# Massive primordial tensor perturbations?

• 302021895
In summary, the energy-momentum tensor for the inflaton is diagonal, which means that the previous equation leads to zero.
302021895
I am studying the generation of tensor perturbations during inflation, and I am trying to check every statement as carefully as possible. Starting from the metric

$ds^2 = dt^2 - a^2(\delta_{ij}+h_{ij})dx^idx^j$

I make use of Einstein's equations to find the equation of motion for the perturbation ##h_{ij}##. For the Ricci tensor I obtain

$R_{00}=-3(\dot{H}+H^2)$
$R_{0i}=0$
$R_{ij} = \frac{a^2}{2}\left[2(3H^2+\dot{H})(\delta_{ij}+h_{ij})+3H\dot{h}_{ij}+\ddot{h}_{ij}-\frac{1}{a^2}\nabla^2h_{ij}\right]$
and
$R=-(12H^2+6\dot{H})$

This coincides with the results shown in Dodelson's textbook, equations (5.47) and (5.57). Dodelson then claims that since the Ricci scalar contains no perturbation (true), the Einstein tensor can be calculated as

$\delta G^i_j = \delta R^i_j$

(where ##\delta## means the perturbed part), which supposedly leads to the standard result for the massless tensor perturbation. However, if I just blindly substitute,

$G_{ij} = R_{ij}-\frac{1}{2}g_{ij}R = \frac{a^2}{2}\left[\ddot{h}_{ij}+3H\dot{h}_{ij} - \frac{1}{a^2}\nabla^2h_{ij} - 2(2\dot{H}+3H^2)(\delta_{ij}+h_{ij})\right]$

The energy-momentum tensor for the inflaton is diagonal ##T_{ij}\propto\delta_{ij}##, which means that the previous equation leads to

$\ddot{h}_{ij}+3H\dot{h}_{ij} - \frac{1}{a^2}\nabla^2h_{ij} - 2(2\dot{H}+3H^2)h_{ij} = 0$

i.e., an equation for a 'massive' perturbation. Why is this incorrect? I agree with Dodelson in that ##\delta R=0##, but ##\delta g_{ij}\neq0##, and it seems to me that this would introduce the extra mass term. Moreover, by contracting with the metric, ##G^a_b=g^{ac}G_{cb}##, we do 'get rid' of the extra term, since ##g_{ab}\rightarrow\delta^a_b##, but it seems to me that this would turn the enegy-momentum tensor in the right-hand side of Einstein's equation to a non-diagonal form, which would then introduce an extra term proportional to ##h^i_j##.

I'm probably making a stupid mistake, but I am now very frustrated and I would appreciate any help. Thanks.

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That is a good reason to include a cosmological constant.

Chronos said:
That is a good reason to include a cosmological constant.

I'm not sure I follow you...

## 1. What are massive primordial tensor perturbations?

Massive primordial tensor perturbations are fluctuations in the fabric of space-time that occurred during the early universe. These perturbations are believed to be responsible for the formation of large-scale structures such as galaxies and galaxy clusters.

## 2. How are massive primordial tensor perturbations different from other types of perturbations?

Massive primordial tensor perturbations are different from other types of perturbations, such as scalar and vector perturbations, because they involve gravitational waves rather than density or velocity fluctuations. These gravitational waves have a distinct signature that can be observed in the cosmic microwave background radiation.

## 3. What is the significance of studying massive primordial tensor perturbations?

Studying massive primordial tensor perturbations can provide valuable insights into the early universe and the processes that led to the formation of structures we see today. It can also help us understand the nature of gravity and the fundamental laws of physics.

## 4. How are massive primordial tensor perturbations detected?

Massive primordial tensor perturbations are detected through their effects on the polarization of the cosmic microwave background radiation. This polarization is measured by specialized telescopes, such as the Planck satellite, which can detect subtle changes in the direction of light waves caused by gravitational waves.

## 5. What current research is being done on massive primordial tensor perturbations?

There is ongoing research into massive primordial tensor perturbations, including efforts to improve our understanding of their properties and to develop more sensitive detection methods. Some researchers are also exploring the possibility that these perturbations may have originated from exotic sources, such as cosmic strings or inflationary models.

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