clustering may provide such a test: it depends mainly on , with weak sensitivity to other parameters. Thus, comparing a BAO-based estimate of to a CMB-based measurement (approximately ) can provide a new probe of the radiation density which is largely independent of existing methods. A strong point of this method is that the CMB can measure using only the first three acoustic peaks; for models near concordance parameter values, the ratio of the third to first peak height is especially sensitive to (Hu et al 2001; Page et al 2003), and the third peak at is only weakly affected by Silk damping which dominates at . Thus, while we need CMB data at , this method is not strongly dependent on the CMB damping tail and other possible early-time nuisance parameters.
3 Estimating baryon fraction from the BAO peak
3.1 The BAO equivalent width
We noted above that the CMB power spectrum from WMAP constrains to percent, (and this improves to with Planck data); therefore, an estimate of derived from the BAO amplitude can translate into a probe of or equivalently .
However, deriving from the BAO feature is affected by several complications listed below: firstly there is galaxy bias, which may be scale-dependent. Here we choose to work with the correlation function rather than the power spectrum, since the former makes the BAO feature a single hump which simplifies the analysis. In the linear-bias approximation, the galaxy and matter correlation functions are related by , therefore a suitable ratio of correlation functions near the BAO bump vs outside the bump can cancel galaxy bias if it is scale-independent. This is believed to be a good approximation at linear scales (Angulo et al, 2008; Baugh, 2013), but a better understanding of galaxy formation may be required to clarify this.
Secondly, the BAO bump is blurred by the non-linear growth of structure, mainly due to peculiar motions (Eisenstein et al, 2007); this both lowers its height and broadens its shape, and causes a small shift in central position. However, it is shown by Orban & Weinberg (2011) that non-linear growth almost conserves the total area of the bump; thus, measuring the bump area rather than its height is relatively insensitive to the non-linear growth of structure.
Thirdly, the global shape of , with in units, depends on other quantities including and dark energy equation of state, which are not tightly constrained by the CMB. However, we show later that if we define to be the ratio of comoving separation to the sound horizon scale, then the broad-band shape of on intermediate scales depends mostly on : nearly all shape dependence on other parameters is collapsed into , which is already well determined by the CMB.
Therefore, we define the following observable from the measured galaxy correlation function , as a measure of the BAO “equivalent width”: we define
where is the observed galaxy correlation function in units of , is the bump scale (here the value of at the peak in ), and is a smooth “no bump” curve, here a polynomial fitted to the regions of just outside the BAO bump. Then are arbitrary dimensionless limits of integration, where span almost the full area of the bump; while are intermediate scales non-overlapping with the bump, used for normalization. There is a compromise here, since we want to avoid the non-linear regime , while at the measurement noise in generally increases, and becomes more sensitive to systematic errors. In the following we choose as simple values which give a well-measured signal in the linear regime, and are not too far below the bump scale to minimise the possible effects of scale-dependent bias.
In the above definition, a constant bias in cancels out as long as it is scale-independent on large scales ; while a multiplicative stretch of cosmic distance scales (e.g. from varying dark energy) also cancels in , since we are measuring at fixed fractions of the comoving scale which is fitted from the data. The ratio between and the horizon size at matter-radiation equality is determined almost entirely by , so we expect defined as above to be mainly sensitive to the baryon fraction as desired.
To verify this and test parameter dependences,
we next evaluate from the linear matter power spectrum
for some example theoretical models
generated by CAMB.
3.2 Dependence of on and
Here, we evaluate (defined above) for a set of six representative models which are all consistent with CMB data up to 2012. All models are flat CDM (, ), and have fixed to 0.96 and in accordance with WMAP. We vary and , and also adjust to preserve the CMB acoustic scale .
For our “base” model (hereafter C3) we set , , ; therefore and . For model C4 we add a fourth light neutrino species, but retain identical values of and ; therefore C4 has and increased by and respectively relative to C3. (Here is held at 0.0226 for both models, so model C4 has dark matter density increased by slightly more than 13.4%, while is reduced by a factor of 0.882).
The overall shape of the correlation function also depends significantly on : to explore this dependence, we choose two models (labelled L3, L4) with fixed to 5% lower than C3, and respectively and ; likewise another two models (H3, H4) with fixed 5% higher than C3. For these models, is adjusted in order to preserve the CMB acoustic scale . The resulting parameter values are shown in Table 1. Since these models are all flat, they do not quite follow the CMB geometrical degeneracy, but they do follow the related degeneracy of constant or horizon angle as outlined in Percival et al (2002).
We used CAMB to evaluate the CMB temperature power spectra for the above six models; these are shown in Figure 2, normalized to match model C3 at . Clearly the CMB spectra are very similar for all our models, since the acoustic scales are matched by construction, and the variations in are only percent. Minor differences are apparent, notably around the third peak (which is positively correlated with ), while the effects of appear mainly in the damping tail and are small at . We repeat here that and have been held fixed in all models for simplicity, in order to highlight the effects of and . Clearly, allowing and to float to fit CMB data would result in model spectra that are even more similar, especially if a running spectral index is also allowed.
We took the linear-theory matter power spectra for the above six models generated by CAMB, and then Fourier transformed them to obtain the real-space matter correlation functions; these are shown in Figure 2, with the axis in units of corresponding to the observable from a low- redshift survey.
For the matter correlation functions in Figure 2, the differences between models are much more obvious than in the CMB: the position of the BAO bump (in units) is insensitive to for fixed , but it does shift with . In fact, as explained in Appendix A, the BAO bump location is more sensitive to than ; but changing required us to adjust to conserve the CMB acoustic scale, and it is actually the change in which dominates the shift of the bump location. The other notable feature in Figure 2 is that all the models have a slightly reduced BAO peak amplitude, as qualitatively expected given their smaller .
To highlight the effects of varying , in Figure 3 we plot the matter correlation functions as a function of , so that the BAO bump appears at . This Figure shows clearly that the bump amplitude is mainly sensitive to , while the broad-band shape (the ratio of power at to that at ) is governed mainly by . This is understandable since the broad-band shape is determined by the scale of the turnover in the matter power spectrum, which is directly proportional to the particle horizon size at . This scale in observable units depends on several cosmological parameters. However, as noted in e.g. Eq. B2 of Sutherland (2012), the ratio of the BAO sound horizon to the particle horizon at (both in comoving units) has a simpler dependence: this ratio is well approximated by simply
since the sound speed is well constrained by the WMAP baryon density. The dependence on other parameters such as , , is almost entirely compressed into , and the ratio is completely independent of late-time parameters such as , . Thus the changes in in Figure 3 are largely driven by the differing and between the six models, and adding optional parameters such as , will have minimal effect.
Here it is also noteworthy that the zero-crossing in occurs close to for all the models; this offers an interesting possible consistency test for the CDM framework which is largely insensitive to galaxy bias. However, this is observationally challenging to measure since the zero-crossing is much more sensitive than the BAO bump position and amplitude to broad-band systematic errors in the observed .
The denominator of Eq. 8 is mainly sensitive to the broad-band large-scale power at , which as above depends on the turnover scale in the matter power spectrum. If we measured this in a fixed range of Mpc or , this would depend on quantities such as and , which would seriously degrade our ability to measure ; but since we chose our mid-scale power estimate as a fixed fraction of the BAO length rather than a fixed range in , this mostly cancels the dependence on low-redshift parameters such as , and ; the broad-band shape of at depends almost entirely on and , which are already well constrained by the CMB. Therefore, we anticipate that should depend mainly on and only weakly on .
To quantify this, we evaluated the ratio for our six models, and the results are given in the last column of Table 1: the table shows that is close to for all three models, and close to for all three models, consistent with our expectations above. The dependence of on is below 1% and nearly negligible, while adding a fourth neutrino species or equivalent reduces by a factor close to (i.e. 8.5% suppression) in each case. This reduction is slightly less than we would expect from linear scaling , since our models have reduced by a factor relative to the corresponding model. The probable explanation is that baryons, in addition to causing oscillations, also affect the broad-band shape of the power spectrum (Eisenstein & Hu, 1998) by suppressing power on all scales smaller than the sound horizon. Therefore, reducing the baryon fraction slightly increases power on intermediate scales, and changes the broad-band shape of , which slightly counteracts the reduction in the bump area.
The conclusion is that, if galaxy bias is scale-independent on large scales and the area of the BAO peak is conserved under non-linear evolution (or can be recovered by reconstruction methods), then measurements of can offer a potential new probe of . Estimates from large numerical simulations could be used to test these assumptions, and possibly attempt to correct for any resulting biases.
The largest current redshift surveys provide a detection of the BAO peak (Anderson et al, 2012), which would translate to approximately 16 percent uncertainty in ; this is twice as large as the 8.5 percent shift predicted above for , so at present the precision on looks uncompetitive with other methods. However, future next-generation large redshift surveys can potentially offer a large improvement, and thus an interesting test of which is complementary to the better-known methods from the CMB and nucleosynthesis.
3.3 Effect of Planck data
Most of this paper studies models with parameter choices based on the WMAP-9 cosmological parameter results (Hinshaw et al, 2012); the C3 model is near the best-fit, and L and H models have shifted by in WMAP units. After this paper was nearly completed, the first Planck cosmology data release occurred in 2013 March. While there are many interesting consequences for inflation and the early universe, for the present purposes, two results are most notable: firstly concerning , the evidence for has generally weakened (Ade et al, 2013), but the strength of this conclusion is somewhat dependent on the choice of additional data sets.
The fit CDM + varying to the dataset “Planck + WMAP polarisation + high-L + BAO” (the right column of Table 10 of Ade et al 2013) gives , which is above the standard value and excludes at the level. However, there remains the well-publicised tension that Planck with vanilla CDM (and ) prefers a value of , which is below the range given by recent local measurements (Riess et al 2011; Freedman et al 2012). There are many possible explanations, but this tension can be ameliorated by increasing : e.g. fitting Planck + data allowing variable gives and , i.e. above the standard . In summary, is somewhat disfavoured by Planck, but a value of is completely allowed or perhaps even preferred by combining all current data. There are interesting possible models with extra relativistic species other than neutrinos leading to (e.g. Weinberg 2013).
Secondly, concerning and , the Planck data imply values somewhat higher than WMAP; for the vanilla CDM model, fits to Planck+BAO data give and (and for standard ). The Planck constraints on are especially robust: in the many extensions of CDM considered by the Planck team, the bounds are generic, i.e. values outside this range are excluded at for all of the added-parameter models and data combinations. (Clearly, still more complicated models with even more non-vanilla parameters might widen this range; but there appears little motivation at present for adding two or more new parameters beyond the basic six).
Comparing to our models above, the Planck central value is near the mid-point between our model pairs C and H above, but slightly closer to H. Our two L models () are now firmly excluded by Planck, at around the level for the base model or for extended models. Also, Planck prefers which is just 2 percent below our default; and which is nearly identical. Thus, while Planck has narrowed the allowed range of and , our models C3/C4/H3/H4 approximately bracket the range of and allowed by Planck; and the main conclusions of this paper regarding the BAO amplitude are essentially unaffected.
We have shown that a measurement of the BAO peak amplitude via the observable in Eq. 8 may provide an interesting measurement of the cosmic baryon fraction; this observable has been constructed so as to cancel galaxy bias, non-linearity and dark energy effects to leading order, thus being sensitive mostly to .
Comparing this BAO-based measurement to the measurement of (approximately) from the CMB then gives an interesting probe of ; this is largely complementary to the better-known method based on fitting the CMB damping tail. Here, the key inputs required from the CMB are constraints on and . Assuming standard gravity and standard recombination, these two parameters are very robust against extra-parameter extensions to vanilla CDM.
There are two main assumptions used here: firstly that galaxy bias is nearly scale-independent on the large scales between , and secondly that the area (not height) of the BAO bump is conserved during the non-linear evolution of structure. Both of these assumptions are reasonably well-motivated, but much more detailed numerical simulations would be needed to see how well these approximations are expected to hold in practice.
A measurement of to useful precision will require a substantial advance on current data: the current precision on the BAO bump area is around 16%, while we would need to reach around to get a useful distinction between or 4; this appears a challenging proposition. However, given that the CMB temperature measurements are now approaching the limits set by cosmic variance and foregrounds, other independent probes of are highly desirable, and the test here should become feasible at no extra cost from planned next-generation BAO redshift surveys.
We thank the anonymous referee for helpful comments which significantly clarified this paper.
We acknowledge the use of WMAP data from the Legacy Archive for Microwave Background Data Analysis (LAMBDA) at GSFC (lambda.gsfc.nasa.gov), supported by the NASA Office of Space Science.
Appendix A Parameter dependence of CMB peaks and BAOs
In this section we give some accurate approximations for the dependence of CMB acoustic scale and BAO distance ratios on cosmological parameters, especially and used as basic parameters above. This helps to understand the parameter choices in Table 1, and the resulting shifts in BAO bump position observed in § 3.2.
Firstly, we find that a very good approximation to the CMB acoustic wavenumber for models fairly close to standard CDM is
where , and this allows for small neutrino mass, weak curvature and constant . (This is for ; however, changing to the Planck value gives only around 0.1 percent reduction in ). This has almost negligible dependence on , since varying (at fixed as above) results in both and shrinking by a factor very close to , but these cancel almost exactly in .
Since is measured to high precision percent by WMAP+ACT+SPT, if we vary (as in the L/H models in Table 1 above), then to remain consistent with CMB data we must adjust other parameter(s) to preserve . Given our assumption of flat models and minimal neutrino mass in § 3, the only available parameter above is . Forcing a 5% reduction in (as chosen for models L3/L4) requires a 12% reduction in to keep the same as the baseline model C3; this corresponds to an increase in by 3.6% (at fixed ). Shifts from C to H models are basically the opposite of this. We note one counter-intuitive feature: when varying parameters to conserve , it turns out that changes in the opposite sense to ; this is distinct from the common case of fixing and varying , when varies .
We can also understand the resulting shifts in the BAO bump location as follows: if we copy approximation (12) from Sutherland (2012) for low-redshift BAO ratios, which is
here the LHS is a direct observable from a BAO survey at effective redshift , is the usual BAO dilation length (Eisenstein et al, 2005), , and is a small cosmology-dependent correction term (Sutherland, 2012), which is typically and effectively negligible at modest redshift . Approximation 11 is accurate to at , comparable to the cosmic variance limit, and again this is almost independent of . At low redshift, Eq. 11 is only weakly sensitive to additional non-vanilla parameters such as curvature and varying via the term; this explains why low-redshift BAO observations provide a very robust constraint on .
In the limit , the above simplifies to
The LHS is equivalent to a hypothetical BAO measurement at ; this is not strictly observable since cosmic variance prevents us measuring the BAO feature at ; but it is a modest extrapolation from real low- BAO surveys. The main point is since a galaxy redshift survey of course measures redshifts not distances, the apparent BAO bump “length” presented in units, as in Figure 2, is really measuring the “BAO velocity” in units of . Although this quantity contains , in the case of varying this gets cancelled: if we vary while holding fixed (as appropriate for fitting CMB data), then and both depend on the radiation density as and respectively; so their product is almost independent of and only depends on the dimensionless parameters and , plus a very weak dependence on which is negligible at the current level of accuracy. Therefore, the observed velocity scale of the BAO feature at low redshift is primarily measuring , not , which explains why the BAO feature does not shift between the 3 and 4 neutrino model pairs in § 3.2.
Since all of the approximations above are nearly independent of , this was the rationale for choosing and as two of the basic parameters: observations of CMB and BAOs give us direct constraints on and , nearly independent of . These two directly give a constraint on from Eq. 5, but give almost no ability to measure , separately; this explains why WMAP+BAO alone currently have very weak leverage on , unless further dimensionful data such as or is added.
Finally, the fact that appears with a power in Eq. 12 explains why the BAO feature shifts to smaller (larger) velocity scale for the models H (L) above.
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