# How to determine the total energy in the universe

1. Aug 18, 2010

### Tipi

How to determine the total energy of matter in the universe

Hi all,

while looking at a table of order of magnitudes of energies in nature, I just realize that I really don't know how to determine the total energy contained in the universe (near 10 to the 69).

Can someone get this number with a simple argument?

Thanks a lot,

TP

Last edited: Aug 18, 2010
2. Aug 18, 2010

### phyzguy

It's not known. Many physicist believe (Feynman was one of them) that the total energy content of the universe is exactly zero, with the positive energy of the matter in the universe being exactly cancelled by the negative energy of the gravitational field.

3. Aug 18, 2010

### Tipi

Re: How to determine the total energy of matter in the universe

Ok. That was not the answer I was looking for but it's interesting.

So, how could we determine the positive energy of matter, for instance?

Thanks,

TP

 Enhanced title...

4. Aug 18, 2010

### phyzguy

Well, we don't know the extent of the universe, or even whether it is finite or infinite. You could calculate the energy content of the matter in the observable universe by adding up all of the mass and multiplying by c^2. The observable universe has been estimated (http://en.wikipedia.org/wiki/Observable_universe) to contain about 10^80 baryons, so, since a proton has an energy content of about 2 GeV, you could estimate a baryon energy content of ~10^89 eV. Then there is the dark matter......

5. Aug 18, 2010

### nicksauce

The total energy in the universe is a vague concept that might not have a well-defined answer. So let's simplify your question to "what is the total rest energy of all the matter in the observable universe?". This is easier to answer now.

Some assumptions (ask if you need elaboration on any of them): The universe is at the critical density, and matter makes up 25% of the universe. The Hubble constant today is 72km/s/Mpc. The observable universe is 46 billion light years in radius.

We can then get the total energy, as we've defined it, as
$$E = \frac{1}{4}\rho\,c^2\,V$$
$$E = \frac{1}{4}\frac{3H_0^2c^2}{8\pi\,G}\frac{4\pi}{3}R^3$$

If we put in the numbers, we get about 10^71 joules.

6. Aug 18, 2010

### Tipi

Thanks for your answer. Could you explain a little were the first equation come from and how you go to the second?

Thanks,

TP

 OK. I get the first one, I didnt saw its evidence on first look. And for the second one, you used $$H_\text{crit} =30\sqrt (\rho (m_p/m^3))$$?

Last edited: Aug 18, 2010
7. Aug 18, 2010

### nicksauce

Energy = (1/4 of energy is matter) * energy density * volume

I used 1/4 of energy is matter because the universe is thought to be roughly 75% dark energy, and 25% matter, where the 25% matter can be further decomposed into dark matter and dark energy.

Then I used that
$$\rho_{crit} = \frac{3H_0^2}{8\pi\,G}$$

8. Aug 20, 2010

### Kevin_Axion

I think nicksauce meant the 25% of matter can be further classified by two groups, dark matter and Intergalactic Dust/Stars, just a clarification.

9. Aug 21, 2010

### Chronos

Agreed, dark energy is part of the 'missing' ~75% energy component. This makes perfect sense for reasons involving a lot of weird math.

10. Aug 21, 2010

### nicksauce

Errr yes. Of course.

11. Aug 22, 2010

### Orion1

Total amount of energy in the Universe...

Affirmative.

These are my equations for the total Universe_mass-energy equivalence based upon the Lambda-CDM model parameters and the Hubble Space Telescope (HST) and WMAP observational parameters and the Hubble radius in Systeme International units.

$$H_0 = 2.3298 \cdot 10^{- 18} \; \text{s}^{- 1}$$ - Hubble parameter (WMAP)
$$\Omega_s = 0.005$$ - Lambda-CDM stellar Baryon density parameter
$$dN_s = 10^{22}$$ - Hubble Space Telescope observable stellar number
$$dV_s = 3.3871 \cdot 10^{78} \; \text{m}^3 \; \; \; (4 \cdot 10^{30} \; \text{ly}^3)$$ - Hubble Space Telescope observable stellar volume
$$M_{\odot} = 1.9891 \cdot 10^{30} \; \text{kg}$$ - solar mass

Hubble Space Telescope observable stellar density:
$$\rho_s = M_{\odot} \left( \frac{dN_s}{dV_s} \right)$$

$$R_0 = \frac{c}{H_0}$$

Hubble sphere volume:
$$V_0 = \frac{4 \pi R_0^3}{3} = \frac{4 \pi}{3} \left( \frac{c}{H_0} \right)^3$$

Observable Universe_mass-energy equivalence total mass integration by substitution:
$$M_t = \frac{\rho_s V_0}{\Omega_s} = \frac{4 \pi M_{\odot}}{3 \Omega_s} \left( \frac{dN_s}{dV_s} \right) \left( \frac{c}{H_0} \right)^3 = 1.048 \cdot 10^{55} \; \text{kg}$$

Observable Universe_mass-energy equivalence total mass
$$\boxed{M_t = \frac{4 \pi M_{\odot}}{3 \Omega_s} \left( \frac{dN_s}{dV_s} \right) \left( \frac{c}{H_0} \right)^3}$$

$$\boxed{M_t = 1.048 \cdot 10^{55} \; \text{kg}}$$

However, only a fraction of this total mass exists in the form of mass in the Universe, which is composed of 22.8% cold dark matter and 4.56% ordinary baryonic matter.

$$\Omega_m = \Omega_{c} + \Omega_b = 0.2736$$ - Lambda-CDM total matter density

Universe total matter mass integration by substitution:
$$M_u = \Omega_m M_t = \frac{4 \pi M_{\odot}}{3} \left( \frac{\Omega_m}{\Omega_s} \right) \left( \frac{dN_s}{dV_s} \right) \left( \frac{c}{H_0} \right)^3 = 2.867 \cdot 10^{54} \; \text{kg}$$

Universe total matter mass:
$$\boxed{M_u = \frac{4 \pi M_{\odot}}{3} \left( \frac{\Omega_m}{\Omega_s} \right) \left( \frac{dN_s}{dV_s} \right) \left( \frac{c}{H_0} \right)^3}$$

Universe total matter mass:
$$\boxed{M_u = 2.867 \cdot 10^{54} \; \text{kg}}$$

Universe_mass-energy equivalence integration by substitution:
$$E_t = M_t c^2 = \left[ \frac{4 \pi M_{\odot}}{3 \Omega_s} \left( \frac{dN_s}{dV_s} \right) \left( \frac{c}{H_0} \right)^3 \right] c^2 = \frac{4 \pi c^5 M_{\odot}}{3 \Omega_s H_0^3} \left( \frac{dN_s}{dV_s} \right) = 9.419 \cdot 10^{71} \; \text{j}$$

Universe_mass-energy equivalence total energy:
$$\boxed{E_t = \frac{4 \pi c^5 M_{\odot}}{3 \Omega_s H_0^3} \left( \frac{dN_s}{dV_s} \right)}$$

Total amount of energy in the Universe:
$$\boxed{E_t = 9.419 \cdot 10^{71} \; \text{j}}$$

However, only a fraction of this total energy exists in the form of energy in the Universe, which is composed of dark energy at 72.6%.

$$\Omega_{\Lambda} = 0.726$$ - Lambda-CDM dark energy density

Total amount of dark energy in the Universe integration by substitution::
$$E_{\Lambda} = \Omega_{\Lambda} E_t = \frac{4 \pi c^5 M_{\odot}}{3 H_0^3} \left( \frac{\Omega_{\Lambda}}{\Omega_s} \right) \left( \frac{dN_s}{dV_s} \right) = 6.838 \cdot 10^{71} \; \text{j}$$

Universe dark energy total energy:
$$\boxed{E_{\Lambda} = \frac{4 \pi c^5 M_{\odot}}{3 H_0^3} \left( \frac{\Omega_{\Lambda}}{\Omega_s} \right) \left( \frac{dN_s}{dV_s} \right)}$$

Total amount of dark energy in the Universe:
$$\boxed{E_{\Lambda} = 6.838 \cdot 10^{71} \; \text{j}}$$

Reference:
http://en.wikipedia.org/wiki/Hubble%27s_law" [Broken]
http://en.wikipedia.org/wiki/Lambda-CDM_model" [Broken]
http://en.wikipedia.org/wiki/Universe" [Broken]
http://en.wikipedia.org/wiki/Observable_universe" [Broken]
http://en.wikipedia.org/wiki/Hubble's_law#Interpretation"
http://en.wikipedia.org/wiki/Hubble_volume" [Broken]
http://en.wikipedia.org/wiki/Dark_matter" [Broken]
http://en.wikipedia.org/wiki/Dark_energy" [Broken]

Last edited by a moderator: May 4, 2017
12. Aug 22, 2010

### Kevin_Axion

Excellent, response.

13. Aug 22, 2010

### Orion1

Total amount of energy in the Universe...

The distribution of galaxies beyond the Milky Way.

$$R_0 = \frac{c}{H_0}$$

$$\boxed{R_0 = 1.286 \cdot 10^{26} \; \text{m}}$$

However, the present observable Universe radius exceeds the Hubble radius due to cosmic inflation:
$$\boxed{R_u \geq R_0}$$

$$R_u = 3.419 \cdot R_0 = \frac{3.419 c}{H_0} = 4.399 \cdot 10^{26} \; \text{m} \; \; \; (46.5 \cdot 10^9 \; \text{ly})$$

$$\boxed{R_u = 4.399 \cdot 10^{26} \; \text{m}}$$

Hubble critical density:
$$\rho_c = \frac{3 H_0^2}{8 \pi G}$$

Universe sphere volume:
$$V_u = \frac{4 \pi R_u^3}{3}$$

For the post #5 equation for the Universe total energy integration by substitution:
$$E_t = \rho_c c^2 V_u = \frac{4 \pi c^2}{3} \left( \frac{3 H_0^2}{8 \pi G} \right) R_u^3 = \frac{H_0^2 c^2 R_u^3}{2 G} = 3.112 \cdot 10^{71} \; \text{j}$$

$$\boxed{E_t = \frac{H_0^2 c^2 R_u^3}{2 G}}$$

$$\boxed{E_t = 3.112 \cdot 10^{71} \; \text{j}}$$

These are my equations for the total Universe_mass-energy equivalence based upon the Lambda-CDM model parameters and the Hubble Space Telescope (HST) and WMAP observational parameters and the observable Universe radius in Systeme International units.

$$R_u = 4.399 \cdot 10^{26} \; \text{m}$$ - observable Universe radius
$$H_0 = 2.3298 \cdot 10^{- 18} \; \text{s}^{- 1}$$ - Hubble parameter (WMAP)
$$\Omega_s = 0.005$$ - Lambda-CDM stellar Baryon density parameter
$$dN_s = 10^{22}$$ - Hubble Space Telescope observable stellar number
$$dV_s = 3.3871 \cdot 10^{78} \; \text{m}^3 \; \; \; (4 \cdot 10^{30} \; \text{ly}^3)$$ - Hubble Space Telescope observable stellar volume
$$M_{\odot} = 1.9891 \cdot 10^{30} \; \text{kg}$$ - solar mass

Hubble Space Telescope observable stellar density:
$$\rho_s = M_{\odot} \left( \frac{dN_s}{dV_s} \right)$$

$$\boxed{R_u = 4.399 \cdot 10^{26} \; \text{m}}$$

Universe sphere volume:
$$V_u = \frac{4 \pi R_u^3}{3}$$

Observable Universe_mass-energy equivalence total mass integration by substitution:
$$M_t = \frac{\rho_s V_u}{\Omega_s} = \frac{4 \pi M_{\odot}}{3 \Omega_s} \left( \frac{dN_s}{dV_s} \right) R_u^3 = 4.188 \cdot 10^{56} \; \text{kg}$$

Observable Universe_mass-energy equivalence total mass
$$\boxed{M_t = \frac{4 \pi M_{\odot}}{3 \Omega_s} \left( \frac{dN_s}{dV_s} \right) R_u^3}$$

$$\boxed{M_t = 4.188 \cdot 10^{56} \; \text{kg}}$$

However, only a fraction of this total mass exists in the form of mass in the Universe, which is composed of 22.8% cold dark matter and 4.56% ordinary baryonic matter.

$$\Omega_m = \Omega_{c} + \Omega_b = 0.2736$$ - Lambda-CDM total matter density

Universe total matter mass integration by substitution:
$$M_u = \Omega_m M_t = \frac{4 \pi M_{\odot}}{3} \left( \frac{\Omega_m}{\Omega_s} \right) \left( \frac{dN_s}{dV_s} \right) R_u^3 = 1.146 \cdot 10^{56} \; \text{kg}$$

Universe total matter mass:
$$\boxed{M_u = \frac{4 \pi M_{\odot}}{3} \left( \frac{\Omega_m}{\Omega_s} \right) \left( \frac{dN_s}{dV_s} \right) R_u^3}$$

Universe total matter mass:
$$\boxed{M_u = 1.146 \cdot 10^{56} \; \text{kg}}$$

Universe_mass-energy equivalence integration by substitution:
$$E_t = M_t c^2 = \left[ \frac{4 \pi M_{\odot}}{3 \Omega_s} \left( \frac{dN_s}{dV_s} \right) R_u^3 \right] c^2 = \frac{4 \pi c^2 M_{\odot}}{3 \Omega_s} \left( \frac{dN_s}{dV_s} \right) R_u^3 = 3.764 \cdot 10^{73} \; \text{j}$$

Universe_mass-energy equivalence total energy:
$$\boxed{E_t = \frac{4 \pi c^2 M_{\odot}}{3 \Omega_s} \left( \frac{dN_s}{dV_s} \right) R_u^3}$$

Total amount of energy in the Universe:
$$\boxed{E_t = 3.764 \cdot 10^{73} \; \text{j}}$$

However, only a fraction of this total energy exists in the form of energy in the Universe, which is composed of dark energy at 72.6%.

$$\Omega_{\Lambda} = 0.726$$ - Lambda-CDM dark energy density

Total amount of dark energy in the Universe integration by substitution::
$$E_{\Lambda} = \Omega_{\Lambda} E_t = \frac{4 \pi c^2 M_{\odot}}{3} \left( \frac{\Omega_{\Lambda}}{\Omega_s} \right) \left( \frac{dN_s}{dV_s} \right) R_u^3 = 2.733 \cdot 10^{73} \; \text{j}$$

Universe dark energy total energy:
$$\boxed{E_{\Lambda} = \frac{4 \pi c^2 M_{\odot}}{3} \left( \frac{\Omega_{\Lambda}}{\Omega_s} \right) \left( \frac{dN_s}{dV_s} \right) R_u^3}$$

Total amount of dark energy in the Universe:
$$\boxed{E_{\Lambda} = 2.733 \cdot 10^{73} \; \text{j}}$$

Reference:
http://en.wikipedia.org/wiki/Hubble%27s_law" [Broken]
http://en.wikipedia.org/wiki/Lambda-CDM_model" [Broken]
http://en.wikipedia.org/wiki/Universe" [Broken]
http://en.wikipedia.org/wiki/Observable_universe" [Broken]
http://en.wikipedia.org/wiki/Dark_matter" [Broken]
http://en.wikipedia.org/wiki/Dark_energy" [Broken]
http://en.wikipedia.org/wiki/Hubble's_law#Interpretation"
http://en.wikipedia.org/wiki/Hubble_volume" [Broken]
http://en.wikipedia.org/wiki/Friedmann_equations#Density_parameter"

Last edited by a moderator: May 4, 2017