CosmoCoffee

Antony Lewis · Posted: April 14 2005

In this paper they appear to be claiming that by using the LSS-CMB correlation to constrain the primodial scalar amplitude A

s

, they can improve limits on the tensor amplitude ratio r. They claim ultimately 0.5% constraints on r may be possible, which is much lower than previously thought.

If all they are getting from the LSS observation is a measurement of A

s

, I don't see how they can possibly get such a good limit on r from CMB scales 10 < l < 40. If you assume perfect knowledge of all the cosmological parameters (except r), the cosmic variance limit is around 0.05 - see e.g.

http://cosmologist.info/notes/tensors.ps

- so how can you do better with a much worse than perfect measurement of A

s

?

Maybe I misunderstood, but there does seem to be a clear statement in the paper about how they actually made Fig. 1 or what is happening with cosmic variance. The only way out I can think of to get better constraints than expected is if the value of A

s

you get is larger than you would expect from the measured CMB power. In this case I could imagine r would get squeezed to be as small as possible, but only at the expense of the r=0 model still not fitting very well.

Other comments:

* r is defined implicitly, presumably as some C

l

ratio at large scales (otherwise the CMB observation would not measure A

s

(1+r). Note that some people (e.g. me) usually quote values for primordial amplitude ratios A

t

/ A

s

which are different.

* The amplitude enters 3 times in equations 5,6, via A

s

,

Δ

R

2

and

Φ

. Presumably they only want it once, which requires odd definitions of

Δ

R

2

and

Φ

where the amplitude is scaled out.

* On page 2 they state that the tensor perturbations are generated at recombination or before. I don't think this is correct: the effect of tensors is to induce anisotropies because you are looking at the smooth recombination surface through distorting gravitational waves entering the horizon along the line of sight. The contributions from recombination are small on large scales (comparable to the polarization signal, which does only come from recombination and reionization).

Alessandro Melchiorri · Posted: April 16 2005

Hi Anthony,

The ISW is related to Omega_Lambda and A

S

. In few words, in a flat universe, if you lower Omega_l you lower the ISW signal and you need an higher A

S

to fit the data. But you can't have A

s

higher than the WMAP normalization which contains scalar modes but also tensor, foregrounds, etc etc (plus some cosmic variance). So, exactly as you mention, lowering Omega_l you squeeze the models with r until even the model with r=0 does'nt fit the data. This explains our contours in Fig.1 using current data and why flat models with Omega_l=0.5 and gravity waves don't provide a good fit. I consider this is a quite important result!
It is also quite independent from neutrinos masses and many other parameters and complementary to other results.

Regarding the forecasts you don't need to take just up to l<40 but you can go further and beat some of the cosmic variance on (1+r)A

s

(not A

t

) and also include data on Omega_l.
Our biggest concern however is not cosmic variance but determination of bias and systematics.

Thanks for the valuable comments !
We will implement them in a revised version.

ciao
Alessandro

Antony Lewis · Posted: April 21 2005

Thanks for the clarification. Just to clarify further: this method can never detect very small tensor amplitudes, you can only get small limits (non-detections) by conditioning on an inconsistent value of

Ω

Λ

(e.g. if the datasets turn out to be inconsistent even with r=0).

In fact you could do this with other parameters — I expect the constraint on the neutrino mass is very good if you condition on n

s

= 1.2. How useful this is I'm not so sure!

In particular the comparison with ground based B-mode missions is potentially misleading: these experiments actually could detect small tensor amplitudes, unlike the method in this paper (if the datasets are consistent).

If the datasets are not consistent, then I'm not sure it's very valid to perform a usual parameter analysis. Instead one should probably be looking for what's gone wrong with the observations or model.

The amount of extra information in the CMB-LSS correlation seems to be small (for consistent parameters). Doing a simple Fisher estimate for perfect full sky observations, assuming all the power spectra are proportional, and that all parameters including A

s

but excluding r are known perfectly, gives

$\sigma_r \sim \sqrt{\frac{2(1-c^2)}{N}}$

where c is the correlation and N is the number of observed modes. This differs by (1-c

2

)

1/2

from the result without using the correlation. Taking very optimistically an average c=0.2 gives $\text{[math]}$ 3% improvement on the error (much smaller because the correlation actually falls off rapidly with l). Even a perfect cross-correlation using lensing reconstruction only gives ~ 5% improvement on the error in r.

So overall, the amount by which you can improve sensitivity to tensors by using CMB-LSS correlation seems to be very small when you are cosmic variance limited from the temperature (i.e. can constrain to r ~ 0.05 without CMB-LSS).

Alessandro Melchiorri · Posted: July 19 2005

Hi Antony,

There is a new version on the web today that matches the PRD accepted version.
We revised the section on our forecast. Indeed it was too optimistic and now we state things in a more clear and (perhaps too much!) conservative way.

Regarding the limit on the neutrino mass, I think that if you have n

s

=1.2 then you need
a quite high mass for the neutrinos to match LSS so I think you mean the opposite (low n

s

-> low neutrino mass to match LSS). Knowing that neutrinos must have zero mass if n

s

=0.9 is already important !

Concerning the detection of B modes... this would be in any case an indirect evidence for gravity waves...and for constraining inflation you need to know A

s

quite well too !

cheers and thanks for your comments
Alessandro